Formalité Opéradique Et Homotopie Des Espaces De Configuration

Université Lille 1 Thèse de Doctorat Laboratoire Paul Painlevé Mathématiques École Doctorale SPI No 42424

Formalité opéradique et homotopie des espaces de configuration

Najib Idrissi Kaïtouni

Thèse dirigée par Benoit Fresse et soutenue le 17 novembre 2017

Jury Directeur de thèse : Benoit Fresse Université Lille 1 Rapporteurs : Damien Calaque Université de Montpellier Kathryn Hess Bellwald École Polytechnique Fédérale de Lausanne Examinateurs : Pascal Lambrechts Université catholique de Louvain Patrick Popescu-Pampu Université Lille 1 Thomas Willwacher École Polytechnique Fédérale de Zurich

Remerciements

Même si cette thèse n’a qu’un seul auteur, de nombreuses personnes y ont contribué plus ou moins indirectement, et je ne serai pas arrivé là où j’en suis sans eux. Je voudrais tout d’abord remercier mon directeur de thèse, Benoit Fresse. Il a toujours su me donner des pistes de recherches, me laisser chercher de façon indépendante tout en m’aidant quand j’avais des difficultés, me conseiller et me guider dans mes activités mathématiques, consacrer du temps à mes questions... Je suis particulièrement reconnaissant envers Damien Calaque et Kathryn Hess, qui ont accepté d’être rapporteurs de ma thèse. Je voudrais également remercier les autres membres du jury : Pascal Lambrechts et Thomas Willwacher, avec qui les collaborations se sont révélées fructueuses, ainsi que Patrick Popescu-Pampu, dont le sens de la pédagogie inspire. L’Université Lille 1 et le Laboratoire Paul Painlevé m’ont fourni un cadre pro- pice à la réalisation de cette thèse. Je remercie en particulier les topologues Lillois, sans qui ces années au labo auraient été bien moins intéressantes. Je suis égale- ment reconnaissant envers l’ENS, qui finance ma thèse et où j’ai auparavant pu étudier dans des conditions favorables, ainsi qu’envers tous mes professeurs de mathématiques, du lycée jusqu’à aujourd’hui. Je dois également remercier tous mes amis. Tout d’abord, mes collègues doc- torants, qu’ils soient déjà partis ou encore ici : les pauses du midi, les cafés, les conférences... auraient été bien mornes sans eux. Également mes amis Lillois, et plus particulièrement ceux de la BDM : entre les hauts et les bas, ils ont toujours été là. Et aussi mes amis d’ailleurs, que je vois moins souvent mais que je n’oublie pas. Enfin et surtout, mes parents, ma sœur et mon frère me soutiennent etme supportent (dans tous les sens du terme!) depuis toujours. Je leur dois beaucoup, et je les remercie pour tout.

iii

Résumé. Dans une première partie, nous étudions l’opérade SC2 «Swiss-Cheese» de Voronov, qui gouverne l’action d’une algèbre D2 sur une algèbre D1. Nous construisons un modèle en groupoïdes de cette opérade et nous décrivons les algèbres sur ce modèle de manière similaire à la description classique des algèbres sur 퐻∗(SC2). Nous étendons notre modèle en un modèle rationnel dépendant d’un associateur de Drinfeld, et nous le comparons au modèle qui existerait si l’opérade SC2 était formelle. Dans une seconde partie, nous étudions les espaces de configurations des variétés compactes, lisses, sans bord et simplement connexes. Nous démontrons sur ℝ une conjecture de Lambrechts–Stanley qui décrit un modèle de tels espaces de configurations, avec comme corollaire leur invariance homotopique réelle. En nous fondant sur la preuve par Kontsevich de la formalité des opérades D푛, nous obtenons en outre que ce modèle est compatible avec l’action de l’opérade de Fulton–MacPherson quand la variété est parallélisée. Cela nous permet de calculer explicitement l’homologie de factorisation d’une telle variété. Enfin, dans une troisième partie, nous élargissons ce résultat à une large classede variétés à bord. Nous utilisons d’abord une dualité de Poincaré–Lefschetz au niveau des chaînes pour calculer l’homologie des espaces de configurations de ces variétés, puis nous reprenons les méthodes du second chapitre pour obtenir le modèle, qui est compatible avec l’action de l’opérade Swiss-Cheese SC푛. Mots-clés : opérades, espaces de configuration (topologie), variétés topologiques, topologie algébrique Cette œuvre est mise à disposition selon les termes de la Licence Creative Commons Attribution – Pas d’Utilisation Commerciale – Pas de Modification 4.0 International.

Abstract. In a first part, we study Voronov’s “Swiss-Cheese” operad SC2, which governs the action of a D2-algebra on a D1-algebra. We build a model in groupoids of this operad and we describe algebras over this model in a manner similar to the classical description of algebras over 퐻∗(SC2). We extend our model into a rational model which depends on a Drinfeld associator, and we compare this new model to the one that we would get if the operad SC2 were formal. In a second part, we study configuration spaces of closed smooth simply connected manifolds. We prove over ℝ a conjecture of Lambrechts–Stanley which describes a model of such configuration spaces, and we obtain as corollary their real homotopy invariance. Moreover, using Kontsevich’s proof of the formality of the operads D푛, we obtain that this model is compatible with the action of the Fulton–MacPherson operad when the manifold is framed. This allows us to explicitly compute the factorization homology of such a manifold. Finally, in a third part, we expand this result to a large class of manifolds with boundary. We first use a chain-level Poincaré–Lefschetz duality result to compute the homology of the configuration spaces of these manifolds, then we reuse the methods of the second chapter to obtain our model, which is compatible with the action of the Swiss-Cheese operad SC푛. English title: Operadic Formality and Configuration Spaces

v Remerciements

Keywords: operads, configuration spaces (topology), topological manifolds, algebraic topology This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

vi Organisation de cette thèse

Outre le Chapitre 0 qui sert d’Introduction, chacun des trois autres chapitres est tirée d’un article :

• le Chapitre 1 est l’article «Swiss-Cheese Operad and Drinfeld Center», publié au Israel J. Math en 2017 [Idr17];

• le Chapitre 2 constitue la prépublication «The Lambrechts–Stanley Model of Configuration Spaces»Idr16 [ ];

• le Chapitre 3 est tiré d’un travail en cours, en collaboration avec Pascal Lambrechts.

Tous ces chapitres sont en anglais. Un résumé substantiel en français de cette thèse est disponible sur les pages qui suivent. Le Chapitre 1 est en grande partie indépendant des Chapitres 2 et 3 et peut être lu séparément.

Organization of this thesis

Besides the Introduction in Chapter 0, each of the other three chapters is drawn from an article:

• Chapter 1 is the article “Swiss-Cheese Operad and Drinfeld Center”, pub- lished at the Israel J. Math in 2017 [Idr17];

• Chapter 2 is the preprint “The Lambrechts–Stanley Model of Configuration Spaces” [Idr16];

• Chapter 3 is based on a current work-in-progress joint with Pascal Lam- brechts.

All of these chapters are in English. A substantial summary of this thesis in French can be found in the next pages. Chapter 1 is largely independent from Chapters 2 and 3 and can be read separately.

vii

Résumé

Dans cette thèse, nous étudions le type d’homotopie des espaces de configuration de variétés en utilisant des idées venant de la théorie des opérades. Ces espaces de configuration consistent en des collections de points deux àdeux distincts dans une variété donnée. Nous répondons aux problèmes de l’invariance homotopique et de la définition de modèles rationnels de ces espaces. Nous adaptons et généralisons des constructions de Kontsevich, qui donnent la formalité des espaces de configuration (compactifiés) des espaces euclidiens en tant qu’opérade [Kon99]. La notion d’opérade fut initialement introduite dans le but d’étudier les espaces de lacets itérés en théorie de l’homotopie [May72; BV73]. La théorie a connu une renaissance considérable au milieu des années 90 quand, inspirés par un article de Kontsevich [Kon93], Ginzburg et Kapranov [GK94] ont démontré dans des travaux fondateurs que certains phénomènes de dualité en algèbre pouvaient s’interpréter en termes d’opérades. Depuis, de nombreuses nouvelles applications des opérades ont été découvertes dans plusieurs domaines des mathématiques.

Introduction Opérades Une opérade est un objet qui gouverne une catégorie d’algèbres. L’idée centrale de la théorie peut s’expliquer par analogie avec la théorie des représentations de groupes. Dans cette analogie, l’opérade correspond au groupe, et les algèbres sur l’opérade correspondent aux représentations du groupe. Si un groupe est défini par une présentation par générateurs et relations, alors la catégorie de ses représentations peut se définir en termes des actions des géné- rateurs sujettes aux relations. Dans notre analogie, une catégorie d’algèbres est définie par des opérations génératrices et des relations. Par exemple, la structure d’une algèbre associative se définit par une opération génératrice, le produit, et une relation, l’associativité. La structure d’une algèbre commutative se définit de façon similaire, avec la condition supplémentaire de symétrie du produit. Les algèbres de Lie sont définies par un crochet antisymétrique et la relation de Jacobi, etc. Nous pouvons interpréter chacune de ces définitions comme la défi-

ix Résumé nition d’une opérade par générateurs et relations qui gouvernerait la catégorie d’algèbres en question. Tout comme nous étudions les groupes, il est intéressant d’étudier l’opérade elle-même, indépendamment de toute présentation par générateurs et relations. Cette nouvelle perspective nous permet de parler de morphismes, de sous- opérades, de quotients, d’extensions, etc., et de traduire des informations sur une opérade en des informations sur la catégorie des algèbres sur cette opérade. Expliquons plus précisément ce qu’est une opérade. 푖 ... 푖 + 푙 − 1 Les structures algébriques associées aux opérades sont celles qui peuvent être décrites en termes d’opéra- 1 ... 푞 ... 푘 + 푙 − 1 tions avec un nombre fini d’entrées et exactement une 1 P P P 푝 sortie. Une opérade est une collection = { (푘)}푘≥0 «d’opérations» abstraites. On peut voir un élément de P(푘) comme une opération à 푘 entrées et une sortie. Le groupe symétrique Σ푘 agit sur P(푘), ce qui correspond à la permutation des entrées d’une opération. Il faut également se donner des opérations d’insertion

∘푖 ∶ P(푘) ⊗ P(푙) → P(푘 + 푙 − 1), 1 ≤ 푖 ≤ 푘, qui modélisent la composition des opérations (tout comme la multiplication dans un groupe 퐺 correspond à la composition des actions sur les représentations). Enfin, une identité id ∈ P(1) est un élément neutre pour la composition. Pour illustrer cette définition, considérons l’exemple prototypique, l’opérade des endomorphismes. Étant donné un objet 푋 dans une catégorie monoïdale symétrique, ⊗푛 l’opérade des endomorphismes End푋 est définie par End푋(푛) ≔ Hom(푋 , 푋). L’action du groupe symétrique permute les entrées :

(푓 ⋅ 휎)(푥1, … , 푥푛) ≔ 푓 (푥휎(1), … , 푥휎(푛)), et les opérations d’insertion sont données par la composition des morphismes :

(푓 ∘푖 푔)(푥1, … , 푥푘+푙−1) ≔ 푓 (푥1, … , 푥푖−1, 푔(푥푖, … , 푥푖+푙−1), 푥푖+푙, … , 푥푘+푙−1).

Enfin, l’identité id ∈ End푋(1) est simplement l’identité de 푋. Une algèbre sur une opérade P est, par définition, un morphisme d’opérades P → End푋, exactement comme une représentation d’un groupe 퐺 est la donnée d’un morphisme de monoïdes 퐺 → End(푉). Comme autre exemple, une opérade n’ayant que des opérations d’arité 1 est exactement la même chose qu’un monoïde, et une algèbre sur une telle opérade n’est autre qu’une représentation du monoïde associé. Nous renvoyons aux livres [LV12; Fre17] pour un traitement plus détaillé des opérades.

1Pour nous, une «opérade» sans autre qualificatif est une opérade symétrique à une couleur. Il existe d’autres variantes : opérades non symétriques, opérades cycliques, opérades colorées, etc.

x Introduction

Opérades des petits disques Une famille d’opérades topologiques est d’un intérêt particulièrement grand : les opérades des petits 푛-disques D푛 (ou, de manière équivalente, les opérades des petits 푛-cubes). Ce sont les opérades qui ont fait leur apparition dans l’étude originelle des espaces de configuration. Un élément de D푛(푘) est une configuration ordonnée de 푘 petits 푛-disques aux intérieurs disjoints dans le disque unité 푛 퐷 . Chaque disque de la configuration s’obtient comme l’image 1 d’un plongement de 퐷푛 dans lui-même obtenu par la compo- D sition d’une translation et d’une homothétie. L’ensemble 푛(푘) 2 3 est muni de la topologie compacte-ouverte des plongements. L’action du groupe symétrique réordonne les disques d’une configuration, et l’insertion est donnée par la composition des plongements.

1 1 1

∘2 2 = 2 2 3 3 4

푛 Un espace de lacets itéré Ω 푋 est, presque par définition, une algèbre sur D푛. Le «principe de reconnaissance» [May72; BV73] dit que la réciproque est vraie : sous des hypothèses techniques, une D푛-algèbre «group-like» est faiblement équivalente à un espace de 푛-lacets. Les opérades des petits disques se sont révélées, depuis cette première application, être utiles dans de nombreux contextes. Mentionnons la conjecture de Deligne [KS00; MS02], qui dit que les cochaînes de Hochschild 퐶∗(퐴; 퐴) d’une algèbre associative sont munies d’une action de D2 ; le théorème de formalité des cochaînes de Hochschild et ses applications à la quantification des variétés de Poisson [Kon99; Tam98; Kon03]; le calcul de Goodwillie–Weiss et le calcul des espaces de plongements et des espaces de longs nœuds [Sin06; LTV10; AT14; DH12; BW13], ainsi que l’homologie de factorisation, en quelque sorte la «version covariante» du calcul des plongements [BD04; Lur09; Lur16; AF15; CG17]. Un résultat fondamental au sujet des opérades des petits disques est qu’elles sont formelles sur ℚ [Kon99; Tam03; LV14; FW15], c.-à-d. que l’opérade des chaînes 퐶∗(D푛; ℚ) est quasi-isomorphe à son homologie e푛 ≔ 퐻∗(D푛; ℚ). Il suffit

xi Résumé

donc, sur ℚ, d’étudier les opérades e푛, qui ont une description combinatoire simple : e1 = Ass gouverne les algèbres associatives, et pour 푛 ≥ 2, e푛 gouverne les (푛−1)-algèbres de Poisson [Coh76]. Quand 푛 = 2, la formalité de l’opérade D2 dépend du choix d’un associateur de Drinfeld, un objet qui définit une manière universelle de construire une catégorie monoïdale tressée à partir de données qui viennent de la théorie de Lie (voir p.ex. [Fre17, Chapter I.10] pour plus de détails).

Opérades «Swiss-Cheese» Les opérades colorées (aussi connues sous le nom de multicatégories) géné- ralisent les opérades et sont utilisées pour décrire des structures algébriques à plusieurs objets. Dans ce contexte, les entrées et la sortie d’une opération sont toutes étiquetées par une «couleur», et l’insertion n’est définie que si les couleurs correspondent. Dans la définition d’une algèbre sur une opérade colorée, chaque couleur correspond à un objet donné. L’opérade «Swiss-Cheese» SC = SC2 [Vor99] est une opérade à deux couleurs qui gouverne l’action d’une al- 1 gèbre D2 sur une algèbre D1 par un morphisme central. Une opération de SC est donné par le plongement de 3 2 disques et de demi-disques dans le demi-disque unité supérieur. Ces opérades sont liées aux OCHA de Kajiu- ra et Stasheff [KS06b; Hoe09], et il existe une version Swiss-Cheese de la conjecture de Deligne [DTT11]. Des variantes en dimension supérieur SC푛 gouvernent l’action d’une algèbre D푛 sur une algèbre D푛−1. Contrairement aux opérades des petits disques, les opérades Swiss-Cheese ne sont pas formelles [Liv15].

Espaces de configuration Soit 푀 une variété. Son 푘ième espace de configuration est donné par :

푘 Conf푘(푀) ≔ {푥 ∈ 푀 ∣ ∀푖 ≠ 푗, 푥푖 ≠ 푥푗}.

Les espaces de configuration sont intimement liés aux opérades des petits disques. 푛 Par exemple, l’application D푛(푘) → Conf푘(ℝ ) qui associe à une configuration de disques la configuration constituée des centres des disques est une équivalence d’homotopie. Cette construction n’est évidemment pas un invariant d’homotopie pour les 푛 푛−1 variétés ouvertes dès que 푘 ≥ 2. Par exemple, pour 푛 ∈ ℕ, Conf2(ℝ ) ≃ 푆 . Même en se restreignant aux variétés compactes sans bord, Conf푘(−) n’est pas un invariant d’homotopie [LS05b]. Le contre-exemple (donné par des espaces

xii Résultats du Chapitre 1

lenticulaires) n’est pas simplement connexe, et la question de savoir si Conf푘(−) est un invariant d’homotopie pour les variétés compactes, sans bord et simplement connexes reste ouverte. Quelques résultats sont connus. Le type d’homotopie de Ω Conf푘(푀) ne dé- pend que de celui de (푀, 휕푀) pour les variétés compactes connexes [Lev95]. Les espaces de configuration sont des invariants stables d’homotopie des variétés compactes sans bord [AK04]. Si 푀 est une variété projective lisse, alors le type d’homotopie rationnel de Conf푘(푀) ne dépend que de celui de 푀. Le même résultat est valable avec 푘 = 2 pour les variétés compactes sans bord qui sont soit 2-connexes [LS04], soit simplement connexes et de dimension paire [Cor15].

Résultats du Chapitre 1

Comme mentionné précédemment, l’opérade Swiss-Cheese SC = SC2 n’est pas formelle. En d’autres termes, il n’est pas possible de récupérer le type d’homotopie de SC à partir de son homologie sc = 퐻∗(SC). Le but du premier chapitre de cette thèse est de trouver un modèle de SC qui corrige ce défaut de formalité. Les trois opérades D1, D2 et SC sont asphériques, c.-à-d. que l’homotopie de chacun des espaces qui les composent est concentrée en degrés 0 et 1. Il y a donc, pour chacune de ces opérades (notons les P), une équivalence d’homotopie canonique P ∼ 퐵휋P, où 퐵 est le foncteur «espace classifiant» et 휋 le foncteur «groupoïde fondamental» (les deux étant fortement monoïdaux, ils préservent la structure d’opérade). À homotopie près, il suffit donc d’étudier le groupoïde fondamental 휋P de chacune de ces trois opérades P ∈ {D1, D2, SC} à «équivalence catégorique» (morphisme d’opérade induisant une équivalence de catégorie en chaque arité) près. Il existe des descriptions classiques de modèles en groupoïdes des deux opé- rades D1 et D2.

• En chaque arité, D1(푟) est discret à homotopie près (avec 푟! composantes connexes), et il est facile de voir que l’opérade 휋D1 est faiblement équiva- lente à l’opérade PaP des permutations parenthésées. La catégorie PaP(푟) = Σ푟 est discrète, avec comme objets les permutations de {1, … , 푟}, et la structure d’opérade est donnée par la composition par blocs des permutations.

• Les composantes D2(푟) de l’opérade D2 sont des espaces classifiants des groupes de tresses pures 푃푟. L’opérade 휋D2 est ainsi faiblement équivalente à l’opérade PaB des tresses parenthésées (cf. [Fre17, Chapter I.3], voir aussi [Fie96]). Les objets de PaB(푟) sont encore les permutations de {1, … , 푟}. Les morphismes entre deux permutations 휎, 휎′ ∈ PaB(푟) sont les tresses

xiii Résumé

colorées à 푟 brins entre ces deux permutations : chacun des brins de la tresse est «coloré» par un des nombres {1, … , 푟}, et l’ordre des brins au début (resp. à la fin) de la tresse est donné par 휎 (resp. 휎′). La composition des tresses est donnée par l’insertion d’une tresse dans un voisinage tubulaire d’un brin.

Le premier résultat de ce premier chapitre (Theorem A) est la définition d’une opérade en groupoïdes PaPB des tresses et permutations parenthésées. Cette opérade combine, en quelque sorte, les deux opérades PaP et PaB pour obtenir une opérade colorée, et est faiblement équivalente au groupoïde fondamental 휋SC. La définition de PaPB est motivée par le résultat suivant. L’homologie de l’opé- rade Swiss-Cheese se scinde en un «produit de Voronov» sc = e2 ⊗ e1, où e1 = 퐻∗(D1) = Ass gouverne les algèbres associatives et e2 = 퐻∗(D2) = Ger gouverne les algèbres de Gerstenhaber [Vor99]. Concrètement, cela signifie qu’une algèbre sur sc est la donnée d’un triplet (퐴, 퐵, 푓 ) où 퐴 est une algèbre associative, 퐵 est une algèbre de Gerstenhaber, et 푓 ∶ 퐵 → 푍(퐴) est un morphisme d’algèbres de 퐵 dans le centre de 퐴.

Les algèbres (dans la catégorie des catégories) sur l’opérade PaP ≃ 휋D1 sont les catégories monoïdales, et les algèbres sur PaB ≃ 휋D2 sont les catégories monoïdales tressées. En analogie avec le théorème de Voronov, nous démontrons que les algèbres (toujours dans la catégorie des catégories) sur PaPB ≃ 휋SC sont les triplets (M, N, 퐹) où M est une catégorie monoïdale, N est une catégorie monoïdale tressée, et 퐹 ∶ M → 풵(N) est un foncteur monoïdal tressé de M dans le centre de Drinfeld de N – un analogue catégorique du centre d’une algèbre associative. Ce résultat est la contrepartie pour les opérades en groupoïdes de résultats ∞-catégoriques sur les algèbres à factorisation du demi-plan supérieur (cf. [Gin15, Proposition 31] et [AFT17, Example 2.13]) Dans une deuxième étape, nous fixons un associateur de Drindeld, que nous pouvons voir comme une équivalence rationnelle (au sens de la théorie de l’ho- ̂ motopie rationnelle) PaB+ → CD̂+ où CD̂+ = B 핌픭̂ + est l’opérade complétée des diagrammes de cordes. À partir de cette donnée, nous construisons une opérade 휑 PaPCD̂+ rationnellement équivalente à la complétion de l’opérade Swiss-Cheese. Cette opérade étend la construction utilisée par Tamarkin [Tam03] pour démon- trer la formalité de l’opérade D2 et peut s’interpréter informellement comme un produit de Voronov tordu d’un modèle pour D2 et d’un modèle pour D1.

xiv Résultats du Chapitre 2 Résultats du Chapitre 2

Dans le second chapitre, nous étudions les espaces de configuration des variétés compactes sans bord simplement connexes. Dans la suite, on note 푀 une telle variété. Nous nous plaçons dans le cadre de la théorie de l’homotopie rationnelle de Sullivan, où les espaces sont modélisés par des algèbres différentielles graduées commutatives (CDGA en anglais) et nous cherchons à obtenir un modèle de Conf푘(푀) pour 푘 ≥ 0. La cohomologie d’une variété orientée compacte sans bord satisfait la dualité de Poincaré, il y a un accouplement non dégénéré 퐻푘(푀)⊗퐻푛−푘(푀) → ℚ donné par 훼 ⊗ 훽 ↦ 훼훽 ⌢ [푀]. Lambrechts et Stanley [LS08b] démontrent que si la variété est simplement connexe, alors cette propriété peut se traduire sur les modèles : 푀 admet un modèle rationnel 퐴 à «dualité de Poincaré», c.-à-d. il y a un accouplement non-dégénéré 퐴푘 ⊗ 퐴푛−푘 → ℚ induit par une «augmentation» 푛 휀퐴 ∶ 퐴 → ℚ. À partir d’un tel modèle à dualité de Poincaré, ils introduisent une CDGA G퐴(푘) et démontrent que les nombres de Betti rationnels de G퐴(푘) coïncident avec ceux de Conf푘(푀) [LS08a]. Ils conjecturent que cette CDGA est en fait un modèle rationnel de Conf푘(푀), ce qui est connu pour les variétés projectives lisses complexes [Kri94], ainsi que quand 푘 = 2 et que 푀 est 2- connexe [LS04] ou de dimension paire [Cor15]. Cette CDGA avait déjà été étudiée dans certains cas particuliers [CT78; BG91; Kri94; Tot96; FT04]. Elle admet une description simple pour les premières valeurs de 푘 : G퐴(0) = ℚ et G퐴(1) = 퐴, ce qui est cohérent avec le fait que Conf0(푀) = {∗} ⊗2 et Conf1(푀) = 푀, et G퐴(2) est quasi-isomorphe au quotient de 퐴 par l’idéal engendré par la «classe diagonale» de 퐴 (le dual de Poincaré de 푀 ⊂ 푀 × 푀). Plus généralement, G퐴(푘) est en quelque sorte une version «étiquetée par 퐴» de 푛 la description classique de la cohomologie de Conf푘(ℝ ) [Arn69; Coh76]. La formalité de l’opérade des petits disques entraîne en particulier que les 푛 ∗ 푛 espaces de configuration Conf푘(ℝ ) sont formels : la CDGA 퐻 (Conf푘(ℝ )) ≕ ∨ 푛 e푛 (푘) (avec la différentielle nulle) est un modèle rationnel de Conf푘(ℝ ). La preuve par Kontsevich de cette formalité fournit des morphismes explicites [Kon99; LV14]. L’idée de ce chapitre est d’adapter cette preuve aux espaces de configuration de 푀 pour obtenir le fait que G퐴(푘) est un modèle. Cette preuve fait intervenir la compactification de Fulton–MacPherson des espaces de configuration [FM94; AS94]. Étant donnée une variété 푀, l’espace FM푀(푘) est une variété stratifiée dont l’intérieur est Conf푘(푀). Le bord de FM푀(푘) est obtenu en autorisant les points d’une configuration à devenir infinitésima- 푛 lement proches les uns des autres. Il est possible de compactifier Conf푘(ℝ ) de 푛 manière similaire en un espace FM푛(푘), en tenant compte du fait que ℝ lui-même n’est pas compact.

xv Résumé

La preuve passe par la définition d’un certain complexe de graphes intermé- Graphs ∗ 푛 ∗ FM diaires 푛(푘). On peut voir 퐻 (Conf푘(ℝ )) = 퐻 ( 푛(푘)) comme un quo- Graphs tient de 푛(푘) par un idéal acyclique. Dans l’autre direction, il est nécessaire ∗ d’introduire le complexe des formes semi-algébriques par morceaux ΩPA(FM푛(푘)), FM Graphs qui est un modèle sur ℝ de 푛(푘). Un quasi-isomorphisme de 푛(푘) dans ∗ ΩPA(FM푛(푘)) est alors donné par des intégrales le long des fibres des projections canoniques FM푛(푘 + 푙) → FM푛(푘). Nous adaptons toutes ces constructions à Conf푘(푀) pour obtenir un zigzag explicite de quasi-isomorphismes, pour 푀 une variété lisse compacte sans bord simplement connexe de dimension au moins 4 : G ∼ Graphs ∼ ∗ FM 퐴(푘) 푅(푘) ΩPA( 푀(푘)), 푛 ce qui démontre que G퐴(푘) est un modèle sur ℝ de FM푀(푘) ≃ Conf푘(ℝ ) (pre- mière partie du Theorem C). Le point clé de la preuve consiste à démontrer que pour les variétés simplement connexes de dimension ≥ 4, la fonction de partition Z휑, définie à l’aide d’intégrales sur FM푀, est (presque) triviale, ce qui s’obtient à l’aide d’un argument de comptage de degré. ′ Si 퐴 et 퐴 sont deux modèles de 푀, nous démontrons directement que G퐴(푘) ≃ℝ G퐴′ (푘), d’où l’invariance homotopique réelle de Conf푘(푀) par rapport à 푀 (Co- rollary 2.4.36). On peut insérer une configuration de FM푛 dans une autre, ce qui donne une structure d’opérade faiblement équivalent à l’opérade des petits disques D푛. De même, quand 푀 est parallélisée, on peut insérer une configuration de FM푛 dans une configuration de FM푀 et obtenir ainsi une structure de FM푛-module à droite sur FM푀. Quand 휒(푀) = 0 (en particulier quand 푀 est parallélisée), il est facile d’observer que la collection G퐴 = {G퐴(푘)} est dotée d’une structure de como- ∗ ∨ dule de Hopf à droite sur 퐻 (FM푛) = e푛 . Nous démontrons que le zigzag de quasi-isomorphismes que nous construisons est compatible avec la structure de comodule quand 푀 est parallélisée (deuxième partie du Theorem C). En d’autres termes, nous obtenons un modèle réel du FM푛-module à droite FM푀. Cela nous permet de calculer l’homologie de factorisation, un invariant des variétés. Étant données une 푛-variété parallélisée 푀 et une FM푛-algèbre 퐵, l’homologie de factorisation de 푀 à coefficients dans 퐵 peut se calculer comme le produit de composition relatif dérivé (cf. [AF15] et [Tur13, Section 5.1]) :

핃 ∫ 퐵 ≔ FM푀 ∘FM 퐵. 푀 푛 Notre résultat implique que si 푀 est une variété lisse compacte sans bord simplement connexe de dimension au moins 4 et que 퐵 est une e푛-algèbre, alors l’homologie de factorisation de 푀 à coefficients dans 퐵 est donnée, sur ℝ, par un complexe explicite : ∨ 핃 ∫ 퐵 ≃ G ∘e 퐵. 푀 퐴 푛 xvi Résultats du Chapitre 3

Dans le cas où 퐵 = 푆(Σ1−푛픤) est l’algèbre enveloppante supérieure d’une algèbre de Lie, nous reprenons des arguments de Félix et Thomas [FT04] pour démontrer que ∫ 푆(Σ1−푛픤) se calcule avec un complexe de Chevalley–Eilenberg (Propo- 푀 sition 2.5.6), un résultat précédemment obtenu par Knudsen [Knu16] par des méthodes complètement différentes. Enfin, en utilisant la preuve de Giansiracusa et Salvatore [GS10] de la formalité de l’opérade des petits disques à repère, nous généralisons notre résultat à 푆2, la seule surface simplement connexe (Theorem 2.6.6).

Résultats du Chapitre 3

Dans le troisième chapitre, nous étendons les résultats précédents à une large classe de variétés à bord. Nous nous concentrons sur le cas des variétés admettant un «modèle à dualité de Poincaré–Lefschetz». Cette notion est une généralisation de «joli modèle surjectif» (“surjective pretty model” en anglais), une notion due à Cordova Bulens, Lambrechts et Stanley [CLS15a] (voir aussi [LS05a]). L’intuition derrière cette notion provient de la dualité de Poincaré–Lefschetz. Soit 푁 une variété orientée compacte sans bord de dimension 푛 et 퐾 ⊂ 푁 un sous-polyhèdre, et soit 푀 = 푁 − 퐾̃ est la variété à bord obtenue en retirant un épaississement de 퐾. Alors la cohomologie de 푀 se calcule à partir de la cohomologie de 푁 en «tuant» les classes duales des classes d’homologie provenant de 퐾. Concrètement, un joli modèle surjectif est construit à partir des données suivantes : • une CDGA à dualité de Poincaré 푃 (correspondant à un modèle de 푁);

• une CDGA connexe 푄 satisfaisant 푄≥푛/2−1 = 0 (correspondant à un modèle de 퐾);

• un morphisme surjectif de CDGAS 휓 ∶ 푃 → 푄. ∨ La dualité de Poincaré de 푃 induit un isomorphisme 휃푃 ∶ 푃 → 푃 [−푛] entre 푃 et la désuspension de son dual. On définit l’application 휓! ∶ 푄∨[−푛] → 푃 comme −1 ∨ ! la composition 휃푃 ∘ 휓 [−푛]. Notons que 휓휓 = 0 pour des raisons de degré. Le joli modèle surjectif (correspondant à un modèle de l’inclusion 휕푀 ⊂ 푀 = 푁−퐾) associé à 휓 ∶ 푃 → 푄 est alors donné par :

∨ ∨ 휆 = 휓 ⊕ id ∶ 퐵 = 푃 ⊕휓! 푄 [1 − 푛] → 퐵휕 = 푄 ⊕ 푄 [1 − 푛],

∨ ! où 푃 ⊕휓! 푄 [1 − 푛] est le cône (conoyau homotopique) de 휓 . La CDGA 퐵휕 est une CDGA à dualité de Poincaré de dimension formelle 푛 − 1, et 퐵 est quasi- isomorphe au quotient 퐴 = 푃/퐼 de la CDGA à dualité de Poincaré 푃 par son

xvii Résumé idéal 퐼 = im 휓!. La dualité de Poincaré–Lefschetz se traduit en l’existence d’un isomorphisme 퐴 ≅ 퐾∨[−푛], où 퐾 = ker 휓 ≅ ker 휆 est un modèle des formes relatives sur (푀, 휕푀). Un exemple éclairant de joli modèle surjectif est celui de (푀, 휕푀) = (퐷푛, 푆푛−1). On peut voir 푀 = 퐷푛 comme une sphère 푁 = 푆푛 à laquelle on a retiré l’épaissis- sement d’un point 퐾 = {∗}. En appliquant le dictionnaire ci-dessus, on prend ∗ 푛 2 ∗ donc 푃 = 퐻 (푆 ) = 푆(vol푛)/(vol푛) comme modèle de 푁 et 푄 = 퐻 ({∗}) = ℝ comme modèle de 퐾, l’application 휓 ∶ 푃 → 푄 étant simplement l’augmenta- ! ∨ ! ∨ tion. L’application 휓 ∶ 푄 [−푛] → 푃 est donnée par 휓 (1푄) = vol푛. Le modèle ∨ ∨ 푃 ⊕휓! 푄 [1 − 푛] est de dimension 3, engendré par 1푃 en degré 0, 1푄 en degré ∨ 푛 − 1 et vol푛 en degré 푛. Tous les produits non-triviaux sont nuls et 푑(1푄) = vol푛. Nous ne savons pas si toutes les variétés orientées à bord admettent un joli modèle surjectif, contrairement aux variétés simplement connexes (donc orien- tées) sans bord qui admettent toutes un modèle à dualité de Poincaré. Si 푀 et 휕푀 sont simplement connexes, nous pouvons, sous l’hypothèse supplémen- taire que dim 푀 ≥ 7, construire ce que nous appelons un «modèle à dualité de Poincaré–Lefschetz» 휆 ∶ 퐵 → 퐵휕 de (푀, 휕푀). La caractéristique principale de ces modèles est l’existence d’un accouplement non-dégénéré entre un quotient 퐴 ≔ 퐵/ ker 휃 ≃ 퐵 et le noyau de 휆 qui modélise l’accouplement entre 퐻∗(푀) et 퐻∗(푀, 휕푀), comme pour les jolis modèles surjectifs. L’existence d’un modèle à DPL est suffisante pour reprendre toutes les constructions précédentes et obtenir ∗ 휑 le même modèle G퐴(푘) de ΩPA(Conf푘(푀)), ainsi que le modèle SGraphs푅 de ∗ ΩPA(SFM푀) (tout en étant compatible avec les structures de comodule si 푀 est parallélisée). À partir d’un tel joli modèle surjectif, il est possible de reprendre presque mot 푛 pour mot la définition de G퐴(푘) du Chapitre 2. Si par exemple 푀 = 퐷 , nous ∨ obtenons G퐴 ≅ e푛 vu comme un comodule à droite sur lui-même. En réutilisant les techniques du Chapitre 2, nous démontrons que G퐴(푘) a les mêmes nombres de Betti sur ℚ que Conf푘(푀). Nous définissons également une version «perturbée» G퐴̃ de G퐴, qui est isomorphe à G퐴 comme dg-module mais pas comme algèbre, et nous montrons que G퐴(푘) est un modèle sur ℝ de Conf푘(푀). Il est possible d’adapter la compactification de Fulton–MacPherson FM푀 aux variétés compactes à bord, en autorisant des points à devenir infinitésimalement proches les uns des autres, mais aussi en autorisant des points de l’intérieur de la variété à devenir infinitésimalement proches du bord. On obtient ainsi une compactification de l’espace de configuration coloré : ̊ Conf푘,푙(푀) ≔ {(푥1, … , 푥푘, 푦1, … , 푦푙) ∈ Conf푘+푙(푀) ∣ 푥푖 ∈ 휕푀, 푦푗 ∈ 푀} ∼ ,−→ SFM푀(푘, 푙).

Si 푀 est parallélisée, la collection SFM푀 = {SFM푀(푘, 푙)}푘≥0,푙≥0 est un module à

xviii Résultats du Chapitre 3

droite sur une opérade SFM푛 qui est construite de manière similaire à FM푛 et est faiblement équivalent à l’opérade Swiss-Cheese 푛-dimensionnelle. SGraphs Willwacher [Wil15] a construit un modèle en graphes 푛 de l’opérade SFM Graphs FM 푛, similaire au modèle 푛 de Kontsevich pour l’opérade 푛. Comme dans le Chapitre 2, nous adaptons la construction de Willwacher au «cas étiqueté» 푐 ,z푆 SGraphs 휑 휑 SGraphs pour construire un comodule de Hopf à droite 푅 sur 푛, et nous obtenons des quasi-isomorphismes de CDGAs (compatibles avec les structures de comodule le cas échéant) :

휑 ∼ ∗ SGraphs푅(푘, 푙) ΩPA(SFM푀(푘, 푙)), 푆 ∼ 푐휑,z휑 ∼ ∗ ∗ G퐴(푙) SGraphs푅 (0, 푙) ΩPA(SFM푀(0, 푙)) ≃ ΩPA(Conf푙(푀)).

xix

Contents

Remerciements iii

Résumé / Abstract v

Organisation / Organization vii

Résumé ix Introduction ...... ix Résultats du Chapitre 1 ...... xiii Résultats du Chapitre 2 ...... xv Résultats du Chapitre 3 ...... xvii

0 Introduction and Context 1 0.1 Operads ...... 1 0.1.1 Little disks operads ...... 3 0.1.2 The Swiss-Cheese operads ...... 4 0.2 Configuration spaces ...... 4 0.2.1 Closed manifolds ...... 4 0.2.2 Compact manifolds with boundary ...... 6

1 Swiss-Cheese Operad and Drinfeld Center 9 1.1 Background ...... 10 1.2 Permutations and braids ...... 12 1.2.1 Colored version ...... 12 1.2.2 Magmas ...... 16 1.2.3 Parenthesized version ...... 18 1.3 Drinfeld center ...... 19 1.3.1 Algebras over PaPB ...... 19 1.3.2 Unitary versions ...... 25 1.4 Chord diagrams ...... 26 1.4.1 Drinfeld associators and chord diagrams ...... 27 1.4.2 Shuffle of operads ...... 28 1.4.3 Non-formality ...... 34

xxi Contents

2 The Lambrechts–Stanley Model of Configuration Spaces 39 2.1 Background and recollections ...... 41 2.1.1 Conventions ...... 41 2.1.2 Little disks and related objects ...... 43 2.1.3 Formality of the little disks operad ...... 45 2.1.4 Poincaré duality CDGA models ...... 49 2.1.5 The Lambrechts–Stanley CDGAs ...... 50 2.2 The Hopf right comodule model G퐴 ...... 51 2.3 Labeled graph complexes ...... 53 2.3.1 Graphs with loops and multiple edges ...... 53 2.3.2 External vertices: Gra푅 ...... 54 2.3.3 The propagator ...... 58 2.3.4 Twisting: Tw Gra푅 ...... 59 Graphs 2.3.5 Reduction: 푅 ...... 64 2.4 From the model to forms via graphs ...... 67 2.4.1 Construction of the morphism to G퐴 ...... 67 2.4.2 The morphisms are quasi-isomorphisms ...... 71 2.5 Factorization homology of universal enveloping 퐸푛-algebras .. 83 2.5.1 Factorization homology and formality ...... 83 2.5.2 Higher enveloping algebras ...... 84 2.6 Outlook: The case of the 2-sphere and oriented manifolds .... 88 2.6.1 Framed little disks and framed configurations ...... 89 2.6.2 Cohomology of fFM푛 and potential model ...... 89 ∗ 2.6.3 Connecting fG퐴 to ΩPA(fFM푆2 ) ...... 90 3 Configuration Spaces of Manifolds with Boundary 93 3.1 Background and recollections ...... 95 3.1.1 Colored (co)operads and (co)modules ...... 95 3.1.2 Pretty models for compact manifolds with boundary .. 96 3.1.3 Compactified configuration spaces ...... 99 3.1.4 Graphs and the Swiss-Cheese operad ...... 100 3.1.5 Graph complexes ...... 103 3.2 Poincaré–Lefschetz duality models ...... 104 3.2.1 Motivation ...... 104 3.2.2 Definition and existence ...... 105 3.2.3 Diagonal classes ...... 109 3.3 The model and its cohomology ...... 110 3.3.1 The dg-module model of Conf푘(푀) ...... 110 3.3.2 Computing the homology ...... 111 3.3.3 The perturbed model ...... 116 3.4 Computation of Swiss-Cheese graph cohomology ...... 118

xxii Contents

3.5 A model of right Hopf comodules ...... 122 3.5.1 The propagator ...... 122 3.5.2 Colored labeled graphs ...... 125 3.5.3 Labeled graphs and proof of Theorem D ...... 132 3.5.4 Proof of Theorem F ...... 137

Bibliography 141

Index 151

xxiii

0 Introduction and Context

In this thesis, we study the homotopy of configuration spaces of manifolds using ideas coming from the theory of operads. These configuration spaces consist of collections of pairwise distinct points in a given manifold. We address the problems of their homotopy invariance and the definition of rational models for them. We adapt constructions of Kontsevich [Kon99], who proved the formality of (compactified) configuration spaces of Euclidean spaces as an operad. The notion of an operad was initially introduced for the study of iterated loop spaces in homotopy theory [May72; BV73]. The theory of operads has been considerably renewed since the mid-nineties, when, after insights of Kontse- vich [Kon93], Ginzburg and Kapranov [GK94] proved in a seminal work that some duality phenomena in algebra could be interpreted in terms of operads. Since then, a number of new applications of operads have been discovered in several fields of mathematics. In this introduction, we first explain what operads are in Section 0.1. Then we describe two important families of operads, the little disks operads (Section 0.1.1) and the Swiss-Cheese operads (Section 0.1.2). In Section 0.2, we turn to configuration spaces of manifolds. Section 0.2.1 is devoted to closed manifolds, whereas Section 0.2.2 is about compact manifolds with boundary.

0.1 Operads

Roughly speaking, an operad is an object that governs a category of algebras. The central idea can be understood by analogy with the theory of group representations. In this analogy, the operad corresponds to the group, and the algebras over the operad correspond to the representations of the group. If a group is given with a presentation by generators and relations, then the associated category of representations can also be defined in terms of the actions of the generators, which have to satisfy the relations. In our analogy, a category of algebras is defined in terms of generating operations and relations. For example, the structure of an associative algebra is defined in terms of a product asa generating operation together with associativity as a relation. The structure of a commutative algebra is defined similarly, with the condition that the product is symmetric, Lie algebras are defined in terms of an antisymmetric Lie bracket

1 0 Introduction and Context and the Jacobi relation, and so on. Each of these definitions can be interpreted as the definition of an operad by generators and relations which governs the given category of algebras. It is interesting to study the operad itself, independently of any presentation by generators and relations, just like we study groups. This new perspective enables us to consider morphisms, sub-operads, quotients, extensions... Knowledge about an operad then translates into knowledge about algebras over that operad. Let us explain more precisely what operads are. 푖 ... 푖 + 푙 − 1 The algebraic structures governed by operads are those that can be described in terms of operations with 1 ... 푞 ... 푘 + 푙 − 1 a finite number of inputs and exactly one output. An 1 P P P 푝 operad is a collection = { (푘)}푘≥0 of abstract “operations”. An element of P(푘) can be viewed as a operation with 푘 inputs (“of arity 푘”) and one output. There is an action of the symmetric group Σ푘 on P(푘) which models the permutations of the input of an operation. There are insertion morphisms:

∘푖 ∶ P(푘) ⊗ P(푙) → P(푘 + 푙 − 1), 1 ≤ 푖 ≤ 푘, which model composition of operations, similarly to how multiplication in a group 퐺 corresponds to composition of operations on representations. Finally, there is an identity id ∈ P(1) which is a neutral element for composition. To illustrate this definition, one can look at the prototypical example, the endomorphism operad. Given some object 푋 in a symmetric monoidal category, ⊗푛 the endomorphism operad End푋 is defined by End푋(푛) ≔ Hom(푋 , 푋). The symmetric group action acts by permuting inputs:

(푓 ⋅ 휎)(푥1, … , 푥푘) = 푓 (푥휎(1), … , 푥휎(푘)), and the insertion operation is given by composition of morphisms:

(푓 ∘푖 푔)(푥1, … , 푥푘+푙−1) = 푓 (푥1, … , 푥푖−1, 푔(푥푖, … , 푥푖+푙−1), 푥푖+푙, … , 푥푘+푙−1).

Finally, the identity id ∈ End푋(1) is simply the identity of 푋. An algebra 푋 over an operad P is by definition a morphism of operads P → End푋, just like a group representation is a morphism 퐺 → End(푉). As another example, an operad with only operations of arity 1 is the same thing as a monoid, and an algebra over such an operad is the same thing as a representation of the corresponding monoid. We refer to the books [LV12; Fre17] for a more extensive treatment of operads.

1To be precise, by an “operad” we mean a symmetric, one-colored operad. There exists other variants: non-symetric operads, cyclic operads, colored operads, etc.

2 0.1 Operads

0.1.1 Little disks operads There exists a certain family of topological operads of particular interest, the little 푛-disks operads D푛 (or equivalently, the little 푛-cubes operads). These are precisely the operads which appeared in the original study of iterated loop spaces. An element of D푛(푘) is an ordered configuration of 푘 little 푛-disks with disjoint interiors in the unit disk 퐷푛. Each disk 푛 of the configuration represents an embedding of 퐷 into itself, 1 obtained by the composition of a translation and a positive D homothety. The set 푛(푘) is endowed with the compact-open 2 3 topology of embeddings. The symmetric group action renum- bers the disks in a configuration, and insertion is given by composition of embeddings.

1 1 1

∘2 2 = 2 2 3 3 4

Figure 0.1.1: Composition of embeddings

푛 Almost by definition, an 푛-fold loop space Ω 푋 is a D푛-algebra. The “recogni- tion principle” [May72; BV73] asserts that the converse is true: under technical conditions, a (grouplike) D푛-algebra is weakly homotopy equivalent to an 푛-fold loop space. Since this first homotopy-theoretical application, the little disks operads have proved to be useful in a number of settings. Let us mention the Deligne conjecture [KS00; MS02], which asserts that the Hochschild cochains 퐶∗(퐴; 퐴) of an associative algebra are equipped with an action of D2; the formality theorem for Hochschild cochains and its application to deformation quantization of Poisson manifolds [Kon99; Tam98; Kon03]; Goodwillie–Weiss calculus and the computation of embedding spaces and long knots [Sin06; LTV10; AT14; DH12; BW13], as well as the “covariant version” of embedding calculus that is factorization homology [BD04; Lur09; Lur16; AF15; CG17]. A fundamental result about the little disks operads is that they are formal over ℚ [Kon99; Tam03; LV14; FW15]: the operad of chains 퐶∗(D푛; ℚ) is quasi- isomorphic to its homology e푛 ≔ 퐻∗(D푛; ℚ). Thus it suffices to study the operads

3 0 Introduction and Context

e푛, which have a simple combinatorial description: e1 = Ass governs associative algebras, and for 푛 ≥ 2, e푛 governs (푛−1)-Poisson algebras [Coh76]. When 푛 = 2, the formality of D2 depends on the choice of a Drinfeld associator, something which gives a universal way to build a braided monoidal category out of some Lie-algebraic data (see e.g. [Fre17, Chapter I.10] for details).

0.1.2 The Swiss-Cheese operads

Colored operads (a.k.a. multicategories), a generalization of operads, are used to describe algebraic structures shaped on multiple objects. In this setting, the inputs and the output of an operation are all labeled by “colors”, and composition is only defined if the colors match. In the definition of an algebra oversucha colored operad, each color corresponds to a given object. The Swiss-Cheese operad SC = SC2 [Vor99] is a 2-colored operad which governs the action of a D2- 1 algebra on a D1-algebra by a central morphism. An operation in SC is given by the embedding of disks and 3 2 upper half-disks in the unit upper half-disk. It is related to the OCHAs of Kajiura–Stasheff [KS06b; Hoe09] and there is a Swiss-Cheese version of the Deligne conjecture [DTT11]. Higher dimensional variants SC푛 govern the action of a D푛-algebra on a D푛−1-algebra. Unlike the little disks operads, the Swiss-Cheese operads are not formal [Liv15] and one cannot recover SC푛 from sc푛 = 퐻∗(SC푛) over ℚ. Our first goal in this thesis was to find a model for the Swiss-Cheese operad SC2 that fixes this lack of formality. The homology sc2 splits as a “Voronov product” e2 ⊗ e1 and governs the action of a Gerstenhaber algebra on an associative algebra via a central map. Theorem A describes a model in groupoids for SC2 which governs the action of a braided monoidal category on a monoidal category via a central monoidal functor. Theorem B describes a model over ℚ for SC2, which uses a Drinfeld associator to construct a kind of “twisted” Voronov product between a rational model for e2 (built out of chord diagrams) and a model for e1.

0.2 Configuration spaces

0.2.1 Closed manifolds

Let 푀 be an 푛-dimensional manifold; its 푘th configuration space is:

푘 Conf푘(푀) = {푥 ∈ 푀 ∣ ∀1 ≤ 푖 < 푗 ≤ 푘, 푥푖 ≠ 푥푗}.

4 0.2 Configuration spaces

This construction is obviously not homotopy invariant for open manifolds 푛 푛−1 when 푘 ≥ 2, as for example Conf2(ℝ ) ≃ 푆 for 푛 ∈ ℕ. Even if we restrict our attention to closed manifolds, Conf푘(−) is still not a homotopy invariant [LS05b]. The counterexample (lens spaces) is not simply connected, and the homotopy invariance for simply connected closed manifolds remains an open question.

Some results are known. The homotopy type of Ω Conf푘(푀) only depends on the homotopy type of (푀, 휕푀) for connected compact manifolds [Lev95]. Configuration spaces are also known tobea stable homotopy invariant [AK04] of closed manifolds. If 푀 is a smooth projective complex manifold, then the rational homotopy type of Conf푘(푀) only depends on the one of 푀 [Kri94]. The same result holds for 푘 = 2 when 푀 is a closed manifold and either 2-connected [LS04] or simply connected and even dimensional [Cor15]. We prove, as a corollary of the main result of Chapter 2 (Theorem C), that the real homotopy type of Conf푘(푀) only depends on the real homotopy type of 푀 when the manifold is closed, smooth, simply connected and of dimension at least 4. When working over ℚ (or ℝ) with simply connected manifolds, we can use Sullivan’s [Sul77] theory of rational models to study topological spaces up to rational equivalence. Closed, simply connected manifolds are known to have models satisfying a kind of chain-level Poincaré duality [LS08b]. Out of a Poincaré duality model of 푀, we give an explicit real model for Conf푘(푀). This model had been conjectured by Lambrechts–Stanley [LS08a] in the general case, and had previously been studied in some special cases [CT78; BG91; Kri94; Tot96; FT04; LS04; Cor15]. Our proof relies on Kontsevich’s proof of the formality of the little disks oper- 푛 ads. The configuration spaces Conf푘(ℝ ) admit compactifications FM푛(푘) due to Fulton–MacPherson [FM94] (and Axelrod–Singer [AS94] in the real case). 푛 An element of FM푛(푘) can roughly be seen as a configuration of 푘 points in ℝ , where the points are allowed to become “infinitesimally close”. One can insert such an infinitesimal configuration into another, and the spaces FM푛(−) assemble into an operad weakly equivalent to the little disks operads. Kontsevich’s proof uses piecewise semi-algebraic (PA) forms and integrals along the fiber to build an explicit equivalence between 퐶∗(FM푛) and e푛. Similarly, if 푀 is a closed manifold, then Conf푘(푀) can be compactified into FM푀(푘). If 푀 happens to be framed, then the spaces FM푀(−) assemble to form a right module over the operad FM푛. This extra structure is what allows us to prove Theorem C. When 푀 is framed, we then obtain that the action of the little disks operad on the configuration spaces is compatible with our model. We are able, using this additional result and a comparison statement between the Cohen–Taylor spectral sequence [CT78] and the Bendersky–Gitler spectral

5 0 Introduction and Context sequence [BG91] established by Félix–Thomas [FT04], to recover a theorem of Knudsen [Knu16] about the factorization homology of a framed manifold with coefficients in the higher enveloping algebra of a Lie algebra.

0.2.2 Compact manifolds with boundary

In the last chapter, we study configuration spaces of compact manifolds with boundary. As for closed manifolds, homotopy invariance of configuration spaces of compact manifolds with boundary is an open question. It is known that if the manifold and its boundary are both 2-connected, then the rational homotopy type of Conf2(푀) only depends on the rational homotopy type of the pair (푀, 휕푀) [CLS15b]. The manifolds we consider are those which admit a “Poincaré–Lefschetz duality model” These are a generalization of surjective pretty models as defined in [CLS15a]. Briefly, if 푁 is a closed manifold and 퐾 ⊂ 푁 is a sub-polyhedron, to obtain a model for 푀 = 푁 − 퐾 one takes a Poincaré duality model for 푁 and then “kills” (using a mapping cone) the cohomology classes coming from 퐾. We can apply these ideas to manifolds with boundary which are obtained by removing an open neighborhood of a sub-polyhedron from a closed manifold. Manifolds admitting a surjective pretty model include closed manifolds, 2-connected manifolds with 2-connected boundary satisfying an algebraic retraction property, disk bundles of even rank of simply connected closed manifolds, and spaces obtained by removing a high-codimensional sub-polyhedron from a 2-connected manifold [CLS15a; CLS15b] (see Theorem 3.1.12 for details). More generally, if 푀 is simply connected with simply connected boundary, we can prove that it admits a Poincaré–Lefschetz duality model as soon as 푛 ≥ 7, and we recover the same results. We first use a kind of chain-level Poincaré–Lefschetz duality and cubical diagrams to compute the Betti numbers of Conf푘(푀), as was done for closed manifolds in [LS08a]. We then reuse the methods of Chapter 2 to provide an explicit real model for Conf푘(푀). One must adapt the Fulton–MacPherson compactifications to deal with the boundary of 푀, and we obtain as Theorem D that the model conjectured in [CLS15b] is a real model for Conf푘(푀). This model only depends on a model of 푀, therefore, as a corollary, we obtain the real homotopy invariance of Conf푘(−) for smooth simply connected manifolds with simply connected boundary of dimension ≥ 5 (either admitting a pretty model or of dimension ≥ 7). The compactified configuration spaces on a framed manifold with boundary inherit an action of an operad SFM푛, weakly equivalent to the Swiss-Cheese operad SC푛. In Theorem F, we describe a model for the resulting module over SFM푛, using

6 0.2 Configuration spaces

a graphical model of SC푛 found by Willwacher [Wil15].

1 Swiss-Cheese Operad and Drinfeld Center

The little disks operads D푛 of Boardman–Vogt and May [BV73; May72] govern algebras which are associative and (for 푛 ≥ 2) commutative up to homotopy. For 푛 = 2, one can see that the fundamental groupoid of D2 forms an operad 휋D2 equivalent to an operad in groupoids PaB, called the operad of parenthesized braids, which governs braided monoidal categories [Fre17, §I.6]. Since the homotopy of D2 is concentrated in degrees ≤ 1, this is enough to recover D2 up to homotopy. For 푛 = 1, one can also easily see that 휋D1 is equivalent to an operad PaP, called the operad of parenthesized permutations, which governs monoidal categories. The Swiss-Cheese operad SC = SC2 of Voronov [Vor99] governs the action of a D2-algebra on a D1-algebra by a central morphism. As explained by Hoe- fel [Hoe09], the Swiss-Cheese operad is intimately related to the “Open-Closed Homotopy Algebras” (OCHAs) of Kajiura and Stasheff [KS06a], which are of great interest in string field theory and deformation quantization. We aim to study the fundamental groupoid of SC, which is still an operad. This fundamental groupoid is again enough to recover SC up to homotopy. In a first step, we established the following theorem: Theorem A (See Theorem 1.2.11 and Corollary 1.3.4). The fundamental groupoid operad 휋SC is equivalent to an operad PaPB whose algebras (in the category of categories) are triples (M, N, 퐹), where N is a monoidal category, M is a braided monoidal category, and 퐹 ∶ M → 풵(N) is a strong braided monoidal functor from M to the Drinfeld center 풵(N) of N. In this theorem, the monoidal categories have no unit. We also consider the unitary version SC+ of the Swiss-Cheese operad, and we obtain (Proposition 1.3.9) an extension PaPB+ of the model where the monoidal categories have a strict unit and the functor strictly preserves the unit. The result of Theorem A is a counterpart for operads in groupoids of statements of [Gin15, Proposition 31] and [AFT17, Example 2.13] about the ∞-category of factorization algebras on the upper half plane.

Based on “Swiss-Cheese Operad and Drinfeld Center”, Israel J. Math. (to appear), arXiv: 1507.06844.

9 1 Swiss-Cheese Operad and Drinfeld Center

In a second step, we rely on the result of Theorem A to construct an operad rationally equivalent (in the sense of rational homotopy theory) to the completion of the Swiss-Cheese operad. To this end, we use Drinfeld associators, which we see as morphisms PaB+ → CD̂+, where CD̂+ is the completed operad of chord diagrams. The existence of such a rational Drinfeld associator is equivalent to the rational formality of D2, but the inclusion D1 → D2 is not formal; equivalently, the constant morphism PaP+ → CD̂+ does not factor through a Drinfeld associator. We prove the following theorem:

Theorem B (See Theorem 1.4.21). Given a choice of Drinfeld associator 휙, there is an 휙 operad in groupoids PaPCD̂+ built using chord diagrams, parenthesized permutations, and parenthesized shuffles, which is rationally equivalent to 휋SC+. The Swiss-Cheese operad is not formal [Liv15], thus it cannot be recovered from its homology 퐻∗(SC). We use the splitting of 퐻∗(SC) as a product [Vor99] to build an operad in groupoids CD̂ ×+ PaP, and we compare it to our rational model of SC. Independently of the author, Willwacher [Wil15] found a different model for the Swiss-Cheese operad in any dimension 푛 ≥ 2 that uses graph complexes. His Graphs ∼ ∗ D model extends Kontsevich’s [Kon99] quasi-isomorphism 푛 Ω ( 푛) from the proof of the formality of D푛, whereas our model extends (after passing ̂ to classifying spaces) Tamarkin’s [Tam03] model B 핌픭̂ = B CD̂ of D2. Thus, in contrast to Willwacher’s model, our own model is related to Drinfeld’s original approach to quantization. It would be interesting to compare the two, as was done by Ševera and Willwacher [ŠW11] for the little 2-disks operad. This paper is organized as follows: in Section 1.1, we recall some background on the Swiss-Cheese operad and relative operads; in Section 1.2, we construct two algebraic models for the Swiss-Cheese operad; in Section 1.3, we describe what the algebras over these models are, using Drinfeld centers; and in Section 1.4, we construct a rational model in groupoids for the Swiss-Cheese operad using chords diagrams and Drinfeld associators.

1.1 Background

The little 푛-disks operad D푛 is built out of configurations of embeddings of little 푛-disks (whose images have disjoint interiors) in the unit 푛-disk, and operadic composition is given by composition of such embeddings – see [BV73; May72] for precise definitions. The fundamental groupoid 휋D2 of the little disks operads is weakly equivalent to an operad in groupoids called CoB, the operad of colored braids [Fre17, §I.5],

10 1.1 Background which we now recall. Given an integer 푟 ≥ 0, the object set of CoB(푟) is the ′ ′ symmetric group Σ푟. For 휎, 휎 ∈ Σ푟, the morphisms in HomCoB(푟)(휎, 휎 ) are given by isotopy classes of braids with 푟 strands which respect the permutations, i.e. if we label the beginning of each strand as 휎(1), 휎(2), … , 휎(푟) and the end of each strand with 휎′(1), 휎′(2), … , 휎′(푟), then the labelings match (see Figure 1.1.1 for an example). This way, given a source permutation and a braid, the target permutation is fixed and we will not draw it on pictures.

1 2

2 1

Figure 1.1.1: Example of element in CoB(2)

We will also consider colored operads, which are also called multicategories. These are used to study several algebraic structures at once, together with possible relations between them. Compared with standard operads, a colored operad has an extra data, the set of its “colors”, and for a given operation, each input and the output is assigned a color. There is an identity for each color, and composition is only possible if the input and output match. An “algebra” over a colored operad is in fact given by several objects, one for each color, and the operations act on them according to how their inputs and outputs are colored. For example, given an operad P, there is a canonical operad P⃗ with two colors, whose algebras are triplets (퐴, 퐵, 푓 ) where 퐴 and 퐵 are P-algebras and 푓 ∶ 퐴 → 퐵 is a morphism of P-algebras. The Swiss-Cheese operad SC is an operad with two colors, 픠 and 픬 (standing for “closed” and “open”). The space of operations SC(푥1, … , 푥푛; 픠) with a closed output is equal to D2(푛) if 푥1 = ⋯ = 푥푛 = 픠, and it is empty otherwise. The space SC(푥1, … , 푥푛; 픬) is the space of configurations of embeddings of full disks (corresponding to the color 픠) and half disks (corresponding to the color 픬), with disjoint interiors, inside the unit upper half disk (see Figure 1.1.2 for an example). Composition is again given by composition of embeddings. The Swiss-Cheese operad is an example of a relative operad [Vor99]: it can be seen as an operad in the category of right modules (in the sense of [Fre09]) over another operad.

11 1 Swiss-Cheese Operad and Drinfeld Center

2 1

1 2 3

Figure 1.1.2: Example of an element in SC(3, 2)

Definition 1.1.1. Let P be a (symmetric, one-colored) operad. A relative operad over P is an operad Q in the category of right modules over P. Equivalently, it is a two-colored operad Q (where the two colors are called 픠 and 픬) such that: ⎧ {P(푛), if 푥1 = ⋯ = 푥푛 = 픠; Q(푥1, … , 푥푛; 픠) = ⎨ ⎩{∅, otherwise. Q(푛, 푚) ∶= Q(픬,⏟ … , 픬, 픠,⏟ … , 픠; 픬) = (Q(푛))(푚). 푛 푚 If Q is such a relative operad, we will write Q픠(푚) ≔ P(푚).

The Swiss-Cheese operad SC is a relative operad over the little disks operad D2. We also consider the unitary version of the Swiss-Cheese operad SC+, which is a + relative operad over the unitary version of the little disks operad D2 , and which satisfies SC+(0, 0) = ∗. Composition with the nullary elements simply forgets half disks or full disks of the configuration. Remark 1.1.2. We consider a variation of the Swiss-Cheese operad, where we allow operations with only closed inputs and an open output, whereas in Voronov’s definition these configurations are forbidden. We write SCvor for Voronov’s vor version, so that SC (0, 푚) = ∅ while SC(0, 푚) ≃ D2(푚) ≠ ∅.

1.2 Permutations and braids 1.2.1 Colored version We first define an operad in groupoids CoPB, the operad of colored permutations and braids. It is an operad relative over CoB, the operad of colored braids [Fre17, §I.5]. Let 퐷+ = {푧 ∈ ℂ ∣ ℑ푧 ≥ 0, |푧| ≤ 1} be the upper half disk, and let + Conf(푛, 푚) = {(푧1, … , 푧푛, 푢1, … , 푢푚) ∈ 퐷 ∣ ℑ푧푖 = 0, ℑ푢푗 > 0, 푧푖 ≠ 푧푗, 푢푖 ≠ 푢푗}

12 1.2 Permutations and braids be the set of configurations of 푛 points on the real interval [−1, 1] and 푚 points in the upper half disk. The disk-center mapping 휔 ∶ SC(푛, 푚) ∼ Conf(푛, 푚), sending each disk to its center, is a weak equivalence [Vor99]. Let Σ푘 be the 푘th symmetric group, and let Sh푛,푚 be the set of (푛, 푚)-shuffles:

Sh푛,푚 = {휇 ∈ Σ푛+푚 ∣ 휇(1) < ⋯ < 휇(푛), 휇(푛 + 1) < ⋯ < 휇(푛 + 푚)}.

0 For every 휇 ∈ Sh푛,푚, we choose a configuration 푐휇 ∈ Conf(푛, 푚) with 푛 “terrestrial” points (on the real axis) and 푚 “aerial” points (with positive imaginary part), in the left-to-right order given by the (푛, 푚)-shuffle 휇. For example we can choose:

0 1 2 3 휇 = (14|235) ∈ Sh2,3 푐휇 = ∈ Conf(2, 3), (1.2.1) 1 2

We consider the set:

퐶0(푛, 푚) = {휎 ⋅ 푐0 } ⊂ Conf(푛, 푚), 휇 휎∈Σ푛×Σ푚, 휇∈Sh푛,푚 where Σ푛 × Σ푚 acts by permuting labels. Example 1.2.2. For example, 퐶0(2, 1) can be chosen to be:

⎧ ⎫ { 1 2 2 1 1 2 } { } { } { } { 1 1 1 } 0 { } 퐶 (2, 1) = ⎨ ⎬ { } { } { 2 1 1 2 2 1 } { } { } { 1 1 1 } ⎩ ⎭

The precise position of the points does not matter for our purposes, only their left-to-right order.

Definition 1.2.3. The groupoid CoPB(푛, 푚) is the restriction of the fundamental groupoid of Conf(푛, 푚) to the set 퐶0(푛, 푚) ⊂ Conf(푛, 푚) (i.e. it is its full subcategory with these objects):

CoPB(푛, 푚) ≔ 휋 Conf(푛, 푚)|퐶0(푛,푚).

13 1 Swiss-Cheese Operad and Drinfeld Center

0 The set ob CoPB(푛, 푚) = 퐶 (푛, 푚) is isomorphic to Sh푛,푚 × Σ푛 × Σ푚. We represent these objects by sequences of 푛 “terrestrial” points (drawn in white and labeled by {1, … , 푛}) and 푚 “aerial” points (drawn in black and labeled by {1, … , 푚}) on the interval 퐼 = [−1, 1]; the order in which terrestrial and aerial points appear is given by the shuffle. For example, the element in Equation (1.2.1) is represented by: 1 1 2 2 3 .

Morphisms between two such configurations are given by isotopy classes of bicolored braids, where strands between terrestrial points never go behind any other strand, including other terrestrial strands (indeed, they represent paths in the interval [−1, 1], and points cannot move over one another in Conf푛([−1, 1]), nor can they go behind the paths in the open upper half disk). See Figure 1.2.1 for an example of an element in CoPB(2, 3), and Figure 1.2.2 for the corresponding path in Conf(2, 3).

1 3 1 2 2

Figure 1.2.1: Element in CoPB(2, 3) Figure 1.2.2: Corresponding path in Conf(2, 3)

To not confuse objects of CoPB and objects of CoB, and to be coherent with the graphical representation of ΩΩ in Section 1.2.2, we draw the objects of CoB with ends in the shape of chevrons:

1 2 ∈ ob CoB(2).

The symmetric sequence CoPB(푛) = {CoPB(푛, 푚)}푚≥0 is a right module over CoB by inserting a colored braid in a tubular neighborhood of an aerial strand (Figure 1.2.3). Similarly, the operad structure inserts a colored braid in a tubular neighborhood of a terrestrial strand (Figure 1.2.4). One can easily check that this gives a relative operad over CoB (in the same manner that one checks that CoB itself is an operad, cf. [Fre17, §I.5]).

14 1.2 Permutations and braids

1 3 1 2 2 1 2 1 4 1 2 2 3

픠 ∘1 =

CoPB(2, 3) CoB(2) CoPB(2, 4)

Figure 1.2.3: Definition of the right CoB-module structure

1 3 1 2 2 1 1 1 4 3 1 2 2

픬 ∘1 =

CoPB(2, 3) CoPB(1, 1) CoPB(2, 4)

Figure 1.2.4: Definition of the operad structure

15 1 Swiss-Cheese Operad and Drinfeld Center

1.2.2 Magmas Definition 1.2.4. Let Ω be the magma operad (as in [Fre17]), defined as the free symmetric operad 핆(휇픠) on a single generator 휇픠 = 휇픠(푥1, 푥2) of arity 2 (where Σ2 acts freely on 휇픠). Its algebras are sets endowed with a product satisfying no further conditions.

Elements of Ω(푛) are parenthesizations of a permutation of 푛 elements, for example (((푥1푥3)(푥2푥4))푥5) ∈ Ω(5). The index 픠 of 휇픠 is there to be coherent with the following definition:

Definition 1.2.5. Let ΩΩ = 핆(휇픠, 푓 , 휇픬) be the free colored operad on the three generators 휇픠 ∈ ΩΩ(픠, 픠; 픠), 푓 ∈ ΩΩ(픠; 픬) et 휇픬 ∈ ΩΩ(픬, 픬; 픬). It is a relative operad over Ω.

An algebra over ΩΩ is the data of two magmas 푀, 푁, and of a mere function 푓 ∶ 푀 → 푁 (not necessarily preserving the product).

Lemma 1.2.6. The suboperad of SC generated by the following three elements is free on those generators:

1 2 1 휇픠 = , 휇픬 = 1 2 , 푓 = .

Proof. We would like to show that the induced morphism 푖 ∶ ΩΩ → SC, sending the three generators of ΩΩ to the elements depicted in the lemma, is an embedding, i.e. an isomorphism onto its image. The image of this induced morphism is by definition the suboperad generated by the three elements, hence the lemma. The fact that the suboperad of D2 generated by 휇픠 is free is given by [Fre17, Proposition I.6.2.2(a)]. Let 훼 ∈ SC(푛, 푚) be a configuration, as in Figure 1.2.5. We will build an element of ΩΩ which is sent to 훼 under 푖. This set-level retraction is not necessarily a morphism of operads but still shows that 푖 is injective, which will prove the lemma. We first regroup the 푚 full disks into connected components 퐶1, … , 퐶푟. For each 퐶푖, we consider the center of the middle horizontal interval (in blue on the figure), which we project onto the real line (in red). These points, together with the centers of the 푛 half disks, make up a dyadic configuration on the horizontal diameter of the ambient half disk. By [Fre17, Proposition I.6.2.2(a)], such a dyadic configuration is equivalent to an element 푢 ∈ Ω(푛 + 푟) (which we see as an iterate of 휇픠).

16 1.2 Permutations and braids

1 2 3 1

Figure 1.2.5: Example element of ΩΩ(1, 3)

For each 퐶푖 (corresponding to an input of 푢), we apply the same proposition [Fre17, Proposition I.6.2.2(a)] to get an element 푣푖 ∈ Ω(푘푖) (which we see as a iterate of 휇픬), and ∑ 푘푖 = 푚. If we plug 푓 (푣푖) in the corresponding inputs of 푢, we get an element of ΩΩ(푛, 푚), and by construction this elements gets sent to 훼 by 푖.

We consider the following graphical representation for elements of ΩΩ with open output, where the generators are represented as follows:

1 2 1 2 1 휇픠 , 휇픬 , 푓 .

For example, this is the representation of the element of Figure 1.2.5:

1 2 3 1 휇픬(푓 (휇픠(푥1, 푥2)), 휇픬(푓 (푥3), 푦1)) ∈ ΩΩ(1, 3)

Each 휇픬 is represented by cutting in half the interval; 푓 is represented by parentheses, and 휇픠 is again represented by cutting the interval inside the parentheses in half. Closed inputs are represented by black points, while open inputs are represented by white points.

Remark 1.2.7. The parentheses separating the aerial points are really necessary in

17 1 Swiss-Cheese Operad and Drinfeld Center the representation. For example these are two different objects:

1 2 푓 (휇픠(푥1, 푥2)) = = 1 2

1 2 휇픬(푓 (푥1), 푓 (푥2)) = = 1 2

1.2.3 Parenthesized version We now define PaPB, the operad of parenthesized permutations and braids, a relative operad over PaB. The definition of PaPB is given as a pullback of CoPB, similarly to how PaB is a pullback of CoB. Definition 1.2.8. We consider the morphism 휔 ∶ ΩΩ → ob CoPB, given on generators by:

1 2 1 1 2 휇픠 ↦ , 푓 ↦ , 휇픬 ↦ , and we define PaPB ≔ 휔∗CoPB, the pullback of CoPB along 휔. It is an operad in groupoids such that ob PaPB = ΩΩ and

HomPaPB(푛,푚)(푢, 푣) ≔ HomCoPB(푛,푚)(휔(푢), 휔(푣)) for 푢, 푣 ∈ ΩΩ(푛, 푚). Definition 1.2.9. A categorical equivalence is a morphism of operads in groupoids which is an equivalence of categories in each arity. Two operads P and Q are said to be categorically equivalent (and we write P ≃ Q) if they can be connected by a zigzag of categorical equivalences:

P ∼ ⋅ ∼ … ∼ ⋅ ∼ Q.

Recall that the fundamental groupoid functor 휋 ∶ (Top, ×) → (Gpd, ×) is monoidal, thus the fundamental groupoid of a topological operad is an operad in groupoids. Remark 1.2.10. Since each arity of the operad SC has homotopy concentrated in degrees ≤ 1, it follows that its fundamental groupoid is enough to recover the homotopy type of the operad through the classifying space construction: SC ∼ B 휋SC.

18 1.3 Drinfeld center

Theorem 1.2.11. The operad PaPB is isomorphic to the fundamental groupoid of SC restricted to the image of ΩΩ ↪ SC, and we get a zigzag of categorical equivalences:

∼ ∼ 휋SC 휋SC|ΩΩ ≅ PaPB CoPB.

Proof. The proof of the first part of the proposition is a direct adaptation of the proof of [Fre17, Proposition I.6.2.2(b)]. We note that ΩΩ ⊂ ob 휋SC is a suboperad, thus 휋SC|ΩΩ is also a suboperad of 휋SC. For the second part, we note that ΩΩ(푛, 푚) meets all the connected components of SC(푛, 푚) ∼ Σ푛 × D2(푚), so the first inclusion is a categorical equivalence. Since 휔 ∶ ΩΩ → ob CoPB is surjective, the second morphism is also a categorical equivalence.

1.3 Drinfeld center 1.3.1 Algebras over PaPB Definition 1.3.1. Let C be a (non-unitary) monoidal category. Its suspension ΣC is a bicategory with a single object. The Drinfeld center [Maj91; JS91] of C is the braided monoidal category 풵(C) = End(idΣC). Explicitly, it is given as follows (see also [nLa16] for the more abstract point of view):

• Objects are pairs (푋, Ψ), where 푋 is an object of C and Ψ ∶ (푋⊗−) → (−⊗푋) is a half-braiding, i.e. a natural isomorphism such that for all 푌, 푍 ∈ C the following diagram commutes:

Ψ 푋 ⊗ (푌 ⊗ 푍) 푌⊗푍 (푌 ⊗ 푍) ⊗ 푋 훼 훼

(푋 ⊗ 푌) ⊗ 푍 푌 ⊗ (푍 ⊗ 푋)

Ψ푌⊗1 1⊗Ψ푍 (푌 ⊗ 푋) ⊗ 푍 훼 푌 ⊗ (푋 ⊗ 푍)

• Morphisms between (푋, Ψ) and (푌, Ψ′) are morphisms 푓 ∶ 푋 → 푌 of C such that, for all 푍 ∈ C, the following diagram commutes

푓 ⊗1 푋 ⊗ 푍 푌 ⊗ 푍 ′ Ψ푍 Ψ푍 푍 ⊗ 푋 푍 ⊗ 푌 1⊗푓

19 1 Swiss-Cheese Operad and Drinfeld Center

• The tensor product of two objects (푋, Ψ)⊗(푋′,Ψ′) is given by (푋 ⊗푋′,Ψ″), ″ where Ψ푍 is defined by the following diagram (that can be rearranged as an hexagon by inverting both vertical 훼’s, see Figure 1.3.6):

Ψ″ (푋 ⊗ 푋′) ⊗ 푌 푌 푌 ⊗ (푋 ⊗ 푋′) 훼 훼 ′ ′ ′ ′ 푋 ⊗ (푋 ⊗ 푌) ′ 푋 ⊗ (푌 ⊗ 푋 ) −1 (푋 ⊗ 푌) ⊗ 푋 (푌 ⊗ 푋) ⊗ 푋 1⊗Ψ푌 훼 Ψ푌⊗1

′ ′ ′ ′ • The braiding (푋, Ψ) ⊗ (푋 ,Ψ ) → (푋 ,Ψ ) ⊗ (푋, Ψ) is given by Ψ푋′ and the associator is given by the associator of C. Example 1.3.2. Let 퐻 be a Hopf algebra and consider the category Rep(퐻) of its representations. Then the Drinfeld center 풵(Rep(퐻)) is equivalent to the category Rep(퐷(퐻)) of representations of the “Drinfeld double” 퐷(퐻) of 퐻, which is roughly speaking obtained as a semi-direct product of 퐻 with its dual 퐷(퐻) ≈ 퐻 ⋊ 퐻∨ [Dri87]. We consider the following elements of PaPB:

휇픠 ∈ ob PaB(2) 휇픬 ∈ ob PaPB(2, 0) 푓 ∈ ob PaPB(0, 1) 휏 ∈ PaB(2) 1 2 1 2 1 2 1

푝 ∈ PaPB(0, 2) 휓 ∈ PaPB(1, 1) 훼픠 ∈ PaB(3) 훼픬 ∈ PaPB(3, 0) 1 2 1 2 1 2 3 1 2 3

1 Theorem 1.3.3. Let P be a {픠, 픬}-colored operad in the category of categories, let 푚픠 ∈ 픠 ob P (2), 푚픬 ∈ ob P(2, 0), 퐹 ∈ ob P(0, 1) be objects, and let

푎픠 ∶ 푚픠(푚픠(푥1, 푥2), 푥3) → 푚픠(푥1, 푚픠(푥2, 푥3)), 휋 ∶ 푚픬(푓 (푥1), 푓 (푥2)) → 푓 (푚픠(푥1, 푥2)), 푡 ∶ 푚픠(푥1, 푥2) → 푚픠(푥2, 푥1), Ψ ∶ 푚픬(푓 (푥1), 푦1) → 푚픬(푦1, 푓 (푥1)), 푎픬 ∶ 푚픬(푚픬(푦1, 푦2), 푦3) → 푚픬(푦1, 푚픬(푦2, 푦3)), 1The operad will not necessarily be a relative operad, but we will still use the notation P(푛, 푚) = P(픠푚, 픬푛; 픬) and P픠(푚) = P(픠푚; 픠).

20 1.3 Drinfeld center be isomorphisms. Then there exists a morphism 휃 ∶ PaPB → P such that

휃(휇픠) = 푚픠, 휃(휇픬) = 푚픬, 휃(푓 ) = 퐹, 휃(훼픠) = 푎픠, 휃(훼픬) = 푎픬, 휃(휏) = 푡, 휃(푝) = 휋, 휃(휓) = Ψ, (in which case this morphism is unique) if, and only if, the coherence diagrams of Figures 1.3.1 to 1.3.6 commute.

푚(푚(푚(푥 , 푥 ), 푥 ), 푥 ) 푚(푎,id) 1 2 3 4 푎 푚(푚(푥1, 푚(푥2, 푥3)), 푥4)

푎 푚(푚(푥1, 푥2), 푚(푥3, 푥4))

푚(푥 , 푚(푚(푥 , 푥 ), 푥 )) 1 2 3 4 푎 푚(id,푎) 푚(푥1, 푚(푥2, 푚(푥3, 푥4)))

Figure 1.3.1: Pentagon for (푚, 푎) = (푚픠, 푎픠) and (푚픬, 푎픬)

푚 (푚 (푥 , 푥 ), 푥 ) 푚픠(푡,id) 픠 픠 1 2 3 푎픠

푚픠(푚픠(푥2, 푥1), 푥3) 푚픠(푥1, 푚픠(푥2, 푥3)) 푎픠 푡 푚 (푥 , 푚 (푥 , 푥 )) 푚 (푚 (푥 , 푥 ), 푥 ) 픠 2 픠 1 3 푎픠 픠 픠 2 3 1

푚픠(id,푡) 푚픠(푥2, 푚픠(푥3, 푥1)) 푚 (푥 , 푚 (푥 , 푥 )) −1 픠 1 픠 2 3 푎픠

푚픠(id,푡) 푚픠(푥1, 푚픠(푥3, 푥2)) 푚픠(푚픠(푥1, 푥2), 푥3) −1 푎픠 푡 푚픠(푚픠(푥1, 푥3), 푥2) 푚픠(푥3, 푚픠(푥1, 푥2))

−1 푚픠(푡,id) 푎 푚픠(푚픠(푥3, 푥1), 푥2) 픠 Figure 1.3.2: Hexagons

Recall that an algebra over a colored operad Q is a morphism Q → End(퐴,퐵), where

픠 ⊗푛 ⊗푚 픬 ⊗푛 ⊗푚 End(퐴,퐵)(푛, 푚) = hom(퐵 ⊗ 퐴 , 퐴), End(퐴,퐵)(푛, 푚) = hom(퐵 ⊗ 퐴 , 퐵).

Given a morphism PaPB → End(M,N) with the names as in Theorem 1.3.3 for the images of the generators, the previous coherence diagrams are exactly the

21 1 Swiss-Cheese Operad and Drinfeld Center

푚픬(푚픬(푓 (푥1), 푓 (푥2)), 푓 (푥3)) 푚픬(휋,id) 푎픠

푚픬(푓 (푚픠(푥1, 푥2)), 푓 (푥3)) 푚픬(푓 (푥1), 푚픬(푓 (푥2), 푓 (푥3))) 휋 푚픬(id,휋) 푓 (푚픠(푚픠(푥1, 푥2), 푥3)) 푚픬(푓 (푥1), 푓 (푚픠(푥2, 푥3))) 휋 푓 (푎픠) 푓 (푚픠(푥1, 푚픠(푥2, 푥3))) Figure 1.3.3: 퐹 is monoidal

푚픬(푚픬(푓 (푥1), 푥2), 푥3) 푚픬(Ψ,id) 푎픬

푚픬(푚픬(푥2, 푓 (푥1)), 푥3) 푚픬(푓 (푥1), 푚픬(푥2, 푥3)) 푎픬 Ψ

푚픬(푥2, 푚픬(푓 (푥1), 푥3)) 푚픬(푚픬(푥2, 푥3), 푓 (푥1)) 푎 푚픬(id,Ψ) 픬 푚픬(푥2, 푚픬(푥3, 푓 (푥1))) Figure 1.3.4: Ψ is a half-braiding

Ψ 푚픬(푓 (푥1), 푓 (푥2)) 푚픬(푓 (푥2), 푓 (푥1)) 휋 휋 푓 (푚 (푥 , 푥 )) 푓 (푚 (푥 , 푥 )) 픠 1 2 푓 (푡) 픠 2 1

Figure 1.3.5: 퐹 is braided

푚픬(푓 (푥1), 푚픬(푓 (푥2), 푥3)) −1 푚(id,Ψ) 푎픬

푚픬(푓 (푥1), 푚픬(푥3, 푓 (푥2))) 푚픬(푚픬(푓 (푥1), 푓 (푥2)), 푥3) 푚픬(휋,id) 푚픬(푓 (푚픠(푥1, 푥2)), 푥3) −1 푎픬 Ψ 푚픬(푥3, 푓 (푚픠(푥1, 푥2))) −1 푚픬(id,휋 ) 푚픬(푚픬(푓 (푥1), 푥3), 푓 (푥2)) 푚픬(푥3, 푚픬(푓 (푥1), 푓 (푥2)))

푚 (Ψ,id) −1 픬 푎픬 푚픬(푚픬(푥3, 푓 (푥1)), 푓 (푥2)) Figure 1.3.6: 퐹 is monoidal w.r.t. half-braidings

22 1.3 Drinfeld center

diagrams encoding the fact that (푚픠, 푎픠, 푡픠) is a braided monoidal structure on M, (푚픬, 푎픬) is a monoidal structure on N, and 퐹 is a braided monoidal functor to the Drinfeld center. Corollary 1.3.4. An algebra over PaPB is the data of a (non-unitary) monoidal category (N, ⊗), a (non-unitary) braided monoidal category (M, ⊗, 휏), and a strong braided monoidal functor 퐹 ∶ M → 풵(N). Definition 1.3.5. Between two objects 푥, 푦 ∈ PaPB(푛, 푚) such that the terrestrial (resp. aerial) points of 푥 are numbered in the same order as the terrestrial (resp. aerial) points 푦, there is a unique morphism 휇 ∈ homPaPB(푛,푚)(푥, 푦), called a shuffle-type morphism, such that the aerial strands do not cross each other (see Figure 1.3.7).

Figure 1.3.7: Example of shuffle-type morphism

Proof of Theorem 1.3.3. It is clear (a simple exercise in drawing braid diagrams) that the morphisms of PaPB satisfy the corresponding relations, thus we get the “only if” part of the theorem. Let 푌 ∈ homPaPB(푛,푚)(푥1, 푥2) be a morphism. We want to decompose it as in Figure 1.3.8:

′ ′ • We first arbitrarily choose two objects 푥1, 푥2 which are in the image of the ′ 픬 픠 product PaP(푛) × PaB(푚) by 휇픬(−, 푓 (−)). In other words, 푥푖 = 휇픬(푥푖 , 푓 (푥푖 )) 픬 is the concatenation of an object 푥푖 ∈ PaP(푛) = PaPB(푛, 0) and of the image 픠 by 푓 of an object 푥푖 ∈ PaB(푚). We also require that the aerial points (resp. ′ the terrestrial points) of 푥푖 are numbered in the same order as those of 푥푖. ′ • We take the unique shuffle-type morphism 휇 ∶ 푥1 → 푥1. 픬 픠 ′ ′ • We build a morphism 푋 = 휇픬(푋 , 푓 (푋 )) ∶ 푥1 → 푥2. It is the concatenation of 푋픬 ∈ PaP(푛), and 푋픠 ∈ PaB(푚). Explicitly, 푋픬 is the colored permutation 픠 where all the aerial strands of 휔∗(푌) have been forgotten, and 푋 is the colored braid where all the terrestrial strands of 휔∗(푌) have been forgotten.

23 1 Swiss-Cheese Operad and Drinfeld Center

푥1 휇

′ 푥1

′ ″ = 휇표(푋 ,푓 (푋 )) ′ 푥2

휇′

푥2

Figure 1.3.8: Decomposition in PaPB(2, 3)

′ ′ • Finally, we take the unique shuffle-type morphism 휇 ∶ 푥2 → 푥2. By construction, 푌 = 휇′ ∘푋 ∘휇. Besides, this decomposition is unique given the ′ ′ specified intermediary objects 푥1, 푥2, so it suffices to show that 휃 can be defined ′ ′ unequivocally on each part, that it doesn’t depend on the choice of 푥1 and 푥2, and that it is compatible with operadic composition. ±1 The shuffle-type morphisms are all in the suboperad of PaPB generated by 훼픬 , 푝±1, 휓±1: first one can cut the objects of PaB in the smallest possible pieces with −1 ±1 ±1 푝 , the 훼픬 and 휓 can be used to bring all the aerial points at their positions, and finally 푝 is used to glue back all the aerial pieces. By the theorems I.6.1.7 and I.6.2.4 of [Fre17], the two morphisms 푋픬 ∈ PaP(푛) and 푋픠 ∈ PaB(푚) are ±1 ±1 ±1 ±1 ±1 respectively in the suboperads generated by 휇픬 , 훼픬 and by 휇픠 , 훼픠 , 휏 . It follows that every morphism 푌 of PaPB is in the suboperad generated by all these elements, thus the morphism 휃 ∶ PaPB → P, if it exists, is unique. By the same theorems of [Fre17], the pentagons (Figure 1.3.1) and the hexagons (Figure 1.3.2) show that the morphism 휃 can be defined with no ambiguity on the 픠 픬 ′ ′ two pieces 푋 and 푋 . The possible choices for 푥1 and 푥2 are all related by associators, so the pentagons (Figure 1.3.1) and MacLane’s coherence theorem [Mac98] ′ for monoidal categories show that the image does not depend on the choice of 푥1 ′ and 푥2. Let 휇 ∈ PaPB(푛, 푚) be a shuffle-type morphism; we saw that it could be decomposed in terms of 푝±1, 휓±1 and 훼±1. The coherence theorem of MacLane [Mac98] and the coherence theorem of Epstein [Eps66] on monoidal functors (non-symmetric version) show that, thanks to the pentagons (Figure 1.3.1) and the fact that 퐹 is monoidal (Figure 1.3.3), the image 휃(휇) neither depends on the choice of associator decomposition, nor on the choice of decomposition of 푝±1, nor on the way the 휓±1 are gathered in the parenthesizing. It thus suffices to define 휃 on the underlying morphism of CoPB. This last morphism is actually an element of the braid group 퐵푛+푚 (of course

24 1.3 Drinfeld center not all the elements of the braid group can given a morphism: terrestrial strands cannot cross any other strand). By seeing 휓 as a braiding, and by interpreting the relations of Figures 1.3.4 and 1.3.6 as two hexagon relations, we can adapt the proof of the step 2 of [Fre17, Theorem 6.2.4] to see that the image by 휃 of this braid does not depend on its representation. Finally, 휃 is well-defined on every morphism. It remains to show that it respects operad composition. By adapting the fourth step of the proof of the same theorem of [Fre17] and by using the relation of Figure 1.3.5, we can see that 휃 respects operadic composition. Indeed, shuffle-type morphism are sent by construction on elements decomposed ±1 ±1 픬 in terms of associators, their inverses, 푝 and 휓 , while for example 휓 ∘1 id푓 = 푓 (휏); but thanks to the relation of Figure 1.3.5, both elements are equal in the image. By dropping all mentions of parenthesizing, we get:

Proposition 1.3.6. An algebra over CoPB consists of a strict non-unitary monoidal category N, a strict braided non-unitary monoidal category M, and of a strict braided monoidal functor 퐹 ∶ M → 풵(N).

1.3.2 Unitary versions

We are going to define unitary versions CoPB+ and PaPB+ of the operads we are studying, satisfying CoPB+(0, 0) = PaPB+(0, 0) = {∗픬}. For consistency we will denote ∗픠 the element of the one-colored unitary operads we will consider (CoB+, PaB+, etc.).

Definition 1.3.7. Let CoPB+ be the relative operad over CoB+, defined as a unitary extension of CoPB+. Composition with ∗픠 ∈ CoB+(0) forgets aerial strands, while composition with ∗픬 forgets terrestrial strands.

Definition 1.3.8. Let ΩΩ+ be the relative operad over Ω+, a unitary extension of ΩΩ. Composition with nullary elements is given on generators by (it is not necessary to specify 푓 (∗픬) as ΩΩ+(0, 0) is a singleton anyway):

휇픠(∗픠, id픠) = 휇픠(id픠, ∗픠) = id픠, 휇픬(∗표, id픬) = 휇픬(id픬, ∗픬) = id픬 .

∗ Let also PaPB+ = 휔+CoPB+ be the pullback of CoPB+ along 휔+, where 휔+ is defined as the 휔 of Definition 1.2.8 (it is compatible with the unitary extensions).

′ Proposition 1.3.9. There is a zigzag of categorical equivalences, where ΩΩ+ ⊂ SC+ is the sub–operad generated by 푚픠, 푚픬, 푓 and the nullary elements:

∼ ∼ ∗ ∼ 휋SC 휋SC | ′ PaPB ≔ 휔 CoPB CoPB , + + ΩΩ+ + + + +

25 1 Swiss-Cheese Operad and Drinfeld Center

′ Remark 1.3.10. In ΩΩ+, we have for example:

1 휇픬(id픬, ∗픬) = 1 ≠ = id픬, but 휇픬(∗픬, ∗픬) = ∗픬. In other words, ∗픬 (and similarly ∗픠 for 휇픠) is not a strict unit for ∗픬, but is still idempotent. ′ ′ Proof. There is an evident morphism 휔+ ∶ ΩΩ+ → ΩΩ+ sending generators on generators, and we check directly that 휋SC | ′ is identified with the pullback + ΩΩ+ ′ ′ ′ 휔+PaPB . Since 휔+ and 휔+ are both surjective, we obtain the two categorical equivalences ∼ ∼ 휋SC | ′ PaPB CoPB . + ΩΩ+ + + ′ And since ΩΩ+ meets all connected components of SC+ we also have that the inclusion 휋SC | ′ ↪ 휋SC is a categorical equivalence. + ΩΩ+ + The proof of the following proposition is a direct unitary extension of the proof of Corollary 1.3.4 (one also needs to extend the definition of the Drinfeld center to unitary monoidal categories, cf. the given references):

Proposition 1.3.11. An algebra over PaPB+ is given by:

• a monoidal category (N, ⊗, 1N) with a strict unit;

• a braided monoidal category (M, ⊗, 1M, 휏) with a strict unit;

• a monoidal functor 퐹 ∶ M → 풵(N) satisfying 퐹(1M) = 1N.

An algebra over CoPB+ is given by the same data, but where the two tensors products are strictly associative, strictly braided for the second, and the functor is strict braided monoidal.

1.4 Chord diagrams

Let P an operad in groupoids. Its completion P̂ is defined by the Malcev completion arity by arity: P(푟)̂ = 픾ℚ[̂P(푟)], and it is an operad in complete groupoids [Fre17, §I.9]. Here, ℚ[퐺] is the Hopf algebroid of the groupoid 퐺; it has the same objects as 퐺, and homℚ[퐺](푥, 푦) = ℚ[hom퐺(푥, 푦)] is the free ℚ-module on the hom-set, equipped with a coalgebra structure where every generator is grouplike. It is completed at the augmentation

26 1.4 Chord diagrams ideal, and then the functor 픾 extracts the grouplike elements to define an operad in complete groupoids. This completion is equipped with a canonical completion morphism P → P.̂

Definition 1.4.1. A morphism of operads in groupoids P → Q is called a rational categorical equivalence (denoted P ∼ℚ Q) if the induced morphism P̂ → Q̂ is a categorical equivalence. We write P ∼ℚ Q if P and Q can be connected by a zigzag of rational categorical equivalences.

This definition is motivated by the following remark: if 퐴 is an abelian group, ̂ then 퐴 = 퐴 ⊗ℤ ℚ. Examples of rational categorical equivalences include categorical equivalences and the canonical completion morphisms P → P.̂ We refer to [Fre17, §I.9] for more details.

1.4.1 Drinfeld associators and chord diagrams Definition 1.4.2. The Drinfeld–Kohno operad 픭 is an operad in Lie algebras.2 In each arity we have the presentation by generators and relations:

픭(푟) = Lie(푡푖푗)1≤푖≠푗≤푟 /([푡푖푗, 푡푘푙], [푡푖푘, 푡푖푗 + 푡푗푘]), and operadic composition is given by explicit formulas [Fre17, §I.10.2].

The universal enveloping algebra functor 핌 being monoidal, 핌픭 is an operad in associative algebras. The algebra 핌픭(푟) is generated by chord diagrams with 푟 strands, and composition is given by insertion of a diagram (cf. ibid. for precise definitions). We can complete 픭 with respect to the weight grading (the weight of 푡푖푗 is defined to be 1) to get an operad 픭̂ in complete Lie algebras, and we can consider its completed universal enveloping algebra:

Definition 1.4.3. The operad of completed chord diagrams, CD̂, is the operad ̂ in groupoids given by ob CD̂(푟) = ∗ and HomCD̂(푟)(∗, ∗) = 픾핌픭(푟)̂ . Operadic composition is induced by the one of 픭.

These operads have unitary extensions: restriction operations forget strands of the chord diagrams, and if a chord was attached to the strand, the diagram is sent to 0. We thus get unitary operads 픭+, 픭̂ +, and CD̂+. Definition 1.4.4. A Drinfeld associator (with parameter 휇 ∈ ℚ×) is a morphism 휇푡12/2 휙 ∶ PaB+ → CD̂+ of operads that sends the braiding 휏 ∈ PaB+(2) to 푒 ∈ 휇 CD̂+(2). We let Ass (ℚ) be the set of such associators.

2The monoidal product in the category of Lie algebras is the direct sum.

27 1 Swiss-Cheese Operad and Drinfeld Center

If 휙 ∈ Ass휇(ℚ), then the formal series in two variables

Φ(푡12, 푡13) ≔ 휙(훼픠) ∈ CD̂+(3) ≅ ℚ[[푡12, 푡13]] is a Drinfeld associator in the usual sense, satisfying the usual equations (pentagon, hexagon), and vice versa. A Drinfeld associator 휙 extends to a categorical ∼ 휇 equivalence 휙 ∶ PaB̂+ CD̂+, i.e. 휙 is a rational equivalence. The set Ass (ℚ) is a torsor under the action of the Grothendieck–Teichmüller group 퐺푇1(ℚ), the group of automorphisms of PaB̂+ fixing 휇픠 and 휏. A theorem of Drinfeld [Dri90] states that the set of associators Ass1(ℚ) is nonempty. We can also consider the operad PaCD̂ +, which is the pullback of CD̂+ along the terminal morphism Ω → ob CD̂+ = ∗. It is used to define the pro-unipotent version of the Grothendieck–Teichmüller group 퐺푅푇1(ℚ), under which Ass휇(ℚ) is a pro-torsor. We recall the following statement [Fre17], which is actually a general fact about pullback along morphisms from a free operad: each morphism 휙 ∶ PaB̂+ → CD̂+ admits a unique lifting

PaCD̂ + 휙̃+

휙 PaB̂+ CD̂+ which is given by the identity on the level of objects. If the morphism 휙 came from a Drinfeld associator, then this defines an isomorphism of operads in groupoids:

≅ 휙+̃ ∶ PaB̂+ PaCD̂ +.

1.4.2 Shuffle of operads By analogy with the decomposition of Figure 1.3.8, we define a new rational model in groupoids for 휋SC+ that involves the operad of chord diagrams.

Other description of CoPB+

Definition 1.4.5. Let Π be the permutation operad: Π(푛) = Σ푛, and operadic composition is given by bloc composition of permutations. We also denote Π the same operad seen as an operad in discrete groupoids. We also let Π+ be its obvious unitary extension. The following operad is meant to represent the shuffle-type morphisms of Definition 1.3.5:

28 1.4 Chord diagrams

Definition 1.4.6. We define the relative (unitary) operad in groupoid Sh+ over Π+. The set of objects of Sh+(푛, 푚) is Sh푛,푚 × Σ푛 × Σ푚, the same as CoPB+ (with the same graphical representation). Operadic composition on the object level is the same as that of CoPB+. On the level of morphisms: ⎧ ′ ′ ′ ′ {∗ 휎 = 휏, 휎 = 휏 , HomSh (푛,푚)((휇, 휎, 휎 ), (휈, 휏, 휏 )) = ⎨ + ⎩{∅ otherwise, and we check that this gives a well-defined relative operad over Π+ (i.e. there are no maps ∗ → ∅ to define, and all the maps ⋅ → ∗ are terminal maps).

Graphically, we simply represent morphisms of Sh+ by an arrow between two bicolored configurations on the interval. Such an arrow exists iff the terrestrial (resp. aerial) points of the first configuration are in the same order as the terrestrial (resp. aerial) points of the second configuration, so we do not write the labels for the second configuration: 2 1 1 3 2

Remark 1.4.7. The symmetric groups Σ푛 and Σ푚 act on the left and on the right on Sh+(푛, 푚) (by multiplication on respective factors).

Lemma 1.4.8. The groupoid CoPB+(푛, 푚) is isomorphic to (CoP (푛) × CoB (푚)) × Sh (푛, 푚). + + Σ푛×Σ푚 + Proof. We define: 휁 ∶ (CoP (푛) × CoB (푚)) × Sh (푛, 푚) → CoPB (푛, 푚) + + Σ푛×Σ푚 + + by a graphical calculus: 2 2 3 1 1

1 2 2 1 3 1 2 3 1 2

× ×

29 1 Swiss-Cheese Operad and Drinfeld Center

Concretely, on objects, we define: 휁 ∶ ob(CoP (푛) × CoB (푚) × Sh (푛, 푚)) + + Σ푛×Σ푚 + = (Σ × Σ ) × ob CoPB(푛, 푚) 푛 푚 Σ푛×Σ푚 ≅ ob CoPB(푛, 푚).

On morphisms, 휁[푢, 푥, 휇] (for 푢 ∈ CoP+(푛), 푥 ∈ CoB+(푚), 휇 ∈ Sh+(푛, 푚)) is the composition of the unique shuffle type morphism that brings all terrestrial points to the left, then the concatenation of 푢 and 푥, then the unique shuffle-type morphism that brings ground point to their places. We thus get a well-defined (up to isotopy) braid, and it is easy to see that this gives a bijection on morphisms. On can thus transport the operadic composition, which will serve as inspiration for Equation (1.4.15) to come.

A variation on PaPB+ ′ We first define a new operad PaPB+, a minor variation on PaPB+.

Definition 1.4.9. Let 휔+ ∶ ΩΩ+ → ob CoPB+ ≅ ob Sh+ be the morphism of Definition 1.2.8. We define PaSh+ to be the pullback of Sh+ along 휔.

Remark 1.4.10. There is a function of sets 푈 ∶ ob PaPB+(0, 푚) → ob PaB+(푚) that forgets the second level of parenthesizing. Definition 1.4.11. Let PaPB′ (푛, 푚) ⊂ (PaP (푛) × PaB (푚)) × PaSh (푛, 푚), + + + Σ푛×Σ푚 + be the full subgroupoid whose objects [푢, 푥, 휇] such that there exists a permutation 휏 ∈ Σ푚 satisfying 푈(휇(∗픬, … , ∗픬)) = 푥 ⋅ 휏 (this does not depend on the choice of a representative for the coinvariants). Example 1.4.12. For example, 1 2 1 3 1 1 2 3 [ × × ] ∈ ob PaPB′(1, 3), but

2 1 3 1 2 3 ′ [∗픬 × × ] ∉ ob PaPB (0, 3). ′ Lemma 1.4.13. The symmetric sequence PaPB+(푛) is a right module over PaB+, given by: 픠 ′ ′ ∘푖 ∶ PaPB+(푛, 푚) × PaB+(푘) → PaPB+(푛, 푚 + 푘 − 1) [푢, 푥, 휇] × 푦 ↦ [푢, 푥 ∘ 푦, 휇 ∘픠 1 ], 푖 푖 Σ푘 where 1 is seen as a morphism in 휔∗Π (푘) between the source of 푦 and the target of 푦. Σ푘 +

30 1.4 Chord diagrams

Proof. The operad PaB+ is a right module over itself, and PaSh+(푛) is a right module over Ω+. One can then directly check that the above formula defines a right module structure over PaB+. Definition 1.4.14. Let P be an operad in some symmetric monoidal category. The shifted operad P[⋅] [Fre09, §10.1] is an operad in right modules over P (i.e. an operad relative over P), given by P[푛](푚) = P(푛 + 푚), and the structure maps are induced by the operad structure of P (cf. ibid. for explicit formulas).

′ To define the operad structure of PaPB+, we first define a morphism of colored ′ collections 휌 ∶ PaPB+ → PaB[⋅], that will be similar to the definition of 휁 1.4.8. It is again defined in a graphical way, see Figure 1.4.1 (the starred numbers correspond to shifted entries). The precise definition involves the inclusion 휄 ∶ PaP+ ↪ PaB+, concatenation, the functor PaSh+ → PaB+ (picking the unique shuffle-type morphism when it exists), 휂 ∶ PaP+ → PaP+[⋅] (where 휂 ∶ PaP+(푛) ≅ PaP+[푛](0)), as well as 푈 on objects – see Equation (1.4.20).

1 2 3 1∗ 2∗

1 2 1 2 3 1 2 3 1 2

× ×

′ Figure 1.4.1: Definition of 휌 ∶ PaPB+(2, 3) → PaB+[2](3)

The operad structure is then defined by (where 휎 ∈ Σ푚 is such that

푈(휇(∗픬, … , ∗픬)) = 푥 ⋅ 휎, ′ and where 푦푖 is seen as an element of PaPB+[0](푙푖)):

′ ′ ′ ′ 훾 ∶ PaPB+(푟, 푠) × PaPB+(푘1, 푙1) × ⋯ × PaPB+(푘푟, 푙푟) → PaPB+(∑ 푘푖, 푠 + ∑ 푙푖)

[푢, 푥, 휇] × [푣1, 푦1, 휎1] × ⋯ × [푣푟, 푦푟, 휎푟] −1 ↦ [푢(푣1, … , 푣푟), 휎 ⋅ 휌[푢,⏟⏟⏟⏟⏟ 푥, 휇](푦1, … , 푦푟), 휇(휎1, … , 휎푟)] (1.4.15) ′ PaPB+[푟](푠)

1 1 ′ and where the identity of the operad is id = [ × ∗픠 × ] ∈ PaPB+(1, 0).

31 1 Swiss-Cheese Operad and Drinfeld Center

Proposition 1.4.16. Given this operadic composition, this identity and this right module ′ structure, PaPB+ is an operad relative over PaB+. Proof. The 휎−1 in the formula ensures that this 훾 is well-defined (it does not depend on the representative in the coinvariants). The fact that 훾 is equivariant is a direct consequence of the fact that the operad structures of PaP+, PaSh+ and PaB+[⋅] are equivariant. ′ Let [푢, 푥, 휇] ∈ PaPB+(푛, 푚). The identity id([푢, 푥, 휇]) = [푢, 푥, 휇] is immediate ′ by definition, and from the condition on the objects of PaPB+, one can also show the identity [푢, 푥, 휇](id,…, id) = [푢, 푥, 휇]. To see that 훾 is a morphism of right PaB+-modules, it is enough to have the identity

휌[푢, 푥, 휇](푦 , … , 푦 ∘ 푧, … , 푦 ) = 휌[푢, 푥, 휇](푦 , … , 푦 ) ∘ 푧 1 푗 푖 푟 1 푟 푙1+⋯+푙푗−1+푖 for 푧 ∈ PaB+(푚); but since PaB+[⋅] is an operad, this identity is satisfied. Finally, associativity of 훾 follows from the condition on objects (to show that ob 훾 is associative), and from the fact that PaB+[⋅] is an operad (to show associativity on morphisms).

′ ∼ Proposition 1.4.17. There exists a categorical equivalence PaPB+ CoPB+.

′ Proof. This equivalence is given in arity (푛, 푚) by the restriction to PaPB+(푛, 푚) of the composite:

(PaP (푛) × PaB (푚)) × PaSh (푛, 푚) + + Σ푛×Σ푚 + → (CoP (푛) × CoB (푚)) × Sh (푛, 푚) + + Σ푛×Σ푚 + ≅ 휁 CoPB+(푛, 푚)

′ By construction (the operad structure of PaPB+ is directly mimicked from the ′ operad structure of CoPB+), this yields a morphism of operads PaPB+ → CoPB+. Since PaP+ → CoP+, PaB+ → CoB+ and PaSh+ → Sh+ are categorical, their product is too, thus the above morphism yields a categorical equivalence.

An operad defined from chord diagrams

We choose a Drinfeld associator 휙 ∶ PaB+ → CD̂+; let 휙+̃ ∶ PaB+ → PaCD̂ + be its ′ unique lifting. Similarly to the definition of PaPB+, we will define a relative operad 휙 PaPCD̂+ over PaCD̂ +, that will combine parenthesized shuffles, parenthesized permutations and parenthesized chords diagrams.

32 1.4 Chord diagrams

Definition 1.4.18. Let

휙 PaPCD̂ (푛, 푚) ⊂ (PaP (푛) × PaCD̂ (푚)) × PaSh (푛, 푚) + + + Σ푛×Σ푚 + be the full subgroupoid whose objects are classes [푢, 훼, 휇] such that there exists 휎 ∈ Σ푚 satisfying 푈(휇(∗픬, … , ∗픬)) = 훼 ⋅ 휎.

휙 ′ Remark 1.4.19. The objects of PaPCD̂+ are the same as the objects of PaPB+. 휙 We also define a morphism 휌휙 ∶ PaPCD̂+ → PaCD̂ +[⋅]. Its definition is similar to that of Figure 1.4.1, but one cannot directly use graphical calculus anymore. In the picture, concatenation in PaB+ and PaB+[⋅] corresponded to 푚픠. Pre- and ′ post-composition by shuffles in PaPB+ came from a morphism of operads 휎 ∶ ′ 휇 PaSh+ → PaB+[⋅]. We have ob 휎 = id, and 휎(휇 → 휇 ), denoted 휎휇′ to simplify, is the unique morphism PaB[⋅] of shuffle-type between the corresponding objects. We also recall the canonical morphism 휂 ∶ P → P[⋅], given in arity 푚 by P(푚) ≅ P[푚](0). Finally, we could define 휌 by the following formula (where 푥 ∈ PaB+(푚) is identified with 푥 ∈ PaB+[0](푚)):

푚 (휄(휂(tgt(푢))),tgt(푥)) src(휇) 휌[푢, 푥, 휇] = 휎 픠 ∘ 푚 (휄(휂(푢)), 푥) ∘ 휎 . (1.4.20) tgt(휇) 픠 푚픠(휄(휂(src(푢))),src(푥))

By analogy, we define:

휙 휌휙 ∶ PaPCD̂+(푛, 푚) → PaCD̂ +[푛](푚)

To simplify, we let 휙+̃ (푚픠) = ̃푚픠, 휙+̃ ∘ 휎 = ̃휎, 휙+̃ 휄휂 =, ̃휄 and we again identify 훼 ∈ PaCD̂ +(푚) with 훼 ∈ PaCD̂ +[0](푚). Then 휌휙 is given by:

̃푚 ( ̃휄(tgt(푢)),tgt(훼)) src(휇) [푢, 훼, 휇] ↦ ̃휎 픠 ∘ ̃푚 ( ̃휄(푢), 훼) ∘ 휙̃ ( ̃휎 ). tgt(휇) 픠 + ̃푚픠( ̃휄(src(푢)),src(훼))

Graphically, 휌휙 looks like Figure 1.4.2, where the gray boxes represent applications of the associator. We then define an operadic composition in a similar manner to Equation (1.4.15), replacing 휌 by 휌휙. We also define a right PaCD̂ +- ′ module similar to that of PaPB+.

휙 Theorem 1.4.21. The data PaPCD̂+, equipped with these structures, is a relative operad ′ 휙 over PaCD̂ +, and the morphism PaPB+ → PaPCD̂+ induced by 휙+̃ is a rational categorical equivalence of operads. There is thus a zigzag:

휙 ∼ ∼ ∼ ∼ ′ ∼ℚ ̂ 휋SC+ (휋SC+) ′ PaPB+ CoPB+ PaPB+ PaPCD+. ΩΩ+

33 1 Swiss-Cheese Operad and Drinfeld Center

2 1 3 1∗ 2∗

휙

Figure 1.4.2: Graphical representation of 휌휙

휙 Proof. The proof that PaPCD̂+ is a relative operad is identical to the proof of Proposition 1.4.16, and the fact that the morphism induced by 휙+̃ is a morphism of operads follows by a direct inspection of the definitions. The morphism 휙+̃ ∶ PaB+ → PaCD̂ + is not a categorical equivalence, but it factors as:

휙̃+ PaB+ PaCD̂ +

∼ℚ ≅

PaB̂+ where the morphism PaB̂+ → PaCD̂ + is an isomorphism of operads in groupoids. ′ It follows that 휙+̃ is a rational equivalence of operads in groupoids, thus PaPB+ → 휙 PaPCD̂+ is also a rational categorical equivalence. By combining this fact with Propositions 1.3.9 and 1.4.17, we finally get the zigzag of the theorem.

1.4.3 Non-formality

A theorem of Livernet [Liv15, Theorem 3.1] states that the Swiss-Cheese operad is not formal: its homology 퐻∗(SC) is not equivalent to its operad of chains 퐶∗(SC). We give an interpretation of this fact here. We consider a stronger version of formality, which involves the models of rational homotopy theory of Sullivan (see [Fre17, §II] for the applications to

34 1.4 Chord diagrams operads). Let 핃 op ⟨−⟩ ∶ CDGA+ → 푠Set be the derived Sullivan realization functor,3 that uses commutative dg-algebras as rational models for spaces. We rely on cohomological models to study rational homotopy theory, so we consider dual structures of our objects and we use cooperads rather than operads. For example, the cooperad Com∗ governing cocommutative coalgebras is dual to the operad Com governing commutative algebras. To encode the rational homotopy-theoretic information, we add commutative structures to our objects, and we consider Hopf cooperads, i.e. cooperads in the category of commutative algebras. The Sullivan realization of a Hopf cooperad is a simplicial operad.

Splitting of 퐻∗(SC) as a Voronov product A theorem of Cohen [Coh76] describes the homology of the little disks operads e푛 ≔ 퐻∗(D푛). In low dimensions, e1 ≅ Ass is the operad governing associative algebras, while e2 ≅ Ger is the operad governing Gerstenhaber algebras. These are Hopf operads (i.e. operads in the category of cocommutative coalgebras): the coproduct of the product of either Ass or Ger is Δ(휇) = 휇 ⊗ 휇, while the coproduct of the bracket of Ger is Δ(휆) = 휇 ⊗ 휆 + 휆 ⊗ 휇. Their duals Ass∗ and Ger∗ are Hopf cooperads. The homology of the Swiss-Cheese operad sc ≔ 퐻∗(SC) governs the action of a Gerstenhaber algebra on an associative algebra. A theorem of Voronov [Vor99, Theorem 3.3] (see also [HL12, Theorem 6.1.1] for this particular variant) states that an algebra over 퐻∗(SC) is a triple (퐵, 퐴, 푓 ) where 퐵 is a Gerstenhaber algebra, 퐴 is an associative algebra, and 푓 ∶ 퐵 → 퐴 is a central morphism of associative algebras, which thus makes 퐴 into an associative algebra over the commutative algebra 퐵. Let us note that Corollary 1.3.4 is a categorical analogue of Voronov’s theorem, the Drinfeld center of a monoidal category replacing the center of an associative algebra. This theorem can be interpreted in the following way. Definition 1.4.22. Given two operads P and Q and a morphism Com → P, one can define the Voronov product P ⊗ Q [Vor99]. It is a relative operad over P, defined by (P ⊗ Q)(푛, 푚) = P(푚) ⊗ Q(푛). Insertion of a closed-output operation:

픠 ′ ′ ∘푖 ∶ (P(푚) ⊗ Q(푛)) ⊗ P(푚 ) → P(푚 + 푚 − 1) ⊗ Q(푛) 3The underived realization functor maps a commutative, unitary differential graded algebra 퐴 to the simplicial set ⟨퐴⟩ = hom (퐴, Ω∗ (Δ•)). The derived version takes a cofibrant CDGA+ 푃퐿 replacement first.

35 1 Swiss-Cheese Operad and Drinfeld Center uses the operad structure of P, while insertion of an open-output operation:

픬 ′ ′ ′ ′ ∘푗 ∶ (P(푚) ⊗ Q(푛)) ⊗ (P(푚 ) ⊗ Q(푛 )) → P(푚 + 푚 ) ⊗ Q(푛 + 푛 − 1) uses the operad structure of Q and the commutative product Com → P. Algebras over P ⊗ Q are triplets (퐵, 퐴, 휈) where 퐵 is a P-algebra, 퐴 is a Q-algebra, and 휈 ∶ 퐵⊗퐴 → 퐴 is an action that makes 퐴 into a Q-algebra over the commutative algebra 퐵 (cf. ibid. for the definition). Remark 1.4.23. An Eckmann–Hilton-type argument shows that the algebra structure of P defined by the morphism Com → P has to be commutative for the composition product to even be associative. Voronov’s version of the Swiss-Cheese SCvor operad then satisfies:

vor vor sc ≔ 퐻∗(SC ) ≅ Ger ⊗ Ass. This isomorphism is moreover an isomorphism of Hopf operads. In the case of sc = 퐻∗(SC), one has to use the unital structures of Ass and Ger. We have (and this is still an isomorphism of Hopf operads):

sc+ ≔ 퐻∗(SC+) ≅ Ger+ ⊗ Ass+. If we remove the components with zero closed inputs and zero open inputs, we then get sc = 퐻∗(SC), a relative operad over Ger whose algebras are described above Definition 1.4.22. But one should not forget that:

sc(0, 푚) = Ger+(푚) ⊗ Ass+(0) = Ger(푚) ≠ scvor(0, 푚) = Ger(푚) ⊗ Ass(0) = 0.

Indeed, we still keep the components that have a nonzero number of total inputs. We have in particular that sc(0, 1) ≅ Ger(1) = ℚ is spanned by the morphism between the Gerstenhaber algebra to the associative algebra. We use the notation: sc = Ger+ ⊗0 Ass+ to express the fact that sc is obtained as the Voronov product Ger+ ⊗ Ass+ from which we remove the components with zero closed inputs and zero open inputs.

Comparison By theorems of Kontsevich [Kon99](한 = ℝ, 푛 ≥ 2) and Tamarkin [Tam03] + + (한 = ℚ, 푛 = 2), the little disks operads are formal, i.e. 퐶∗(D푛 ) ≃ 퐻∗(D푛 ). Fresse and Willwacher [FW15] give another proof of this result (한 = ℚ, 푛 ≥ 3),

36 1.4 Chord diagrams and show that it can be enhanced in the rational homotopy context: there is a ∗ 핃 rational equivalence of simplicial operads D푛 ≃ℚ ⟨퐻 (D푛)⟩ , which implies the ∗ 핃 rational formality of D푛. In low dimensions, we thus have D1 ≃ℚ ⟨Ass ⟩ (easy computation). Tamarkin proves that, from the existence of rational Drinfeld ∗ 핃 associators, it follows that D2 ≃ℚ ⟨Ger ⟩ . These rational equivalences are also compatible with the unital structures. ∗ Given two (Hopf) cooperads P푐, Q푐, and a morphism P푐 → Com , one can define the Voronov product P푐 ⊗ Q푐, similarly to Definition 1.4.22. This is a relative (Hopf) cooperad under P푐. It is defined by formulas that are formally dual to the ones defining P ⊗ Q. If these cooperads admit counital extensions, we can + + similarly define P푐 ⊗0 Q푐 . + + The inclusion D1 ↪ D2 induces in cohomology a morphism of Hopf cooper- ∗ ∗ ∗ ∗ ads Ger+ → Ass+. This morphism factors through a morphism Ger+ → Com+ (the coproduct of a Gerstenhaber coalgebra is cocommutative). We then get an isomorphism:

∗ ∗ ∗ ∗ ∗ sc = 퐻 (SC) ≅ (Ger+ ⊗0 Ass+) ≅ Ger+ ⊗0 Ass+.

∗ ∗ The morphism Ger+ → Com+ induces a morphism of simplicial operads Com+ ≅ ∗ 핃 ∗ 핃 4 ⟨Com+⟩ → ⟨Ger+⟩ . Since the realization functor is monoidal, we get:

∗ 핃 ∗ 핃 ∗ 핃 ⟨퐻 (SC)⟩ ≃ ⟨Ger+⟩ ×0 ⟨Ass+⟩ .

It is known that PaP ≃ 휋D1 and CD̂ ≃ℚ 휋D2 [Fre17, §5, §10]. There is an obvious morphism Com → CD̂ sending the generator to the empty chord diagram (see [FW15]), which we can use to build the Voronov product of the operads in groupoids CD̂ and PaP. Since the fundamental groupoid functor is monoidal too, we finally have:

∗ 핃 ∗ 핃 ∗ 핃 휋⟨sc ⟩ ≃ 휋⟨Ger+⟩ × 휋⟨Ass+⟩ ≃ℚ CD̂+ ×0 PaP+.

The operad SC is not formal [Liv15], therefore 휋SC is not equivalent to

∗ 핃 휋⟨sc ⟩ ≃ℚ PaP+ ×0 CD̂+.

휙 Thus, our construction PaPCD̂+ rectifies the model arising from the homology of SC to retrieve, from PaP and CD̂, an operad in groupoids which is actually rationally equivalent to 휋SC.

4The monoidal structure on simplicial sets being the Cartesian product, we denote the Voronov product of two simplicial operads with × instead of ⊗, and we use ×0 to say that we remove the components with zero inputs.

2 The Lambrechts–Stanley Model of Configuration Spaces

Let 푀 be a closed smooth 푛-manifold and consider the ordered configuration space of 푘 points in 푀:

푛 Conf푘(푀) ≔ {(푥1, … , 푥푘) ∈ 푀 ∣ 푥푖 ≠ 푥푗 ∀푖 ≠ 푗}. The homotopy type of these spaces is notoriously hard to compute. For example there exist two homotopy equivalent (non simply connected) closed 3-manifolds such that their configuration spaces are not homotopy equivalentLS05b [ ]. A theorem of Lambrechts–Stanley [LS08b] shows that a simply connected closed manifold 푀 always admits a Poincaré duality model 퐴 (in the sense of rational homotopy theory). They build a CDGA G퐴(푘) [LS08a] out of 퐴 and ∗ show that this CDGA is quasi-isomorphic as Σ푘-dg-modules to APL(Conf푘(푀)). When 푀 is a smooth complex projective variety, Kriz [Kri94] had previously shown that G퐻∗(푀)(푘) is actually a rational CDGA model for Conf푘(푀). The 2 CDGA G퐻∗(푀)(푘) is the E page of a spectral sequence of Cohen–Taylor [CT78] that converges to the cohomology of Conf푘(푀). Totaro [Tot96] has shown that for a smooth complex compact projective variety, the spectral sequence only has one nonzero differential. When 푘 = 2, G퐴(2) is known to be a model of Conf2(푀) either when 푀 is 2-connected [LS04] or when dim 푀 is even [Cor15]. Lambrechts and Stanley conjecture that G퐴(푘) is a rational model of Conf푘(푀) for any simply connected manifold [LS08b]. We prove this conjecture over ℝ for manifolds of dimension at least 4. This proves as a corollary that the real homotopy type of Conf푘(푀) only depends on the real homotopy type of 푀 and its Poincaré duality. We use that the space Conf푘(푀) is homotopy equivalent to its Fulton–Mac- Pherson compactification FM푀(푘) [FM94; AS94; Sin04]. When 푀 is framed, these compactifications assemble into a right module FM푀 over the Fulton–MacPherson operad FM푛, an operad weakly equivalent to the little 푛-disks operad [May72; BV73]. We show that the Lambrechts–Stanley model is compatible with this action of the little disks operad.

Based on “The Lambrechts–Stanley Model of Configuration Spaces”, preprint (submitted), arXiv:1608.08054.

39 2 The Lambrechts–Stanley Model of Configuration Spaces

The little 푛-disks operad D푛 is formal [Kon99; Tam03; LV14; FW15], i.e. its operad of singular chains 퐶∗(D푛) is quasi-isomorphic to its homology e푛 ≔ 퐻∗(D푛). This formality result can be strengthened [LV14; FW15] so that it holds in the category of Hopf cooperads, taking into account the CDGA structures of ∗ ∨ ΩPA(FM푛(푘)) and the dual e푛 (푘). By general arguments, the formality theorem implies that there exists a homotopy class of right e푛-modules e푀 such that the pair (e푀, e푛) is quasi-isomorphic to the chain complex of the pair (FM푀, FM푛). We show that G퐴 = {G퐴(푘)}푘≤0 is a ∨ Hopf right e푛 -comodule whose dual is a representative of e푀. Our results are summarized by:

Theorem C (Theorem 2.4.14). Let 푀 be a smooth simply connected closed 푛-manifold, where 푛 ≥ 4. Then for any Poincaré duality model 퐴 of 푀 and for all 푘 ≥ 0, the CDGA G퐴(푘) defined by Lambrechts and Stanley is Σ푘-equivariantly weakly equivalent ∗ to ΩPA(FM푀(푘)). ∨ If 휒(푀) = 0, the collection G퐴 = {G퐴(푘)}푘≥0 moreover forms a Hopf right e푛 - ∨ ∗ ∗ comodule. If 푀 is framed, then (G퐴, e푛 ) is weakly equivalent to (ΩPA(FM푀), ΩPA(FM푛)) as a Hopf right comodule.

As Corollary 2.4.36, we obtain that if two smooth closed simply connected manifolds of dimension at least 4 have the same real homotopy type, then so do their configuration spaces. In dimension 3, the only simply connected manifold is the 3-sphere, which is framed. The Lambrechts–Stanley conjecture is satisfied over ℚ in this case ∨ (Proposition 2.4.37), and the collection G퐻∗(푆3) is still a Hopf right e3 -comodule. We conjecture that the model is also compatible with the action of the Fulton– MacPherson operad. Factorization homology, an invariant of framed 푛-manifolds defined from an D푛- algebra, may be computed via a derived tensor product over the D푛 operad [AF15]. The Taylor tower in the Goowillie–Weiss calculus of embeddings may similarly be computed via a derived Hom [GW99; BW13]. It follows from a result of [Tur13, Section 5.1] that FM푀 may be used for this purpose. Therefore our theorem shows that G퐴 may be used for computing factorization homology or the Taylor tower. The proof of this theorem is inspired by Kontsevich’s proof of the formality theorem, and is radically different from the ideas of the paper[LS08a]. It in- Graphs volves an intermediary Hopf right comodule of labeled graphs 푅. This comodule is similar to a comodule developed by Campos–Willwacher [CW16], which is isomorphic to our construction applied to 푅 = 푆(퐻̃ ∗(푀)). Despite this similarity, their whole approach is different, and they manage to prove that Graphs ∗ FM 푆(퐻̃ ∗(푀)) is quasi-isomorphic to ΩPA( 푀) even for non simply connected manifolds. They also prove that in the simply connected case, the real homotopy

40 2.1 Background and recollections types of configuration spaces only depend on the real homotopy type ofthe manifold.

Outline In Section 2.1, we recall some background on the little disks operad, the Fulton–MacPherson compactification, the proof of Kontsevich formality, Poincaré duality CDGAs, and the Lambrechts–Stanley CDGAs. In Section 2.2, we ∨ build out of these CDGAs a Hopf right e푛 -comodule G퐴. In Section 2.3, inspired by the proof of the formality theorem, we construct the labeled graph complex Graphs ∗ FM 푅 which will be used to connect this comodule to ΩPA( 푀). In Section 2.4, we prove that the zigzag of Hopf right comodule morphisms between G퐴 and ∗ ΩPA(FM푀) is a weak equivalence. In Section 2.5, we use our model to compute factorization homology of framed manifolds and we compare the result to a complex obtained by Knudsen. Finally, in Section 2.6 we work out a variant of our theorem for the only simply connected surface using the formality of the framed little 2-disks operad, and we present a conjecture about higher dimensional oriented manifolds.

2.1 Background and recollections

We assume basic proficiency with Hopf (co)operads and (co)modules over (co)operads. References include the book [LV12] for (co)operads and the books [Fre09; Fre17] for (co)modules over (co)operads and for Hopf (co)operads.

2.1.1 Conventions All our dg-modules will have a cohomological grading: 푉 = ⨁ 푉푛, 푛∈ℤ and the differentials raise degrees by one: deg(푑푥) = deg(푥) + 1. If 푉, 푊 are dg-modules and 푣 ⊗ 푤 ∈ 푉 ⊗ 푊, we let (푣 ⊗ 푤)21 = (−1)(deg 푣)(deg 푤)푤 ⊗ 푣 and we extend this linearly to 푉 ⊗푊. We will let 푉[푘] be the desuspension, defined by (푉[푘])푛 = 푉푛+푘. We will call CDGAs the (graded) commutative unital algebras in the category of dg-modules. Remark 2.1.1. There is a Quillen adjunction between the category of ℤ-graded CDGAs and the category of ℕ-graded CDGAs. If 퐴 is a cofibrant ℕ-graded CDGA which is connected, i.e. satisfying 퐻0(퐴) = 한, then it is also cofibrant when seen as a ℤ-graded CDGA. Therefore if two connected ℕ-graded CDGAs are weakly equivalent (quasi-isomorphic) in the category of ℤ-graded CDGAs, then they are also weakly equivalent in the category of ℕ-graded CDGAs.

41 2 The Lambrechts–Stanley Model of Configuration Spaces

We index our (co)operads by finite sets instead of integers to ease the writing of some formulas. If 푊 ⊂ 푈 is a subset, we define the quotient 푈/푊 = (푈−푊)⊔{∗}, where ∗ represents the class of 푊; note that 푈/∅ = 푈 ⊔ {∗}. An operad P is given by a functor from the category of finite sets and bijections (also known asa symmetric collection) to the category of dg-modules, a unit 한 → P({∗}), as well as composition maps for every pair of sets:

∘푊 ∶ P(푈/푊) ⊗ P(푊) → P(푈), satisfying the usual associativity, unity and equivariance conditions. Dually, a cooperad C is given by a symmetric collection, a counit C({∗}) → 한, and cocomposition maps for every pair (푈 ⊃ 푊)

∨ ∘푊 ∶ C(푈) → C(푈/푊) ⊗ C(푊). Using the terminology of Fresse [Fre17], we call Hopf cooperads the cooperads in the category of CDGAs. Remark 2.1.2. The constructions of Fresse [Fre17] are done in the category of ℕ-graded CDGAs. They extend to the setting of ℤ-graded CDGAs. The previous remark extends to Hopf cooperads. Let 푘 = {1, … , 푘}. We recover the usual notion of a cooperad indexed by the integers by considering the collection {C(푘)}푘≥0, and the cocomposition maps ∨ ∨ ∘푖 ∶ C(푘 + 푙 − 1) → C(푘) ⊗ C(푙) correspond to ∘{푖,…,푖+푙−1}. We do not generally assume that our (co)operads are trivial in arity zero, but they will satisfy P(∅) = 한 (resp. C(∅) = 한). With this assumption we get (co)operad structures which are actually equivalent to the structure of Λ- (co)operads considered by Fresse [Fre17]. We will consider right (co)modules over (co)operads. Given an operad P, a right P-module is a symmetric collection M equipped with composition maps:

∘푊 ∶ M(푈/푊) ⊗ P(푊) → M(푈) satisfying the usual associativity, unity and equivariance conditions. A right comodule over a cooperad is defined dually. If C is a Hopf cooperad, then a right Hopf C-comodule is a C-comodule N such that all the N(푈) are CDGAs and all ∨ the maps ∘푊 are morphisms of CDGAs. Definition 2.1.3. Let C (resp. C′) be a Hopf cooperad and N (resp. N′) be a Hopf right comodule over C (resp. C′). A morphism of Hopf right comodules 푓 = ′ ′ (푓N, 푓C) ∶ (N, C) → (N , C ) is a pair consisting of a morphism of Hopf cooperads ′ ′ ′ 푓C ∶ C → C , and a map of Hopf right C -comodules 푓N ∶ N → N , where N has the C-comodule structure induced by 푓C. It is said to be a quasi-isomorphism if both 푓C and 푓N are quasi-isomorphisms of dg-modules in each arity.

42 2.1 Background and recollections

If the context is clear, we will allow ourselves to remove the cooperads from the notation in the morphism.

Definition 2.1.4. A Hopf right C-module N is said to be weakly equivalent to a Hopf right C′-module N′ if the pair (N, C) can be connected to the pair (N′, C′) through a zigzag of quasi-isomorphisms.

2.1.2 Little disks and related objects

The little disks operad D푛 is a topological operad initially introduced by May and Boardman–Vogt [May72; BV73] to study iterated loop spaces. Its homology e푛 ≔ 퐻∗(D푛) is described by a theorem of Cohen [Coh76]: it is either the operad encoding associative algebras for 푛 = 1, or the one encoding (푛 − 1)-Poisson algebras for 푛 ≥ 2. For technical reasons, we instead consider the Fulton–MacPherson operad FM푛, introduced by Fulton–MacPherson [FM94] in the complex context and adapted by Axelrod–Singer [AS94] to the real context. Each space FM푛(푘) is a compactifi- 푛 cation of the configuration space Conf푘(ℝ ), where roughly speaking points can become “infinitesimally close”. Using insertion of infinitesimal configurations, they assemble to form a topological operad, weakly equivalent to D푛. We refer to [Sin04] for a detailed treatment. 푛−1 The first two spaces FM푛(∅) = FM푛(1) = ∗ are singletons, and FM푛(2) = 푆 is a sphere. We will let

푛−1 푛−1 푛−1 vol푛−1 ∈ ΩPA (푆 ) = ΩPA (FM푛(2)) (2.1.5) be the top volume form of FM푛(2). For 푘 ≥ 2 and 푖 ≠ 푗 ∈ 푘, we also define the projection maps 푝푖푗 ∶ FM푛(푘) → FM푛(2) (2.1.6) that forget all but two points in the configuration. To follow Kontsevich’s proof of the formality theorem [Kon99], we use the theory of semi-algebraic sets, as developed in [KS00; Har+11]. A semi-algebraic set is a subset of ℝ푁 defined by finite unions of finite intersections of zero setsof ∗ polynomials and polynomial inequalities. There is a functor ΩPA of “piecewise algebraic differential forms”, analogous to the de Rham complex, taking asemi- algebraic set to a real CDGA. If 푀 is a compact semi-algebraic smooth manifold, ∗ ∗ then ΩPA(푀) ≃ ΩdR(푀). The spaces FM푛(푘) are all semi-algebraic stratified manifolds. The dimension of FM푛(푘) is 푛푘 − 푛 − 1 for 푘 ≥ 2, and is 0 otherwise. ∗ ∗ The functor ΩPA is monoidal, but contravariant; it follows that ΩPA(FM푛) is an “almost” Hopf cooperad. It satisfies a slightly modified version of the cooperad

43 2 The Lambrechts–Stanley Model of Configuration Spaces

∗ axioms coming from the fact that ΩPA is not strongly monoidal, as explained in [LV14, Definition 3.1] (they call it a “CDGA model”): the insertion maps ∘푊 become zigzags

∗ ∼ ∗ ∘푊 ∗ ∗ ∗ ΩPA(FM푛(푈)) ΩPA(FM푛(푈/푊)×FM푛(푊)) ΩPA(FM푛(푈/푊))⊗ΩPA(FM푛(푊)), where the second map is the Künneth quasi-isomorphism. If C is a Hopf cooperad, ∗ an “almost” morphism 푓 ∶ C → ΩPA(FM푛) is a collection of CDGA morphisms ∗ 푓푈 ∶ C(푈) → ΩPA(FM푛(푈)) for all 푈, such that the following diagrams commute:

푓푈 ∗ C(푈) ΩPA(퐹푀푛(푈)) ∗ ∘푊 ∘∨ ∗ 푊 ΩPA(FM푛(푈/푊) × FM푛(푊)) ∼ 푓 ⊗푓 푈/푊 푊 ∗ ∗ C(푈/푊) ⊗ C(푊) ΩPA(FM푛(푈/푊)) ⊗ ΩPA(FM푛(푊))

We will generally omit the adjective “almost”, keeping in mind that some commutative diagrams are a bit more complicated than at first glance. If 푀 is a manifold, the configuration space Conf푘(푀) can similarly be compactified to give a space FM푀(푘). When 푀 is framed, these spaces assemble to form a topological right module over FM푛, by inserting infinitesimal configurations. By the Nash–Tognoli Theorem [Nas52; Tog73], any closed smooth manifold is homeomorphic to a semi-algebraic subset of ℝ푁 for some 푁, and in this way FM푀(푘) becomes a stratified semi-algebraic manifold of dimension 푛푘. ∗ By the same reasoning as above, if 푀 is framed, then ΩPA(FM푀) becomes ∗ an “almost” Hopf right comodule over ΩPA(FM푛). As before, if N is a Hopf right C-comodule, where C is a cooperad equipped with an “almost” morphism ∗ ∗ 푓 ∶ C → ΩPA(FM푛), then an “almost” morphism 푔 ∶ N → ΩPA(FM푀) is a collection ∗ of CDGA morphisms 푔푈 ∶ N(푈) → ΩPA(FM푀(푈)) that make the following diagrams commute:

푓푈 ∗ N(푈) ΩPA(FM푀(푈)) ∗ ∘푊 ∘∨ ∗ 푊 ΩPA(FM푀(푈/푊) × FM푛(푊)) ∼ 푔 ⊗푓 푈/푊 푊 ∗ ∗ N(푈/푊) ⊗ C(푊) ΩPA(FM푀(푈/푊)) ⊗ ΩPA(FM푛(푊))

∗ Remark 2.1.7. Following [Fre17, Section II.10.1], there is a construction Ω♯ that turns a simplicial operad P into a Hopf cooperad and such that a morphism of

44 2.1 Background and recollections

∗ Hopf cooperads C → Ω♯(P) is the same thing as an “almost” morphism C → Ω∗(P), where Ω∗ is the functor of Sullivan forms. Moreover there is a canonical ∗ ∗ collection of maps (Ω♯(P))(푈) → Ω (P(푈)), which are weak equivalences if P is a cofibrant operad. Such a functor is built by considering the right adjoint of the functor on operads induced by the Sullivan realization functor, which is monoidal. Although we will not use it in the rest of the paper, a similar construction can ∗ be carried out for ΩPA. The proof follows step by step the constructions carried out in [Fre17, Section II.10.1]. The key points are the two facts that for a given ∗ 푛 푛 ≥ 0, the complex ΩPA(Δ ) is acyclic (a.k.a. the Poincaré lemma), and that for a 푘 • given 푘 ≥ 0, the simplicial vector space ΩPA(Δ ) is acyclic too. These are respectively [Har+11, Lemma 6.3 and Lemma 6.10] (where the last property is called ∗ • the “extendability” of ΩPA(Δ ) and is used to prove the Mayer–Vietoris theorem PA ∗ • for PA forms). Then one can define a functor 퐺• (퐴) ≔ HomCDGA(퐴, ΩPA(Δ )) on CDGAs, and use the right adjoint of its extension to the category of Hopf ∗ cooperads to define and prove the properties of ΩPA,♯. We can also extend this construction to modules over operads, by similar arguments.

2.1.3 Formality of the little disks operad

Kontsevich’s formality theorem [Kon99; LV14] can be summarized by the fact ∗ ∨ that ΩPA(FM푛) is weakly equivalent to e푛 as a Hopf cooperads. We outline here the proof as we will mimic its pattern for our theorem. We will use the formalism of (co)operadic twisting [Wil14; DW15], and we refer to [LV14] for proofs of most of the claims of this section.

The cohomology of D푛 The classical description due to Arnold and Cohen of ∨ ∗ the cohomology e푛 (푈) = 퐻 (D푛(푈)) is:

∨ e푛 (푈) = 푆(휔푢푣)푢≠푣∈푈/퐼 (2.1.8)

where 푆(−) is the free graded commutative algebra, the generators 휔푢푣 have cohomological degree 푛 − 1, and the ideal 퐼 is generated by the relations:

푛 2 휔푣푢 = (−1) 휔푢푣; 휔푢푣 = 0; 휔푢푣휔푣푤 + 휔푣푤휔푤푢 + 휔푤푢휔푢푣 = 0. (2.1.9)

45 2 The Lambrechts–Stanley Model of Configuration Spaces

The cooperadic structure maps are given on generators by:

∨ ∨ ∨ ∨ ∘푊 ∶ e푛 (푈) → e푛 (푈/푊) ⊗ e푛 (푊) ⎧ 1 ⊗ 휔푢푣, if 푢, 푣 ∈ 푊; { {휔∗푣 ⊗ 1, if 푢 ∈ 푊 and 푣 ∉ 푊; 휔푢푣 ↦ ⎨ (2.1.10) {휔푢∗ ⊗ 1, if 푢 ∉ 푊 and 푣 ∈ 푊; { ⎩휔푢푣 ⊗ 1, if 푢, 푣 ∉ 푊.

Graphs with only external vertices The intermediary cooperad of graphs is built in several steps. In the first step, define a cooperad of graphs with only external vertices, with generators 푒푢푣 of degree 푛 − 1:

2 푛 Gra푛(푈) = (푆(푒푢푣)푢,푣∈퐴/(푒푢푣 = 푒푢푢 = 0, 푒푣푢 = (−1) 푒푢푣), 푑 = 0). (2.1.11)

The CDGA Gra (푈) is spanned by words of the type 푒 … 푒 . Such a word 푛 푢1푣1 푢푟푣푟 can be viewed as a graph with 푈 as the set of vertices, and an edge between 푢푖 and 푣 for each factor 푒 . For example, the word 푒 is a graph with a single 푖 푢푖푣푖 푢푣 edge between the vertices 푢 and 푣 (see Figure 2.1.1 for another example). Graphs with double edges or edges between a vertex and itself are set to zero. Given 푈 such a graph, its set of edges 퐸Γ ⊂ ( 2 ) is well-defined. The vertices of these graphs are called “external”, in contrast with the internal vertices that are going to appear in the next part.

3 4 6

푒12푒13푒56 = ∈ Gra푛(6) 1 2 5

Figure 2.1.1: Example of the correspondence between graphs and words

The Hopf product map Gra푛(푈) ⊗ Gra푛(푈) → Gra푛(푈), from this point of view, consists of gluing two graphs along their vertices. The cooperadic structure ∨ map ∘푊 ∶ Gra푛(푈) → Gra푛(푈/푊) ⊗ Gra푛(푊) maps a graph Γ to ±Γ푈/푊 ⊗ Γ푊 such that Γ푊 is the full subgraph of Γ with vertices 푊 and Γ푈/푊 collapses this full subgraph to a single vertex. ∨ On the generators, the formula for ∘푊 is in fact identical to Equation (2.1.10), ∨ replacing 휔?? by 푒??. This implies that the cooperad Gra푛 maps to e푛 by sending 푒푢푣 to 휔푢푣.

46 2.1 Background and recollections

∗ The cooperad Gra푛 also maps to ΩPA(FM푛) using a map given on generators by: ′ ∗ 휔 ∶ Gra푛(푈) → ΩPA(FM푛(푈)) ∗ Γ ↦ ⋀ 푝푢푣(vol푛−1), (2.1.12) (푢,푣)∈퐸Γ 푛−1 where vol푛−1 is the volume form of FM푛(2) ≅ 푆 (2.1.5).

Twisting The second step of the construction is cooperadic twisting, for which our general reference is the appendix of [Wil16] (see also [DW15]). Let hoLie푘 be the operad controlling homotopy Lie algebras shifted by 푘 + 1 (with Lie푘 being the operad controlling shifted Lie algebras). Let C be a cooperad, finite- dimensional in every arity and equipped with a map to the dual of hoLie푘. This map can equivalently be seen as a Maurer–Cartan element 휇 in the deformation ∨ complex Def(hoLie푘 → C ), a convolution dg-Lie algebra. Then define: C(푈) ≔ ⨁(C(푈 ⊔ 푖) ⊗ ℝ[푘]⊗푖) . Tw Σ 푖≥0 푖 The symmetric module Tw C can be given a cooperad structure induced by C. Its differential is the sum of the internal differential of C and a differential coming from the action of 휇, acting on both sides of C. Roughly speaking, Tw C encodes coalgebras over C with a differential twisted by a “Maurer–Cartan element”.1 There is also an obvious inclusion C → Tw C that commutes with differentials and cooperad maps. We refer to [Wil16] for the details – one needs to formally dualize the appendix to twist cooperads instead of operads. If C satisfies C(∅) = 한 and is a Hopf cooperad, then Tw C inherits a Hopf cooperad structure. To multiply an element of C(푈 ⊔ 퐼) ⊂ Tw C(푈) with an element of C(푈 ⊔ 퐽) ⊂ Tw C(푈), first use the maps

∘∨ C(푉) ∅ C(푉/∅) ⊗ C(∅) ≅ C(푉 ⊔ {∗}) several times to map both elements to C(푈 ⊔ 퐼 ⊔ 퐽), and then use the CDGA structure of C(푈 ⊔ 퐼 ⊔ 퐽) to multiply them. ∨ We now turn our attention to graphs. The Hopf cooperad Gra푛 maps into Lie푛 ∨ ∨ as follows. The cooperad Lie푛 is cogenerated by Lie푛 (2), and on cogenerators ∨ the cooperad map is given by sending 푒12 ∈ Gra푛(2) to the cobracket in Lie푛 (2) ∨ ∨ and all the other graphs to zero. This map to Lie푛 yields a map to hoLie푛 by ∨ ∼ ∨ composition with the canonical map Lie푛 hoLie푛 . Roughly speaking, the Maurer–Cartan element 휇 is given by Figure 2.1.2.

1 ∨ ∨ A coalgebra over C is automatically a hoLie푘 -coalgebra due to the fixed morphism C → hoLie푘 , therefore the notion of Maurer–Cartan element in a C-coalgebra is well-defined.

47 2 The Lambrechts–Stanley Model of Configuration Spaces

1 2

∨ ∨ Figure 2.1.2: The Maurer–Cartan element 휇 = 푒12 ∈ Gra푛 (2)

The cooperad Gra푛 satisfies Gra푛(∅) = ℝ. The induced maps Gra푛(푈) → ′ ′ Gra푛(푈 ) (for 푈 ⊂ 푈 ) add new vertices with no incident edges. Thus the general framework makes Tw Gra푛 into a Hopf cooperad, which we now explicitly describe. The dg-module Tw Gra푛(푈) is spanned by graphs with two types of vertices: external vertices, which correspond to elements of 푈, and indistinguishable internal vertices (usually drawn in black). The degree of an edge is 푛 − 1, and the degree of an internal vertex is −푛. The differential sends a graph Γ to the sum:

푑Γ = ∑ ±Γ/푒,

푒∈퐸Γ contractible where Γ/푒 is Γ with the edge 푒 collapsed and 푒 ranges over all “contractible” edges, i.e. edges connecting an internal vertex to another vertex of either kind. See Figure 2.1.3 for an example.

1 1 1 1 ⎜⎛ ⎟⎞ 푑 ⎜ ⎟ = ± ± ± ⎜ ⎟ ⎝ 2 3 ⎠ 2 3 2 3 2 3

Figure 2.1.3: The differential of Tw Gra푛

Remark 2.1.13. An edge connected to a univalent internal vertex is not considered ∨ contractible: the Maurer–Cartan element 휇 ∈ Def(C → hoLie푘 ) “acts” on both sides of Tw C in the definition of the differential, and for such edges thetwo contributions cancel out. Such vertices are called dead ends by Lambrechts– Volić [LV14].2 The product of two graphs glues them along their external vertices only (which is the same thing as adding disjoint internal vertices to both graphs and gluing along all vertices).

2Their definition is slightly different, but since we forbid multiple edges and loops, thetwo definitions are equivalent.

48 2.1 Background and recollections

∨ 휔′ ∗ The two morphisms e푛 ← Gra푛 ΩPA(FM푛) extend along the inclusion ∨ Gra푛 ⊂ Tw Gra푛 as follows. The extended morphism Tw Gra푛 → e푛 simply sends a graph with internal vertices to zero; since dead ends are not contractible, this ∗ commutes with differentials. The extended morphism 휔 ∶ Tw Gra푛 → ΩPA(FM푛) sends a graph Γ ∈ Gra푛(푈 ⊔ 퐼) ⊂ Tw Gra푛(푈) to:

′ ′ 휔(Γ) ≔ (푝푈)∗(휔 (Γ)) = ∫ 휔 (Γ), (2.1.14) FM푛(푈⊔퐼)→FM푛(푈) where 푝푈 is the projection that forgets the points of the configuration corresponding to 퐼, and the integral is an integral along the fiber of this semi-algebraic bundle.

Reduction of the graph complex The cooperad Tw Gra푛 does not have the right homotopy type yet. It is reduced by quotienting out all the graphs with connected components consisting exclusively of internal vertices. This is a Hopf Graphs cooperad bi-ideal and thus the resulting quotient 푛 is still a Hopf cooperad. ∨ ∗ One checks that the two morphisms e푛 ← Tw Gra푛 → ΩPA(FM푛) factor through the quotient (the first one because dead ends are not contractible, the second one because 휔 vanishes on graphs with only internal vertices). The resulting zigzag:

e∨ Graphs ∗ FM 푛 ← 푛 → ΩPA( 푛) (2.1.15) is then a zigzag of weak equivalence of Hopf cooperads by the work of Kontse- vich [Kon99], which proves the formality theorem.

2.1.4 Poincaré duality CDGA models ∗ The model for ΩPA(FM푀) relies on a Poincaré duality model of 푀. We mostly borrow the terminology and notations from [LS08b]. Recall that we only consider simply connected manifolds, which are necessarily orientable and thus satisfy Poincaré duality. Fix an integer 푛 and let 퐴 be a connected CDGA (i.e. 퐴 = 한 ⊕ 퐴≥1). A Poincaré duality pairing on 퐴 is a dg-form 휀 ∶ 퐴 → 한[−푛] (i.e. a linear map 퐴푛 → 한 with 휀 ∘ 푑 = 0) such that the induced pairing

퐴푘 ⊗ 퐴푛−푘 → 한 푎 ⊗ 푏 ↦ 휀(푎푏) (2.1.16) is non-degenerate for all 푘. This implies that 퐴 = 퐴≤푛, and that 휀 ∶ 퐴푛 → 한 is an isomorphism. The pair (퐴, 휀) is called a Poincaré duality algebra.

49 2 The Lambrechts–Stanley Model of Configuration Spaces

If 퐴 is such a Poincaré duality CDGA, then so is its cohomology. The theorem of Lambrechts–Stanley [LS08b] implies that if Ω is a CDGA (over any field) whose minimal model is 1-connected and of finite type,3 and if ∫ ∶ Ω → 한[−푛] is a dg-form inducing a Poincaré duality structure of dimension 푛 on 퐻∗(Ω), then Ω is quasi-isomorphic to a Poincaré duality algebra (퐴, 휀) of the same dimension through a zigzag of CDGA morphisms which respect the dg-forms:

Ω ∼ ⋅ ∼ ∃퐴

∫ ∃휀 한 Remark 2.1.17. Even though it is not written in the statement of the theorem in [LS08b], studying of the proof shows that the diagram above does commute.

Let 퐴 be a Poincaré duality CDGA of finite type and let {푎푖} be a graded ∨ basis of 퐴. Consider the dual basis {푎푖 } with respect to the duality pairing, i.e. ∨ 휀(푎푖푎푗 ) = 훿푖푗 is given by the Kronecker symbol. Then the diagonal cocycle is defined by the following formula and is independent of the chosen graded basis (see e.g. [FOT08]):

|푎푖| ∨ Δ퐴 = ∑(−1) 푎푖 ⊗ 푎푖 ∈ 퐴 ⊗ 퐴. (2.1.18) 푖

The element Δ퐴 is a cocycle of degree 푛 (this follows from 휀 ∘ 푑 = 0). It satisfies 21 푛 Δ = (−1) Δ, and for all 푎 ∈ 퐴 it satisfies the equation (푎 ⊗ 1)Δ퐴 = (1 ⊗ 푎)Δ퐴. Finally, the product 휇퐴 ∶ 퐴 ⊗ 퐴 → 퐴 sends Δ퐴 to 휒(퐴) ⋅ vol퐴, where 휒(퐴) is the 푛 푛 Euler characteristic of 퐴 and vol퐴 ∈ 퐴 is the preimage of 1 ∈ 한 by 휀 ∶ 퐴 → 한.

2.1.5 The Lambrechts–Stanley CDGAs

We will give the definition of the CDGA G퐴(푘), constructed in [LS08a] in the general case of a Poincaré duality CDGA (see the introduction for a more detailed history). Let 퐴 be a Poincaré duality CDGA of dimension 푛 and let 푘 be an integer. For ⊗푘 ⊗푖−1 ⊗푘−푖−1 1 ≤ 푖 ≠ 푗 ≤ 푘, let 휄푖 ∶ 퐴 → 퐴 be defined by 휄푖(푎) = 1 ⊗ 푎 ⊗ 1 , and let ⊗푘 휄푖푗 ∶ 퐴 ⊗ 퐴 → 퐴 be given by 휄푖푗(푎 ⊗ 푏) = 휄푖(푎) ⋅ 휄푗(푏). ∨ Recalling the description of e푛 in Equation (2.1.8), the CDGA G퐴(푘) is then defined by: ⊗푘 ∨ G퐴(푘) = (퐴 ⊗ e푛 (푘)/(휄푖(푎) ⋅ 휔푖푗 = 휄푗(푎) ⋅ 휔푖푗), 푑휔푖푗 = 휄푖푗(Δ퐴)). (2.1.19)

3A chain complex is of “finite type” if it is finite dimensional in each degree. Looking closely at the proof of the theorem, we see that Ω does not need to be of finite type, only its minimal model does.

50 2.2 The Hopf right comodule model G퐴

We will call these CDGAs the Lambrechts–Stanley CDGAs, or LS CDGAs for short. For example G퐴(0) = 한, G퐴(1) = 퐴, and G퐴(2) is isomorphic to:

G퐴(2) ≅ ((퐴 ⊗ 퐴) ⊕ (퐴 ⊗ 휔12), 푑(푎 ⊗ 휔12) = (푎 ⊗ 1) ⋅ Δ퐴 = (1 ⊗ 푎) ⋅ Δ퐴). When 푀 is any simply connected closed manifold, a theorem of Lambrechts and Stanley [LS08a] implies that there exists a Poincaré duality CDGA 퐴 which is a rational model for 푀 and such that

∗ ∗ 퐻 (G퐴(푘); ℚ) ≅ 퐻 (FM푀(푘); ℚ) as graded vector spaces. (2.1.20)

2.2 The Hopf right comodule model G퐴 ∨ In this section we describe the Hopf right e푛 -comodule derived from the LS CDGAs of Section 2.1.5. From now on we work over ℝ, and we fix a simply connected smooth closed manifold 푀 of dimension at least 4. Following Section 2.1.2, we endow 푀 with a fixed semi-algebraic structure. For now we fix an arbitrary Poincaré duality CDGA model 퐴 of 푀; we will choose one in the next section. We then define the right comodule structure of ∨ G퐴 as follows, using the cooperad structure of e푛 given by Equation (2.1.10): Proposition 2.2.1. If 휒(푀) = 0, then the following maps go through the quotients ∨ defining G퐴 = {G퐴(푘)}푘≥0 and endow it with a Hopf right e푛 -comodule structure:

∨ ⊗푈 ∨ ⊗(푈/푊) ∨ ∨ ∘푊 ∶ 퐴 ⊗ e푛 (푈) → (퐴 ⊗ e푛 (푈/푊)) ⊗ e푛 (푊) ∨ (푎푢)푢∈푈 ⊗ 휔 ↦ ((푎푢)푢∈푈−푊 ⊗ ∏ 푎푤) ⊗ ∘푊 (휔) (2.2.2) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟푤∈푊 ⏟ ⊗(푈/푊) ∨ ∨ ∈퐴 ∈e푛 (푈/푊)⊗e푛 (푊) ∨ In informal terms, ∘푊 multiplies together all the elements of 퐴 indexed by 푊 on the 퐴⊗푈 factor and indexes the result by ∗ ∈ 푈/푊, and it applies the cooperadic ∨ ∨ structure map of e푛 on the other factor. Note that if 푊 = ∅, then ∘푊 adds a factor of 1퐴 (the empty product) indexed by ∗ ∈ 푈/∅ = 푈 ⊔ {∗}. We split the proof in three parts: compatibility of the maps with the cooperadic ∨ structure of e푛 , factorization of the maps through the quotient, and compatibility with the differential. Proposition 2.2.1, part 1: Factorization through the quotient. Since the algebra 퐴 is commutative, the maps of the proposition commute with multiplication. The ideals defining G퐴(푈) are multiplicative ideals, hence it suffices to show that the maps (2.2.2) take the generators (휄푢(푎) − 휄푣(푎)) ⋅ 휔푢푣 of the ideal to elements of the ideal in the target. We simply check each case:

51 2 The Lambrechts–Stanley Model of Configuration Spaces

∨ ∨ • If 푢, 푣 ∈ 푊, then ∘푊(휄푢(푎)휔푢푣) = 휄∗(푎)⊗휔푢푣, which is equal to ∘푊(휄푣(푎)휔푢푣). • If 푢 ∈ 푊 and 푣 ∉ 푊, then

∨ ∨ ∘푊(휄푢(푎)휔푢푣) = 휄∗(푎)휔∗푣 ⊗ 1 ≡ ∘푊(휄푣(푎)휔푢푣) are congruent modulo the ideal, and the case 푢 ∉ 푊, 푣 ∈ 푊 is symmetric.

• Finally if 푢, 푣 ∉ 푊 then

∨ ∨ ∘푊(휄푢(푎)휔푢푣) = 휄푢(푎)휔푢푣 ⊗ 1 ≡ ∘푊(휄푣(푎)휔푢푣).

Proposition 2.2.1, part 2: Compatibility with the differential. All the maps involved are respectively morphisms of algebras or derivations of algebras, thus it suffices to check the compatibility on the generators 휄푢(푎) and 휔푢푣. ∨ ∨ The equality ∘푊(푑(휄푢(푎))) = 푑(∘푊(휄푢(푎))) is immediate. For 휔푢푣 we again check the three cases. Recall that since our manifold has vanishing Euler characteristic, 휇퐴(Δ퐴) = 0. • If 푢, 푣 ∈ 푊, then ∨ ∘푊(푑휔푢푣) = 휄∗(휇퐴(Δ퐴)) = 0, ∨ ∨ whereas ∘ (휔 ) = 1 ⊗ 휔 and thus 푑(∘ (휔 )) = 0 (since 푑 ∨ = 0). 푊 푢푣 푢푣 푊 푢푣 e푛 • If 푢 ∉ 푊 and 푣 ∈ 푊, then

∨ ∨ ∘푊(푑휔푢푣) = 휄∗푣(Δ퐴) ⊗ 1 = 푑(휔∗푣 ⊗ 1) = 푑(∘푊(휔푢푣)), and the case 푢 ∈ 푊, 푣 ∉ 푊 is symmetric.

• Finally if 푢, 푣 ∉ 푊, then

∨ ∨ ∘푊(푑휔푢푣) = 휄푢푣(Δ퐴) ⊗ 1 = 푑(휔푢푣 ⊗ 1) = 푑(∘푊(휔푢푣)).

Proposition 2.2.1, part 3: Comodule structure. Although the fact that the cocompo- ∨ sition maps are compatible with the coproduct of e푛 can easily be proved “by hand”, it also follows from general arguments. Let Com∨ be the cooperad of counital cocommutative coalgebras, which is a right comodule over itself, and view 퐴 as an operad concentrated in arity 1. We apply the result of the next lemma (Lemma 2.2.3) to C = Com∨. We get that the ∨ ⊗푘 ∨ collection Com ∘ 퐴 = {퐴 }푘≥0 forms a Com -comodule. Then the arity-wise tensor product (see [LV12, Section 5.1.12], where this operation is called the Hadamard product):

∨ ∨ ⊗푘 ∨ (Com ∘ 퐴) ⊠ e푛 = {퐴 ⊗ e푛 (푘)}푘≥0

52 2.3 Labeled graph complexes

∨ ∨ is the arity-wise tensor product of a Com -comodule and an e푛 -comodule, hence it ∨ ∨ ∨ is a (Com ⊠ e푛 )-comodule. The cooperad Com is the unit of the arity-wise tensor ∨ product of cooperads, hence the result is an e푛 -comodule. It remains to make the easy check that the resulting comodule maps are given by Equation (2.2.2). The next very general lemma can for example be found in [CW16, Section 5.2]. Let C be a cooperad, and see the CDGA 퐴 as an operad concentrated in arity 1. Then the commutativity of 퐴 implies the existence of a distributive law 푡 ∶ C ∘ 퐴 → 퐴 ∘ C, given in each arity by:

푡 ∶ (C ∘ 퐴)(푛) = C(푛) ⊗ 퐴⊗푛 → (퐴 ∘ C)(푛) = 퐴 ⊗ C(푛)

푥 ⊗ 푎1 ⊗ ⋯ ⊗ 푎푛 ↦ 푎1 … 푎푛 ⊗ 푥

Lemma 2.2.3. Let N be a right C-comodule, and see 퐴 as a symmetric collection concentrated in arity 1. Then N ∘ 퐴 is a right C-comodule through the map:

N ∘ 퐴 ΔN∘1 N ∘ C ∘ 퐴 1∘푡 N ∘ 퐴 ∘ C.

2.3 Labeled graph complexes

In this section we construct the intermediary comodule used to prove the theorem. We will construct a zigzag of CDGAs of the form:

G Graphs ∗ FM 퐴 ← 푅 → ΩPA( 푀). Graphs Graphs The construction of 푅 follows the same pattern as the one of 푛 from Section 2.1.3, using labeled graphs. If 휒(푀) = 0, then the collections G퐴 and Graphs e∨ Graphs 푅 are Hopf right comodules respectively over 푛 and a 푛, and the G e∨ Graphs Graphs left arrow is a morphism of comodules between ( 퐴, 푛 ) and ( 푅, 푛). ∗ When 푀 is moreover framed, ΩPA(FM푀) becomes a Hopf right comodule over ∗ ΩPA(FM푛), and the right arrow is a morphism of comodules.

2.3.1 Graphs with loops and multiple edges Graphs Graphs We first define a variant 푛 of 푛, where graphs are allowed to have “loops” (also sometimes known as “tadpoles”) and multiple edges. For a finite set 푈, the CDGA Gra푛 (푈) is given by:

푛 Gra푛 (푈) = (푆(푒푢푣)푢,푣∈퐴/(푒푣푢 = (−1) 푒푢푣), 푑 = 0).

2 The difference with Equation (2.1.11) is that we no longer set 푒푢푢 = 푒푢푣 = 0.

53 2 The Lambrechts–Stanley Model of Configuration Spaces

2 Remark 2.3.1. When 푛 is even, 푒푢푣 = 0 since deg 푒푢푣 = 푛 − 1 is odd; and when 푛 푛 is odd, 푒푢푢 = (−1) 푒푢푢 = −푒푢푢 ⟹ 푒푢푢 = 0.

Like Gra푛, this defines a Hopf cooperad with cocomposition given by an equation similar to Equation (2.1.10):

∨ ∘푊 ∶ Gra푛 (푈) → Gra푛 (푈/푊) ⊗ Gra푛 (푊) ⎧ 푒∗∗ ⊗ 1 + 1 ⊗ 푒푢푣, if 푢, 푣 ∈ 푊; { {푒∗푣 ⊗ 1, if 푢 ∈ 푊 and 푣 ∉ 푊; 푒푢푣 ↦ ⎨ (2.3.2) {푒푢∗ ⊗ 1, if 푢 ∉ 푊 and 푣 ∈ 푊; { ⎩푒푢푣 ⊗ 1, if 푢, 푣 ∉ 푊.

This new cooperad has a graphical description similar to Gra푛. The difference in the cooperad structure is that when we collapse a subgraph, we sum over all ways of choosing whether edges are in the subgraph or not; if they aren’t, then they yield a loop. See Figure 2.3.1 for an example. The cooperad Gra푛 is the quotient of Gra푛 by the ideal generated by the loops and the multiple edges.

3 ⎛ 3 ⎞ ⎛ 3 ⎞ ⎜ ⎟ ⎜ ⎟ ↦ ⎜ ⊗ 1 2 ⎟ + ⎜ ⊗ 1 2 ⎟ 1 2 ∗ ∗ ⎝ ⎠ ⎝ ⎠

Figure 2.3.1: Example of cocomposition in Gra푛

∨ ∨ The element 푒12 ∈ (Gra푛 ) (2) still defines a Maurer–Cartan element

∨ ∨ 휇 ≔ 푒12 ∈ Def(hoLie푛 → (Gra푛 ) ),

which allows us to define the twisted Hopf cooperad Tw Gra푛 . It has a graphical description similar to Tw Gra푛 with internal and external vertices. Finally we can quotient by graphs containing connected component consisting exclusively of Graphs internal vertices to get a Hopf cooperad 푛 .

2.3.2 External vertices: Gra푅

We construct a collection of CDGAs Gra푅, corresponding to the first step in Graphs the construction of 푛 of Section 2.1.3. We first apply the formalism of ∗ Section 2.1.4 to ΩPA(푀) in order to obtain a Poincaré duality CDGA out of 푀:

54 2.3 Labeled graph complexes

Theorem 2.3.3 (Lambrechts–Stanley [LS08b]). There exists a zigzag of weak equivalences of CDGAs 휌 휎 ∗ 퐴 푅 ΩPA(푀), such that:

1. 퐴 is a Poincaré duality CDGA of dimension 푛;

2. 푅 is a quasi-free CDGA generated in degrees ≥ 2;

3. For all 푥 ∈ 푅, 휀 (휌(푥)) = ∫ 휎(푥) (see Remark 2.1.17. 퐴 푀 If 휒(푀) = 0 then the diagonal cocycle of any Poincaré duality model 퐴 of 푀 satisfies 휇퐴(Δ퐴) = 0. We will require the following technical lemma. Proposition 2.3.4. One can choose the zigzag of Theorem 2.3.3 such there exists a symmetric cocycle Δ푅 ∈ 푅 ⊗ 푅 of degree 푛 satisfying (휌 ⊗ 휌)(Δ푅) = Δ퐴. If 휒(푀) = 0 we can moreover choose it so that 휇푅(Δ푅) = 0. We follow closely the proof of [LS08b] to obtain the result.

∗ Proof (case 푛 ≤ 6). When 푛 ≤ 6, the CDGA ΩPA(푀) is formal [NM78, Proposi- ∗ tion 4.6]. We choose 퐴 = (퐻 (푀), 푑퐴 = 0), and 푅 to be the minimal model of 푀. By Künneth’s formula, 푅 ⊗ 푅 → 퐴 ⊗ 퐴 is a quasi-isomorphism. Since 푑Δ퐴 = 0, ′ ′ there exists some cocycle Δ ∈ 푅 ⊗ 푅 such that 휌(Δ ) = Δ퐴 + 푑훼 = Δ퐴 (since 푑퐴 = 0). ′ Let us now assume that 휒(푀) = 0. Then 휌(휇푅(Δ )) = 휇퐴(Δ퐴) = 휒(퐴)vol퐴 = ′ 0. Since 휌 is a quasi-isomorphism and hence injective in cohomology, 휇푅(Δ ) = 푑훽 ″ ′ ″ for some 훽 ∈ 푅 ⊗ 푅. We now let Δ = Δ − 푑훽 ⊗ 1, so that 휇푅(Δ ) = 0, and

″ ′ 휌(Δ ) = 휌(Δ ) − 휌(푑훽) ⊗ 1 = Δ퐴.

″ ′ 1 ″ 푛 ″ 21 If 휒(푀) ≠ 0 we simply let Δ = Δ . We finally set Δ푅 = 2 (Δ + (−1 )(Δ ) ), which is symmetric and still satisfies all the required properties.

Proof (case 푛 ≥ 7). When 푛 ≥ 7, the proof of Lambrechts and Stanley builds a zigzag of weak equivalences:

휌 ′ ∗ 퐴 푅 ← 푅 → ΩPA(푀), where 푅′ is the minimal model of 푀, the CDGA 푅 is obtained from 푅′ by succes- sively adjoining cells of degree ≥ 푛/2 + 1, and the Poincaré duality CDGA 퐴 is a

55 2 The Lambrechts–Stanley Model of Configuration Spaces quotient of 푅 by an ideal of “orphans”. By construction, this zigzag is compatible with 휀 and ∫ . 퐴 푀 The minimal model 푅′ is quasi-free, and since 푀 is simply connected it is generated in degrees ≥ 2. The CDGA 푅 is obtained from 푅′ by a cofibrant cellular extension, adjoining cells of degree greater than 2. It follows that 푅 is cofibrant and quasi-free generated in degrees ≥ 2. Since 푅′ → 푅 is an acyclic cofibration and 휀 ∶ 푅′ → ℝ is a fibration, we can invert 푅′ → 푅 up to homotopy while preserving ′ ∗ ∗ 휀. Composing with 푅 → ΩPA(푀) yields a morphism 휎 ∶ 푅 → ΩPA(푀) that still satisfies 휀 ∘ 휌 = ∫ 휎(−), and we therefore get a zigzag 퐴 ← 푅 → Ω∗ (푀). 퐴 푀 PA The morphism 휌 is a quasi-isomorphism, so there exists some cocycle Δ̃ ∈ 푅⊗푅 ̃ such that 휌(Δ) = Δ퐴 + 푑훼 for some 훼. By surjectivity of 휌 (it is a quotient map) ′ ̃ ′ there is some 훽 such that 휌(훽) = 훼; we let Δ = Δ − 푑훽, and now 휌(Δ ) = Δ퐴. ′ Let us assume for the moment that 휒(푀) = 0. Then the cocycle 휇푅(Δ ) ∈ 푅 ′ satisfies 휌(휇푅(Δ )) = 휇퐴(Δ퐴) = 0, i.e. it is in the kernel of 휌. It follows that the ″ ′ ′ ″ cocycle Δ = Δ − 휇푅(Δ ) ⊗ 1 is still mapped to Δ퐴 by 휌, and satisfies 휇푅(Δ ) = 0. ″ ′ ″ If 휒(푀) ≠ 0 we just let Δ = Δ . Finally we symmetrize Δ to get the Δ푅 of the lemma, which satisfies all the requirements.

From now on we keep the zigzag and the element Δ푅 of the previous lemma fixed until the end of Section 2.4.

Definition 2.3.5. The CDGA of 푅-labeled graphs with loops on the set 푈 is given by: ⊗푈 Gra푅(푈) = (푅 ⊗ Gra푛 (푈), 푑푒푢푣 = 휄푢푣(Δ푅)).

Remark 2.3.6. It follows from the definition that 푑푒푢푢 = 휄푢푢(Δ푅) = 휄푢(휇푅(Δ푅)), which is zero when 휒(푀) = 0.

Proposition 2.3.7. The CDGAs Gra푅(푈) assemble to form a Hopf right Gra푛 -comodule. Proof. The proof of this proposition is almost identical to the proof of Proposi- tion 2.2.1. If we forget the extra differential (keeping only the internal differential ∨ of 푅), then Gra푅 is the arity-wise tensor product (Com ∘ 푅) ⊠ Gra푛, which is automatically a Hopf Gra푛-right comodule. Checking the compatibility with the differential involves almost exactly the same equations as Proposition 2.2.1, except that when 푢, 푣 ∈ 푊 we have:

∨ ∨ ∘푊(푑(푒푢푣)) = 휄∗(휇푅(Δ푅)) ⊗ 1 = 푑(푒∗∗ ⊗ 1 + 1 ⊗ 푒푢푣) = 푑(∘푊(푒푢푣)).

We now give a graphical interpretation of Definition 2.3.5, in the spirit of

Section 2.3.1. We view Gra푅(푈) as spanned by graphs with 푈 as set of vertices, and each vertex has a label which is an element of 푅. The Gra푛 -comodule structure

56 2.3 Labeled graph complexes collapses subgraphs as before, and the label of the collapsed vertex is the product of all the labels in the subgraph.

Let Γ ∈ Gra푅(푈) be some graph. The differential of Γ, as defined in Defini- tion 2.3.5, is the sum over the edges 푒 ∈ 퐸Γ of the graph Γ − 푒 with that edge removed and the labels of the endpoints multiplied by the factors of

′ ″ Δ푅 = ∑ Δ푅 ⊗ Δ푅 ∈ 푅 ⊗ 푅. (Δ푅) In particular if 푒 is a loop, then in the corresponding factor of 푑Γ the vertex incident to 푒 has its label multiplied by 휇푅(Δ푅). We will often write 푑split for this differential, to contrast it with the differential that contracts edges which will occur in the complex Tw Gra푅 defined later on. See Figure 2.3.2 for an example – gray vertices can be either internal or external.

′ ″ 푥 푦 푥Δ푅 푦Δ푅 ↦ ∑

(Δ푅)

Figure 2.3.2: The splitting part of the differential (on one edge)

Finally the Hopf structure glues two graphs along their vertices, multiplying the labels in the process. ∗ If 휒(푀) ≠ 0, we cannot directly map Gra푅 to ΩPA(FM푀), as the Euler class in ∗ ΩPA(푀) would need to be the boundary of the image of the loop 푒11 ∈ Gra푅(1). ∗ We thus define a sub-CDGA which will map to ΩPA(FM푀) whether 휒(푀) vanishes or not.

Definition 2.3.8. For a given finite set 푈, let Gra푅(푈) be the sub-CDGA of

Gra푅(푈) spanned by graphs without loops.

Proposition 2.3.9. The space Gra푅(푈) is a sub-CDGA, and if 휒(푀) = 0 the collection Gra푅 assembles to form a Hopf right Gra푛-comodule. Proof. Clearly, neither the splitting part of the differential nor the internal differential coming from 푅 can create new loops, nor can the product of two graphs without loops contain a loop, thus Gra푅(푈) is indeed a sub-CDGA of Gra푅(푈). Let us now assume that 휒(푀) = 0. The proof that Gra푅 is a Gra푛-comodule is now almost the same as the proof of Proposition 2.3.7, except that we need ∨ ∨ to use the fact that 휇푅(Δ푅) = 0 to check that 푑(∘푊(푒푢푣)) = ∘푊(푑(푒푢푣)) when 푢, 푣 ∈ 푊.

57 2 The Lambrechts–Stanley Model of Configuration Spaces

2.3.3 The propagator

′ 푛−1 To define 휔 ∶ Gra푅 → ΩPA(FM푀), we need a “propagator” 휑 ∈ ΩPA (FM푀(2)), for which a reference is [CM10, Section 4], who refine constructions of Axelrod– Singer [AS94]. For a given 푢 ∈ 푈, we define the projection 푝푢 ∶ FM푀(푈) → 푀 to be the map that forgets all the points of the configuration except the one labeled by 푢. The two projections 푝1, 푝2 ∶ FM푀(2) → 푀 are equal when restricted to the boundary, and form a sphere bundle 휕FM푀(2) → 푀 (FM푀(2) is the blow-up of 푀 × 푀 along the diagonal). When 푀 is framed, this bundle is trivial, and the operadic insertion map

푛−1 ∘1 푀 × 푆 ≅ FM푀(1) × FM푛(2) 휕FM푀(2) is an isomorphism of bundles.

푛−1 Proposition 2.3.10 ([CW16, Proposition 7]). There exists a form 휑 ∈ ΩPA (FM푀(2)) 21 푛 ∗ such that 휑 = (−1) 휑, 푑휑 = (푝1 × 푝2) (휎(Δ푅)) and such that the restriction of 휑 푛−1 to 휕FM푀(2) is a global angular form, i.e. it is a volume form of 푆 when restricted to each fiber. When 푀 is framed one can moreover choose 휑| = 1 × vol 푛−1 ∈ 휕FM푀(2) 푆 푛−1 푛−1 Ω푃퐴 (푀 × 푆 ). One can see from the proofs of [CM10, Section 4] that 푑휑 can in fact be chosen to 푛 be any pullback of a form cohomologous to the diagonal class Δ푀 ∈ ΩPA(푀 ×푀). We will make further adjustments to the propagator 휑 in Proposition 2.3.32.

Proposition 2.3.11. There is a morphism of collections of CDGAs:

휔′ Gra푅 ΩPA(FM푀) ⊗푈 ∗ ⨂ 푥푢 ∈ 푅 ↦ ⋀ 푝푢(휎(푥푢)) 푢∈푈 푢∈푈 ∗ 푒푢푣 ↦ 푝푢푣(휑), where 푝푢푣 was defined in Equation (2.1.6). Moreover, if 푀 is framed, then 휔′ defines a morphism of comodules:

(휔′,휔′) ∗ ∗ (Gra푅, Gra푛) (ΩPA(FM푀), ΩPA(FM푛))

′ ∗ where 휔 ∶ Gra푛 → ΩPA(FM푛) was defined in Section 2.1.3. ∗ ′ Proof. The property 푑휑 = (푝1 × 푝2) (Δ푅) shows that the map 휔 preserves the differential.

58 2.3 Labeled graph complexes

Let us now assume that 푀 is framed to prove that this is a morphism of right comodules. Cocomposition commutes with 휔′ on the generators coming from ⊗푈 ∗ 퐴 , since the comodule structure of ΩPA(FM푀) multiplies together forms that are pullbacks of forms on 푀:

⎧ ∗ ∨ ∗ {푝푢(푥) ⊗ 1 if 푢 ∉ 푊; ∘푊(푝푢(푥)) = ⎨ ∗ ⎩{푝∗(푥) ⊗ 1 if 푢 ∈ 푊.

∨ ′ We now check the compatibility of the cocomposition ∘푊 with 휔 on the generator 휔푢푣, for some 푊 ⊂ 푈.

• If one of 푢, 푣, or both, is not in 푊, then the equality

∨ ′ ′ ′ ∨ ∘푊(휔 (푒푢푣)) = (휔 ⊗ 휔 )(∘푊(푒푢푣))

is clear by the previous relation.

• Otherwise suppose {푢, 푣} ⊂ 푊. We may assume that 푈 = 푊 = 2 (it suffices to pull back the result along 푝푢푣 to get the general case), so that we are considering the insertion of an infinitesimal configuration 푀×FM푛(2) → FM푀(2). This insertion factors through the boundary 휕FM푀(2). We have (recall Definition 2.3.10):

∨ ∗ ∗ ∗ ∗ 푛−1 ∘2 (휑) = 1 ⊗ vol푆푛−1 ∈ ΩPA(푀) ⊗ ΩPA(FM푛(2)) = ΩPA(푀) ⊗ ΩPA(푆 ).

Going back to the general case, we find:

∨ ′ ∨ ∗ ∗ ∘푊(휔 (푒푢푣)) = ∘푊(푝푢푣(휑)) = 1 ⊗ 푝푢푣(vol푆푛−1 ),

∨ ′ ′ which is indeed the image of ∘푊(휔푢푣) = 1 ⊗ 휔푢푣 by 휔 ⊗ 휔 .

2.3.4 Twisting: Tw Gra푅 The general framework of [Wil16, Appendix C] shows that to twist a right (co)module, one only needs to twist the (co)operad. As before, the condition on the arity zero component of the Hopf cooperad provides the Hopf structure on the twisted comodule.

Definition 2.3.12. The twisted labeled graph comodule Tw Gra푅 is a Hopf right comodule over (Tw Gra푛 ), obtained from Gra푅 by twisting with respect to the ∨ Maurer–Cartan element 휇 ∈ (Gra푛 ) (2) of Section 2.1.3.

59 2 The Lambrechts–Stanley Model of Configuration Spaces

We now explicitly describe this comodule in terms of graphs.

The dg-module Tw Gra푅(푈) is spanned by graphs with two kinds of vertices, external vertices corresponding to elements of 푈, and indistinguishable internal vertices (usually drawn in black). The degree of an edge is still 푛 − 1, while the degree of an internal vertex is −푛. All the vertices are labeled by elements of 푅, and their degree is added to the degree of the graph. The Hopf structure glues two graphs along their external vertices, multiplying labels in the process. The differential is a sum of two terms 푑split + 푑contr (in addition to the internal differential coming from 푅). The first part comes from

Gra푅 and splits edges, multiplying by Δ푅 the labels of the endpoints. The second part is similar to the differential of Tw Gra푛 : it contracts edges connecting an internal vertex to another vertex of either kind, multiplying the labels of the endpoints (see Figure 2.1.3).

Remark 2.3.13. Unlike Tw Gra푛, dead ends are contractible in Tw Gra푅. This is ∨ because the Maurer–Cartan element in (Gra푛 ) (2) can only “act” while coming from the right side on Gra푅 in the definition of the differential, and so there is nothing to cancel out the contraction of a dead end.

Finally, the comodule structure is similar to the cooperad structure of Tw Gra푛 : ∨ for Γ ∈ Gra푅(푈 ⊔ 퐼) ⊂ Tw Gra푅(푈), the cocomposition ∘푊(Γ) is the sum over tensors of the type ±Γ푈/푊 ⊗Γ푊, where Γ푈/푊 ∈ Gra푅(푈/푊 ⊔퐽), Γ푊 ∈ Gra푛(푊 ⊔ ′ ′ 퐽 ), 퐽 ⊔ 퐽 = 퐼, and there exists a way of inserting Γ푊 in the vertex ∗ of Γ푈/푊 and reconnecting edges to get back Γ. See Figure 2.3.3 for an example.

푦 푦 ⎛ ⎞ ⎜ ⎟ ⎛ 푥푦 ⎞ ⎛ 푥푦 ⎞ ⎜ 1 ⎟ ⎜ ⎟ ⎜ ⎟ ↦ ⎜ ⊗ ⎟ ± ⎜ ∗ ⊗ ⎟ ± ⎜ ∗ ⊗ ⎟ 푥 ⎜ 푥 ⎟ 1 1 ∗ ⎝ ⎠ ⎝ ⎠ 1 ⎝ ⎠

Figure 2.3.3: Example of cocomposition Tw Gra푅(1) → Tw Gra푅(1)⊗Tw Gra푛 (1)

Lemma 2.3.14. The subspace Tw Gra푅(푈) ⊂ Tw Gra푅(푈) spanned by graphs with no loops is a sub-CDGA.

Proof. It is clear that this defines a subalgebra. We need to check that it ispre- served by the differential, i.e. that the differential cannot create new loops ifthere are none in a graph. This is clear for the internal differential coming from 푅 and for the splitting part of the differential. The contracting part of the differential could create a loop from a double edge; however for even 푛 multiple edges are

60 2.3 Labeled graph complexes zero for degree reasons, and for odd 푛 loops are zero because of the antisymmetry relation (see Remark 2.3.6).

Note that despite the notation, Tw Gra푅 is a priori not defined as the twisting of the Gra푛-comodule Gra푅: when 휒(푀) ≠ 0, the collection Gra푅 is not even a Gra푛-comodule. However, the following proposition is clear and shows that we can get away with this abuse of notation:

Proposition 2.3.15. If 휒(푀) = 0, then Tw Gra푅 assembles to a right Hopf comodule over (Tw Gra푛), isomorphic to the twisting of the right Hopf Gra푛-comodule Gra푅 of Definition 2.3.8.

Remark 2.3.16. We could have defined the algebra Tw Gra푅 explicitly in terms of labeled graphs, and then defined the differential 푑 using an ad-hoc formula. The difficult part would have then been to check that 푑2 = 0 (involving difficult signs), which is a consequence of the general operadic twisting framework.

Proposition 2.3.17. There is a morphism of collections of CDGAs 휔 ∶ Tw Gra푅 → ∗ ′ ΩPA(FM푀) extending 휔 , given on a graph Γ ∈ Gra푅(푈 ⊔ 퐼) ⊂ Tw Gra푅(푈) by: ′ ′ 휔(Γ) ≔ (푝푈)∗(휔 (Γ)) = ∫ 휔 (Γ). FM푀(푈⊔퐼)→FM푀(푈) Moreover, if 푀 is framed, then this defines a morphism of Hopf right comodules: ∗ ∗ (휔, 휔) ∶ (Tw Gra푅, Tw Gra푛) → (ΩPA(FM푀), ΩPA(FM푛)). Remark 2.3.18. It is not a priori possible to integrate any arbitrary form along the fiber of the projection 푝푈, see [Har+11, Section 9.4]. However, following [CW16, ∗ Appendix C], we can assume that the morphism 휎 ∶ 푅 → ΩPA(푀) factors through the quasi-isomorphism sub-CDGA of “trivial forms” and that the propagator is a trivial form. This makes 휔(Γ) well-defined. Proof. The proof of the compatibility with the Hopf structure and, in the framed case, the comodule structure, is formally similar to the proof of the same facts ∗ about 휔 ∶ Tw Gra푛 → ΩPA(FM푛). We refer to [LV14, Sections 9.2, 9.5]. The proof is exactly the same proof, but writing FM푀 or FM푛 instead of 퐶[−] and 휑 instead of vol푆푛−1 in every relevant sentence, and recalling that when 푀 is framed, we ∨ choose 휑 such that ∘2 (휑) = 1 ⊗ vol푆푛−1 . The proof that 휔 is a chain map is different albeit similar. The rest of the section is dedicated to that proof. We recall Stokes’ formula for integrals along fibers of semi-algebraic bundles. If 휋 ∶ 퐸 → 퐵 is a semi-algebraic bundle, the fiberwise boundary 휋휕 ∶ 퐸휕 → 퐵 is the bundle with total space 퐸휕 = ⋃ 휕휋−1(푏). 푏∈퐵

61 2 The Lambrechts–Stanley Model of Configuration Spaces

휕 ⋃ −1 Remark 2.3.19. The space 퐸 is neither 휕퐸 nor 푏∈퐵 휋 (푏) ∩ 휕퐸 in general. Con- sider for example the projection on the first coordinate [0, 1]×2 → [0, 1]. Stokes’ formula is given in the semi-algebraic context by [Har+11, Proposition 8.12]:

푑 (∫ 훼) = ∫ 푑훼 ± ∫ 훼|퐸휕 . 휋∶퐸→퐵 휋∶퐸→퐵 휋휕∶퐸휕→퐵 If we apply this formula to compute 푑휔(Γ), we find that the first part is given by: ′ ′ ∫ 푑휔 (Γ) = ∫ 휔 (푑푅Γ + 푑splitΓ) = 휔(푑푅Γ + 푑splitΓ), (2.3.20) 푝푈 푝푈 since 휔′ was a chain map. It thus remain to check that the second summand satisfies:

′ ′ ∫ 휕 휕 휔 (Γ) = ∫ 휔 (푑contrΓ) = 휔(푑contrΓ). 푝푈∶FM푀(푈⊔퐼)→FM푀(푈) 푝푈

The fiberwise boundary of the projection 푝푈 ∶ FM푛(푈 ⊔ 퐼) → FM푛(푈) is rather complex [LV14, Section 5.7], essentially due to the quotient by the affine group in the definition of FM푛 which lowers dimensions. We will not repeat its explicit decomposition into cells as we do not need it here. The fiberwise boundary of 푝푈 ∶ FM푀(푈⊔퐼) → FM푀(푈) is simpler. Let 푉 = 푈⊔퐼. 휕 The interior of FM푀(푈) is the space Conf푈(푀), and thus FM푀(푉) is the closure −1 of (휕FM푀(푉)) ∩ 휋 (Conf푈(푀)). Let

ℬℱ(푉, 푈) = {푊 ⊂ 푉 ∣ #푊 ≥ 2 and #푊 ∩ 푈 ≤ 1}.

휕 Lemma 2.3.21. The subspace FM푀(푉) ⊂ FM푀(푉) is equal to:

⋃ im(∘푊 ∶ FM푀(푉/푊) × FM푛(푊) → FM푀(푉)). 푊∈ℬℱ(푉,푈)

휕 In the description of FM푛(푉), there was an additional part which corresponds to 푈 ⊂ 푊. But unlike FM푛, for FM푀 the image of 푝푈(− ∘푊 −) is always included in the boundary of FM푀(푈) when 푈 ⊂ 푊. We follow a pattern similar to the one used in the proof of [LV14, Proposition 5.7.1].

Proof. Let cls denote the closure operator. Since Conf푈(푀) is the interior of FM푀(푈) and 푝 ∶ FM푀(푉) → FM푀(푈) is a bundle, it follows that:

휕 휕 −1 FM푀(푉) = cls(FM푀(푉) ∩ 푝 (Conf푈(푀))) −1 = cls(휕FM푀(푉) ∩ 푝 (Conf푈(푀))).

62 2.3 Labeled graph complexes

The boundary 휕FM푀(푉) is the union of the im(∘푊) for #푊 ≥ 2 (note that the case 푊 = 푉 is included, unlike for FM푛). If #푊 ∩ 푈 ≥ 2 ⟺ 푊 ∉ ℬℱ(푉, 푈), then im(푝푈(− ∘푊 −)) ⊂ 휕FM푀(푈), because if a configuration belongs to this image then at least two points of 푈 are infinitesimally close. Therefore:

휕 −1 FM푀(푉) = cls(휕FM푀(푉) ∩ 푝 (Conf푈(푀))) −1 = cls(⋃ im(∘푊) ∩ 휋 (Conf푈(푀))) #푊≥2 −1 = cls(⋃ im(∘푊) ∩ 휋 (Conf푈(푀))) #푊∈ℬℱ(푉,푈) −1 = ⋃ cls(im(∘푊) ∩ 휋 (Conf푈(푀))) 푊∈ℬℱ(푉,푈)

= ⋃ im(∘푊). 푊∈ℬℱ(푉,푈)

Lemma 2.3.22. For a given graph Γ ∈ Tw Gra푅(푈), the integral over the fiberwise boundary is given by: ′ ∫ 휔 (Γ)| 휕 = 휔(푑 Γ). 휕 FM푀(푉) contr 푝푈

Proof. The maps ∘푊 ∶ FM푀(푉/푊) × FM푛(푊) → FM푀(푉) are smooth injective map and their domains are compact, thus they are homeomorphisms onto their images. Recall #푊 ≥ 2 for 푊 ∈ ℬℱ(푉, 푈), hence dim FM푛(푊) = 푛#푊 − 푛 − 1. The dimension of the image of ∘푊 is then:

dim im(∘푊) = dim FM푀(푉/푊) + dim FM푛(푊) = 푛#(푉/푊) + (푛#푊 − 푛 − 1) (2.3.23) = 푛#푉 − 1, i.e. the image is of codimension 1 in FM푀(푉). It is also easy to check that if ′ 푊 ≠ 푊 , then im(∘푊) ∩ im(∘푊′ ) is of codimension strictly bigger than 1. We now fix 푊 ∈ ℬℱ(푉, 푈). Since #푊 ∩ 푈 ≤ 1, the composition 푈 ⊂ 푉 → 푉/푊 is injective and identifies 푈 with a subset of 푉/푊. There is then a forgetful ′ map 푝푈 ∶ FM푀(푉/푊) → FM푀(푈). We then have a commutative diagram:

푝1 FM푀(푉/푊) × FM푛(푊) FM푀(푉/푊) ∘ ′ 푊 푝푈 (2.3.24) 푝푈 FM푀(푉) FM푀(푈)

′ It follows that 푝푈(− ∘푊 −) = 푝푈 ∘ 푝1 is the composite of two semi-algebraic bundles, hence it is a semi-algebraic bundle itself [Har+11, Proposition 8.5].

63 2 The Lambrechts–Stanley Model of Configuration Spaces

Combined with the fact about codimensions above, we can therefore apply the summation formula [Har+11, Proposition 8.11]:

′ ′ ∫ 휔 (Γ) = ∑ ∫ 휔 (Γ)|FM (푉/푊)×FM (푊) (2.3.25) 푝휕 푝 (−∘ −) 푀 푛 푈 푊∈ℬℱ(푉,푈) 푈 푊

Now we can directly adapt the proof of Lambrechts and Volić. For a fixed 푊, by [Har+11, Proposition 8.13], the corresponding summand is equal to ′ ±휔(Γ푉/푊) ⋅ ∫ 휔 (Γ푊), where FM푛(푊)

• Γ푉/푊 ∈ Tw Gra푅(푈) is the graph with 푊 collapsed to a vertex and 푈 ↪ 푉/푊 is identified with its image;

• Γ푊 ∈ Tw Gra푛(푊) is the full subgraph of Γ with vertices 푊 and the labels removed.

The vanishing lemmas in the proof of Lambrechts and Volić then imply that the ′ integral ∫ 휔 (Γ푊) is zero unless Γ푊 is the graph with exactly two vertices FM푛(푊) and one edge, in which case the integral is equal to 1. In this case, Γ푉/푊 is the graph Γ with one edge connecting an internal vertex to some other vertex collapsed. The sum runs over all such edges, and dealing with signs carefully we see that Equation (2.3.25) is precisely equal to 휔(푑contrΓ). End of the proof of Proposition 2.3.17. By Equation (2.3.20) and Lemma 2.3.22, we can apply Stokes’ formula to 푑휔(Γ) to show that it is equal to 휔(푑Γ) = 휔(푑푅Γ + 푑splitΓ) + 휔(푑splitΓ).

Graphs 2.3.5 Reduction: 푅 Graphs Gra The last step in the construction of 푅 is the reduction of Tw 푅 so that it has the right cohomology. We borrow the terminology of Campos– Willwacher [CW16] for the next two definitions.

Definition 2.3.26. Let the full graph complex be:

fGC푅 = Tw Gra푅(∅)[−푛].

It consists of graphs with only internal vertices, and the product is disjoint union of graphs. The degree of a graph 훾 is 푛 + (푛 − 1)#퐸훾 − 푛#푉훾, where 퐸훾 is the set of edges and 푉훾 the set of vertices. Remark 2.3.27. The degree shift is there to be consistent with the definition of the standard graph complex GC푛.

64 2.3 Labeled graph complexes

As an algebra, fGC푅[푛] is free and generated by connected graphs. In general we will call “internal components” the connected components of a graph that only contain internal vertices. The full graph complex naturally acts on Tw Gra푅(푈) by adding extra internal components.

Definition 2.3.28. The partition function Z휑 ∶ fGC푅[푛] → ℝ is the restriction:

∗ ∗ Z휑 = 휔|∅ ∶ Tw Gra푅(∅) = fGC푅[푛] → ΩPA(FM푀(∅)) = ΩPA(pt) = ℝ.

By the double-pushforward formula and Fubini’s theorem, Z휑 is an algebra morphism and

∀훾 ∈ fGC푅[푛], ∀Γ ∈ Tw Gra푅(푈), 휔(훾 ⋅ Γ) = Z휑(훾) ⋅ 휔(Γ). (2.3.29)

Definition 2.3.30. Let ℝ휑 be the fGC푅[푛]-module of dimension 1 induced by 휑 Z휑 ∶ fGC푅[푛] → ℝ. The reduced graph comodule Graphs푅 is the tensor product:

휑 Graphs (푈) = ℝ ⊗ Tw Gra (푈). 푅 휑 fGC푅[푛] 푅 In other words, a graph of the type Γ ⊔ 훾 containing an internal component 훾 ∈ fGC푅[푛] is identified with Z휑(훾) ⋅ Γ. It is spanned by representative classes of graphs with no internal connected component; we call such graphs reduced. 휑 The notation is meant to evoke the fact that Graphs푅 depends on the choice of Graphs휀 the propagator 휑, unlike the collection 푅 that will appear in Section 2.4.1. ∗ Proposition 2.3.31. The map 휔 ∶ Tw Gra푅(푈) → ΩPA(FM푀(푈)) defined in Proposi- 휑 tion 2.3.17 factors through the quotient defining Graphs푅. 휑 If 휒(푀) = 0, the symmetric collection Graphs푅 forms a Hopf right comodule over Graphs 푛. If moreover 푀 is framed, the map 휔 defines a Hopf right comodule morphism. Proof. Equation (2.3.29) immediately implies that 휔 factors through the quotient. The vanishing lemmas shows that if Γ ∈ Tw Gra푛(푈) has internal components, then 휔(Γ) vanishes [LV14, Proposition 9.3.1], so it is straightforward to check that Graphs휑 Graphs if 휒(푀) = 0, then 푅 becomes a Hopf right comodule over 푛. It is also clear that for 푀 framed, the quotient map 휔 remains a Hopf right comodule morphism.

Proposition 2.3.32 ([CM10, Lemma 3]). The propagator 휑 can be chosen such that the additional property (P4) holds:

∗ ∫ 푝2(휎(푥)) ∧ 휑 = 0, ∀푥 ∈ 푅; (P4) 푝1∶FM푀(2)→FM푀(1)=푀

65 2 The Lambrechts–Stanley Model of Configuration Spaces

Remark 2.3.33. The additional property (P5) of the paper mentioned above would 휑 be helpful in order to get a direct morphism Graphs푅 → G퐴, because then the partition function would vanish on all connected graphs with at least two vertices. However we run into difficulties when trying to adapt the proof in the setting ∗ of PA forms, mainly due to the lack of an operator 푑푀 acting on ΩPA(푀 × 푁) differentiating “only in the first slot”. From now on and until the end of the paper, we assume that 휑 satisfies (P4).

Corollary 2.3.34. The morphism 휔 vanishes on graphs containing univalent internal vertices (i.e. dead ends).

Proof. Let Γ ∈ Gra푅(푈 ⊔ 퐼) ⊂ Tw Gra푅(푈) be a graph with a univalent internal vertex 푢 ∈ 퐼, labeled by 푥, and let 푣 be the only vertex connected to 푢. Let Γ̃ be the full subgraph of Γ on the set of vertices 푈 ⊔ 퐼 − {푢}. Then using [Har+11, Propositions 8.10 and 8.15] (in a way similar to the end of the proof of [LV14, Lemma 9.3.9]), we find:

휔(Γ) = ∫ 휔′(Γ) FM푀(푈⊔퐼)→FM푀(푈) ′ ̃ ∗ ∗ = ∫ 휔 (Γ)푝푢푣(휑)푝푢(휎(푥)) FM푀(푈⊔퐼)→FM푀(푈) ′ ̃ ∗ ∗ ∗ = ∫ 휔 (Γ) ∧ 푝푣 (∫ 푝푢푣(휑)푝푢(휎(푥))) , FM푀(푈⊔퐼−{푢})→FM푀(푈) FM푀({푢,푣})→FM푀({푣}) which vanishes by (P4). Almost everything we have done so far works for non-simply connected manifolds. We now prove a proposition which sets simply connected manifolds apart.

Proposition 2.3.35. The partition function Z휑 vanishes on any connected graph with no bivalent vertices labeled by 1푅 and containing at least two vertices.

Remark 2.3.36. If 훾 ∈ fGC푅 has only one vertex, labeled by 푥, then Z휑(훾) = ∫ 휎(푥) which can be nonzero. 푀

Proof. Let 훾 ∈ fGC푅[푛] be a connected graph with at least two vertices and no bivalent vertices labeled by 1푅. By Corollary 2.3.34, we can assume that all the vertices of 훾 are at least bivalent. By hypothesis, if a vertex is bivalent then it is labeled by an element of 푅>0 = 푅≥2. Let 푘 = 푖 + 푗 be the number of vertices of 훾, with 푖 vertices that are at least trivalent and 푗 vertices that are bivalent and labeled by 푅≥2. It follows that 훾 1 has at least 2 (3푖 + 2푗) edges, all of degree 푛 − 1. Since bivalent vertices are

66 2.4 From the model to forms via graphs labeled by 푅≥2, their labels contribute at least 2푗 to the degree of 훾. The (internal) vertices contribute −푘푛 to the degree, and the other labels have a nonnegative contribution. Thus: 3 deg 훾 ≥ ( 푖 + 푗) (푛 − 1) + 2푗 − 푘푛 2 3 3 = ( 푘 − 푗 + 푗) (푛 − 1) + 2푗 − 푘푛 2 2 1 = (푘(푛 − 3) − 푗(푛 − 5)). 2 This last number is always positive for 0 ≤ 푗 ≤ 푘: it is an affine function of 푗, and it is positive when 푗 = 0 and 푗 = 푘 (recall that 푛 ≥ 4). The degree of 훾 ∈ fGC푅[푛] must be zero for the integral defining Z휑(훾) to be the integral of a top form of FM푀(푘) and hence possibly nonzero. But by the above computation, deg 훾 > 0 ⟹ Z휑(훾) = 0.

Remark 2.3.37. When 푛 = 3, the manifold 푀 is the 3-sphere 푆3 thanks to Perel- man’s proof [Per02; Per03] of the Poincaré conjecture. The partition function Z휑 is conjectured to be trivial on 푆3 for a proper choice of framing, thus bypassing the need for the above degree counting argument. See also Proposition 2.4.37.

2.4 From the model to forms via graphs

∗ In this section we connect G퐴 to ΩPA(FM푀) and we prove that the connecting morphisms are quasi-isomorphisms.

2.4.1 Construction of the morphism to G퐴 ′ Proposition 2.4.1. For each finite set 푈, there is a CDGA morphism 휌∗ ∶ Gra푅(푈) → ⊗푈 G퐴(푈) given by 휌 on the 푅 factor and sending the generators 푒푢푣 to 휔푢푣 on the Gra푛 factor. When 휒(푀) = 0, this defines a Hopf right comodule morphism (Gra푅, Gra푛) → ∨ (G퐴, e푛 ).

If we could find a propagator for which property (P5) held (see Remark 2.3.33), then we could just send all graphs containing internal vertices to zero and obtain 휑 an extension Graphs푅 → G퐴. Since we cannot assume that (P5) holds, the definition of the extension is more complex. However we still have Proposition 2.3.35, and homotopically speaking, graphs with bivalent vertices are irrelevant.

67 2 The Lambrechts–Stanley Model of Configuration Spaces

0 Definition 2.4.2. Let fGC푅 be the quotient of fGC푅 defined by identifying a disconnected vertex labeled by 푥 with the number 휀(휌(푥)) = ∫ 휎(푥). 푀 0 It’s clear that Z휑 factors through a map fGC푅[푛] → ℝ, for which we will keep the same notation Z휑.

0 Lemma 2.4.3. The subspace 퐼 ⊂ fGC푅 spanned by graphs with at least one univalent vertex, or at least one bivalent vertex labeled by 1푅, or at least one label in ker(휌 ∶ 푅 → 퐴), is a (shifted) CDGA ideal.

Proof. It is clear that 퐼 is an algebra ideal. Let us prove that it is a differential ideal. If one of the labels of Γ is in ker 휌, then so do all the summands of 푑Γ, because ker 휌 is a CDGA ideal of 푅. If Γ contains a bivalent vertex 푢 labeled by 1푅, then so does 푑푅Γ. In 푑splitΓ, splitting one of the two edges connected to 푢 produces a univalent vertex; and in 푑contrΓ, the contraction of the two edges connected to 푢 cancel each other. Finally let us prove that if Γ has a univalent vertex 푢, then 푑Γ lies in 퐼. It’s clear that 푑푅Γ ∈ 퐼. Contracting or splitting the only edge connected to the univalent vertex could remove the univalent vertex. Let us prove that these two summands cancel each other up to ker 휌. Let 푥 be the label of 푢, and let 푦 be the label of the only vertex incident to 푢. Contracting the edge yields a new vertex labeled by 푥푦. Due to the definition of 0 fGC푅, splitting the edge yields a new vertex labeled by

″ ′ 훼 = ∑ 휀(휌(푦Δ푅))푥Δ푅 (Δ푅) ″ ′ ⟹ 휌(훼) = 휌(푥) ⋅ ∑ ±휀퐴(휌(푦)Δ퐴)Δ퐴. (Δ퐴)

″ ′ It is a standard property of the diagonal class Δ퐴 that ∑ ±휀퐴(푎Δ퐴)Δ퐴 = 푎 (Δ퐴) for all 푎 ∈ 퐴, directly from Equation (2.1.18). Applied to 푎 = 휌(푦), it follows from the previous equation 휌(훼) = ±휌(푥푦); examining the signs, this summand cancels from the summand that comes from contracting the edge.

′ 0 Definition 2.4.4. The algebra fGC푅 is the quotient of fGC푅 by the ideal 퐼. ′ Note that fGC푅[푛] is also free as an algebra, with generators given by connected graphs with no univalent vertices nor bivalent vertices labeled by 1푅, and where the labels in 푅/ ker(휌) = 퐴.

Definition 2.4.5. Let fLoop ⊂ fGC0 be the sub-CDGA generated by graphs with 푅 푅 univalent vertices and by circular graphs (i.e. graphs of the type 푒12푒23 … 푒(푘−1)푘푒푘1).

68 2.4 From the model to forms via graphs

Lemma 2.4.6. The morphism Z ∶ fGC0 [푛] → ℝ vanishes on fLoop [푛]. 휑 푅 푅

Proof. The map Z휑 vanishes on graphs with univalent vertices by Corollary 2.3.34. The degree of a circular graph with 푘 vertices is −푘 < 0, but Z휑 can only be nonzero on graphs of degree zero.

Proposition 2.4.7. The sequence fLoop → fGC0 → fGC′ is a homotopy cofiber 푅 푅 푅 sequence.

Proof. The underlying algebra of fGC0 is a quasi-free extension of fLoop by the 푅 푅 algebra generated by graphs that are not circular and that do not contain any univalent vertices. The homotopy cofiber of the inclusion fLoop → fGC0 is this al- 푅 푅 gebra fGC″ , together with a differential induced by the quotient fGC0 /(fLoop ). 푅 푅 푅 ″ ′ We aim to prove that the induced morphism fGC푅 → fGC푅 is a quasi-isomorphism. ′ Let us define an increasing filtration on both CDGAs by letting 퐹푠fGC푅 (resp. ″ 퐹푠fGC푅) be the submodule spanned by graphs Γ such that #edges−#vertices ≤ 푠 (see also the proof of Proposition 2.4.17 where a similar technique is reused). The splitting part of the differential strictly decreases the filtration, so only 푑푅 and 푑contr remain on the first page of the associated spectral sequences. One can then filter by the number of edges. On the first page of the spectral sequence associated to this new filtration, there is only the internal differential ∗ ∗ 푑푅. Thus on the second page, the vertices are labeled by 퐻 (푅) = 퐻 (푀). The contracting part of the differential decreases the new filtration by exactly one, and so on the second page we see all of 푑contr. We can now adapt the proof of [Wil14, Proposition 3.4] to show that on the part of the complex with bivalent vertices, only the circular graphs contribute to the cohomology (we work dually so we consider a quotient instead of an ideal, but the idea is the same). To adapt the proof, one must see the labels of positive degree as formally adding one to the valence of the vertex, thus “breaking” a line of bivalent vertices. These label break the symmetry (recall the coinvariants in the definition of the twisting) that allow cohomology classes to be produced. ′ Corollary 2.4.8. The morphism fGC푅[푛] → ℝ factors through fGC푅[푛] up to homotopy. ′ ∗ 1 Let 휋 ∶ fGC푅 → fGC푅 be the quotient map. Let also Ω (Δ ) = 푆(푡, 푑푡) be the algebra of polynomials forms on Δ1, which is a path object for ℝ in the model category of CDGAs. ′ The CDGAs fGC푅[푛] and fGC푅[푛] are both quasi-free with a good filtration: ′ the generators are graphs (with some conditions for fGC푅[푛]), and the filtration is given by the number of edges. Therefore they are cofibrant as CDGAs. Thus there

69 2 The Lambrechts–Stanley Model of Configuration Spaces

′ ′ exists some morphism Z휑 ∶ fGC푅[푛] → ℝ and some homotopy ℎ ∶ fGC푅[푛] → Ω∗(Δ1) such that the following diagram commutes:

fGC푅[푛] ′ Z휑 Z휑휋 ℎ

푑1 ∗ 1 푑0 ℝ ∼ Ω (Δ ) ∼ ℝ

∗ 1 Definition 2.4.9. Let Ω (Δ )ℎ be the fGC푅[푛]-module induced by ℎ, and let Graphs′ (푈) = Ω∗(Δ1) ⊗ Tw Gra (푈). 푅 ℎ fGC푅[푛] 푅

Definition 2.4.10. Let Z휀 ∶ fGC푅[푛] → ℝ be the algebra morphism that sends a graph Γ with a single vertex labeled by 푥 ∈ 푅 to ∫ 휎(푥) = 휀(휌(푥)), and that 푀 sends all the other connected graphs to zero. Let ℝ휀 be the one-dimensional fGC푅[푛]-module induced by Z휀, and let Graphs휀 (푈) = ℝ ⊗ Tw Gra (푈). 푅 휀 fGC푅[푛] 푅 Graphs휀 Explicitly, in 푅, all internal components with at least two vertices are identified with zero, whereas an internal component with a single vertex labeled by 푥 ∈ 푅 is identified with the number ∫ 휎(푥) = 휀(휌(푥)). 푀 ′ Lemma 2.4.11. The morphism Z휑휋 is equal to Z휀. Proof. This is a rephrasing of Proposition 2.3.35. Using the same degree counting ′ argument, all the connected graphs with more than one vertex in fGC푅[푛] are ′ of positive degree. Since ℝ is concentrated in degree zero, Z휑휋 must vanish on these graphs, just like Z휀. ′ 0 0 Besides the morphism 휋 ∶ fGC푅 → fGC푅 = fGC푅/퐼 factors through fGC푅, where graphs 훾 with a single vertex are already identified with the numbers Z휑(훾) = Z휀(훾). Proposition 2.4.12. For each finite set 푈, we have a zigzag of quasi-isomorphisms of CDGAs: Graphs휀 ∼ Graphs′ ∼ Graphs휑 푅(푈) 푅(푈) 푅(푈). Graphs′ Graphs휀 Graphs If 휒(푀) = 0, then 푅 and 푅 are right Hopf 푛-comodules, and the zigzag defines a zigzag of Hopf right comodule morphisms. Proof. We have two commutative diagrams: Graphs휀 Graphs′ 푅(푈) 푅(푈) = = 1⊗푑 Tw Gra (푈) ⊗ ℝ 1 Tw Gra (푈) ⊗ Ω∗(Δ1) 푅 fGC푅[푛] 휀 푅 fGC푅[푛] ℎ

70 2.4 From the model to forms via graphs

Graphs′ Graphs휑 푅(푈) 푅(푈)

= = 1⊗푑 Tw Gra (푈) ⊗ Ω∗(Δ1) 0 Tw Gra (푈) ⊗ ℝ 푅 fGC푅[푛] ℎ 푅 fGC푅[푛] 휑

The action of fGC푅[푛] on Tw Gra푅(푈) is quasi-free with a good filtration, thus the functor Tw Gra (푈) ⊗ (−) preserves quasi-isomorphisms. The two 푅 fGC푅[푛] ∗ 1 maps 푑0, 푑1 ∶ Ω (Δ ) → ℝ are quasi-isomorphisms, therefore all the maps in the diagram are quasi-isomorphisms. Graphs′ Graphs휀 Graphs If 휒(푀) = 0, the proof that 푅 and 푅 assemble to 푛-como- 휑 dules is identical to the proof for Graphs푅 (see Proposition 2.3.31). It’s also clear Graphs that the two zigzags define morphisms of comodules: in 푛, all internal components are identified with zero anyway. ′ Proposition 2.4.13. The CDGA morphisms 휌∗ ∶ Gra푅(푈) → G퐴(푈) extend to CDGA Graphs휀 G morphisms 휌∗ ∶ 푅(푈) → 퐴(푈) by sending all the graphs containing internal vertices to zero. If 휒(푀) = 0 this extension defines a Hopf right comodule morphism. Proof. The submodule of graphs containing internal vertices is a multiplicative ideal and a cooperadic coideal, so all we are left to prove is that 휌∗ is compatible ′ with differentials. Since 휌∗ was a chain map, we must only prove that if Γ has internal vertices, then 휌∗(푑Γ) = 0. If a summand of 푑Γ still contains an internal vertex, then it is mapped to zero by definition of 휌∗. The only parts of the differential that can remove all internal vertices are the contraction of dead ends and the splitting off of an internal Graphs휀 component containing all the internal vertices of Γ. By the definition of 푅, splitting off an internal component can yield a nonzero graph only if that internal component contains a single vertex, i.e. if we are dealing with a dead end. Reusing the proof of Lemma 2.4.3, we can see that the contraction of that dead end cancels out with the splitting of that dead end after applying 휌∗. We thus get 휌∗(푑Γ) = 0 as expected.

2.4.2 The morphisms are quasi-isomorphisms In this section we prove that the morphisms constructed in Proposition 2.3.31 and Proposition 2.4.13 are quasi-isomorphisms, completing the proof of Theorem C. Theorem 2.4.14 (Precise version of Theorem C). The following zigzag, where the maps were constructed in Propositions 2.3.31, 2.4.12, and 2.4.13, is a zigzag of quasi- isomorphisms of CDGAs for all finite sets 푈:

G ∼ Graphs휀 ∼ Graphs′ ∼ Graphs휑 ∼ ∗ FM 퐴(푈) 푅(푈) 푅(푈) 푅(푈) ΩPA( 푀(푈)).

71 2 The Lambrechts–Stanley Model of Configuration Spaces

If 휒(푀) = 0, the left-pointing maps form a quasi-isomorphism of Hopf right comodules: G e∨ ∼ Graphs휀 Graphs ∼ Graphs′ Graphs ( 퐴, 푛 ) ( 푅, 푛) ( 푅, 푛). If 푀 is moreover framed, then the right-pointing maps also form a quasi-isomorphism of Hopf right comodules:

Graphs′ Graphs ∼ Graphs휑 Graphs ∼ ∗ FM ∗ FM ( 푅, 푛) ( 푅, 푛) (ΩPA( 푀), ΩPA( 푛)).

The rest of the section is dedicated to the proof of Theorem 2.4.14. Remark 2.4.15. The graph complexes are, in general, nonzero even in negative degrees; see Remark 2.1.1. Graphs휀 G Lemma 2.4.16. The morphisms 푅(푈) → 퐴(푈) factors through a quasi-iso- Graphs휀 Graphs Graphs morphism 푅(푈) → 퐴(푈), where 퐴(푈) is the CDGA obtained Graphs휀 by modding graphs with a label in ker(휌 ∶ 푅 → 퐴) in 푅(푈). Graphs휀 Graphs Proof. The morphism 푅 → 퐴 simply applies the surjective map Graphs휀 G 휌 ∶ 푅 → 퐴 to all the labels. It is clear that 푅 → 퐴 factors through the quotient. We can consider the spectral sequences associated to the filtrations of both Graphs휀 Graphs 푅 and 퐴 by the number of edges, and we obtain a morphism 0Graphs휀 0Graphs 0 E 푅 → E 퐴. On both E pages, only the internal differentials coming from 푅 and 퐴 remain. The chain map 푅 → 퐴 is a quasi-isomorphism, and so the morphism induces an isomorphism on the E1 page. By standard Graphs휀 Graphs spectral sequence arguments, it follows that 푅 → 퐴 is a quasi- isomorphism. Graphs Graphs휀 The CDGA 퐴(푈) has the same graphical description as 푅(푈), except that now vertices are labeled by elements of 퐴. An internal component with a single vertex labeled by 푎 ∈ 퐴 is identified with 휀(푎), and an internal component with more than one vertex is identified with zero. Graphs G Proposition 2.4.17. The morphism 퐴 → 퐴 is a quasi-isomorphism. Before starting to prove this proposition, let us outline the different steps. We filter our complex in such a way that on the E0 page, only the contracting part of the differential remains. Using a splitting result, we can focus on connected graphs. Finally, we use a “trick” (Figure 2.4.2) for moving labels around in a connected component, reducing ourselves to the case where only one vertex Graphs e∨ is labeled. We then get a chain map 퐴 ⊗ 푛 → 퐴 ⊗ 푛 (푈), which is a quasi-isomorphism thanks to the formality theorem.

72 2.4 From the model to forms via graphs

Let us start with the first part of the outlined program, removing the splitting Graphs part of the differential from the picture. Define an increasing filtration on 퐴 by counting the number of edges and vertices in a reduced graph: Graphs 퐹푠 퐴 = {Γ ∣ #edges − #vertices ≤ 푠}. Lemma 2.4.18. This is a filtration of chain complexes. It is bounded below for each finite Graphs set 푈: 퐹−#푈−1 퐴(푈) = 0. The E0 page of the spectral sequence associated to this previous filtration is isomorphic Graphs 0 ′ as a module to 퐴. Under this isomorphism the differential 푑 is equal to 푑퐴 +푑contr, ′ where 푑퐴 is the internal differential coming from 퐴 and 푑contr is the part of the differential that contracts all edges but dead ends. Graphs Proof. If Γ ∈ 퐴(푈) is the graph with no edges and no internal vertices, then it lives in filtration level −#푈. Adding edges can only increase the filtration. Since we consider reduced graphs (i.e. no internal components), each time we add an internal vertex (decreasing the filtration) we must add at least one edge (bringing it back up). By induction on the number of internal vertices, each graph is of filtration at least −#푈. Let us now prove that the differential preserves the filtration and check which parts remain on the associated graded complex. The internal differential 푑퐴 doesn’t change either the number of edges nor the number of vertices and so keeps the filtration constant. The contracting part 푑contr of the differential decreases both by exactly one, and so keeps the filtration constant too. The splitting part 푑split of the differential removes one edge. If the resulting graph is still connected, then nothing else changes and this decreases the filtration by exactly 1. Otherwise, it means that a whole internal component 훾 was connected to the rest of the graph by a single edge, and then split off and identified with a number. If 훾 has a single vertex labeled by 푎 (i.e. we split a dead end), then this number is 휀(푎), and the filtration is kept constant. Otherwise, the summand is zero (and so the filtration is obviously preserved). In all cases, the differential preserves the filtration, and so we get a filtered chain complex. On the associated graded complex, the only remaining parts of the differential are 푑퐴, 푑contr, and the part that splits off dead ends. But by the proof of Proposition 2.4.13 this last part cancels out with the part that contracts the dead ends.

The symmetric algebra 푆(휔푢푣)푢≠푣∈푈 is weight graded, which induces a weight ∨ ′ grading on e푛 (푈). This grading in turn induces an increasing filtration 퐹푠G퐴 on G퐴 (the extra differential strictly decreases the weight). Define a shifted filtration on G퐴 by: ′ 퐹푠G퐴(푈) = 퐹푠+#푈G퐴(푈).

73 2 The Lambrechts–Stanley Model of Configuration Spaces

0 Lemma 2.4.19. The E page of the spectral sequence associated to 퐹∗G퐴 is isomorphic as 0 a module to G퐴. Under this isomorphism the 푑 differential is just the internal differential of 퐴. Graphs G Lemma 2.4.20. The morphism 퐴 → 퐴 preserves the filtration and induces a chain map, for each 푈: 0Graphs 0G E 퐴(푈) → E 퐴(푈).

It maps graphs with internal vertices to zero, an edge 푒푢푣 between external vertices to 휔푢푣, and a label 푎 of an external vertex 푢 to 휄푢(푎). Graphs G Proof. The morphism 퐴(푈) → 퐴(푈) preserves the filtration by construction. If a graph has internal vertices, then its image in G퐴(푈) is of strictly lower filtration unless the graph is a forest (i.e. a product of trees). But treeshave Graphs G leaves, therefore by Corollary 2.3.34 and the formula defining 퐴 → 퐴 they are mapped to zero in G퐴(푈) anyway. It’s clear that the rest of the morphism preserves filtrations exactly, and so is given on the associated graded complex as stated in the lemma. Graphs 0Graphs For a partition 휋 of 푈, define the submodule 퐴⟨휋⟩ ⊂ E 퐴(푈) spanned by reduced graphs Γ such that the partition of 푈 induced by the con- Graphs nected components of Γ is exactly 휋. In particular let 퐴⟨푈⟩ be the submodule of connected graphs. Graphs Lemma 2.4.21. For each partition 휋 of 푈, 퐴⟨휋⟩ is a subcomplex, and the complex 0Graphs E 퐴(푈) splits as the sum over all partitions 휋 of 푈: 0Graphs Graphs E 퐴(푈) = ⨁ ⨂ 퐴⟨푉⟩. 휋 푉∈휋 Proof. Since there is no longer any part of the differential that can split off con- 0Graphs Graphs nected components in E 퐴, it is clear that 퐴⟨푈⟩ is a subcomplex. The splitting result is immediate. 0 The complex E G퐴(푈) splits in a similar fashion. For a monomial in 푆(휔푢푣), we say that 푢 and 푣 are “connected” if the term 휔푢푣 appears in the monomial. Consider the equivalence relation generated by “푢 and 푣 are connected”. The monomial induces in this way a partition 휋 of 푈, and this definition factors ∨ through the quotient defining e푛 (푈) (draw a picture of the 3-term relation). Finally, for a given monomial in G퐴(푈), the induced partition of 푈 is still well- defined. ∨ Thus for a given partition 휋 of 푈, we can define e푛 ⟨휋⟩ and G퐴⟨휋⟩ to be the ∨ 0 submodules of e푛 (푈) and E G퐴(푈) spanned by monomials inducing the partition

74 2.4 From the model to forms via graphs

∨ ∨ 휋. It is a standard fact that e푛 ⟨푈⟩ = Lie푛 (푈) [Sin07]. The proof of the following lemma is similar to the proof of the previous lemma:

0 Lemma 2.4.22. For each partition 휋 of 푈, G퐴⟨휋⟩ is a subcomplex of E G퐴(푈), and there is a splitting as the sum over all partitions 휋 of 푈:

0 E G퐴(푈) = ⨁ ⨂ G퐴⟨푉⟩. 휋 푉∈휋 0Graphs 0G Lemma 2.4.23. The morphism E 퐴(푈) → E 퐴(푈) preserves the splitting. We can now focus on connected graphs to prove Proposition 2.4.17.

∨ Lemma 2.4.24. The complex G퐴⟨푈⟩ is isomorphic to 퐴 ⊗ e푛 ⟨푈⟩. Proof. We define explicit isomorphisms in both directions. ⊗푈 ∨ ∨ Define 퐴 ⊗ e푛 ⟨푈⟩ → 퐴 ⊗ e푛 ⟨푈⟩ using the multiplication of 퐴. This induces 0 ∨ a map on the quotient E G퐴(푈) → 퐴 ⊗ e푛 ⟨푈⟩, which restricts to a map G퐴⟨푈⟩ → ∨ e푛 ⟨푈⟩. Since 푑퐴 is a derivation, this is a chain map. ∨ ⊗푈 ∨ Conversely, define 퐴 ⊗ e푛 ⟨푈⟩ → 퐴 ⊗ e푛 ⟨푈⟩ by 푎 ⊗ 푥 ↦ 휄푢(푎) ⊗ 푥 for some ∨ fixed 푢 ∈ 푈 (it does not matter which one since 푥 ∈ e푛 ⟨푈⟩ is “connected”). This ∨ induces a map 퐴 ⊗ e푛 ⟨푈⟩ → G퐴⟨푈⟩, and it is straightforward to check that this map is the inverse isomorphism of the previous map.

We have a commutative diagram of complexes:

Graphs Graphs′ 퐴⟨푈⟩ 퐴 ⊗ 푛⟨푈⟩ ∼

≅ ∨ G퐴⟨푈⟩ 퐴 ⊗ e푛 ⟨푈⟩

Graphs′ Graphs Here 푛(푈) is defined similarly to 푛(푈) except that multiple edges Graphs′ e∨ are allowed. It is known that the quotient map 푛(푈) → 푛 (푈) (which fac- Graphs tors through 푛(푈)) is a quasi-isomorphism [Wil14, Proposition 3.9]. The Graphs′ subcomplex 푛⟨푈⟩ is spanned by connected graphs. The upper horizontal map in the diagram multiplies all the labels of a graph. Graphs ∼ e∨ The right vertical map is the tensor product of id퐴 and 푛⟨푈⟩ 푛 ⟨푈⟩ (see 2.1.3). The bottom row is the isomorphism of the previous lemma. Graphs Graphs′ It then remains to prove that 퐴⟨푈⟩ → 퐴 ⊗ 푛⟨푈⟩ is a quasi-iso- Graphs′ G morphism to prove Proposition 2.4.17. If 푈 = ∅, then 퐴(∅) = ℝ = 퐴(∅) and the morphism is the identity, so there is nothing to do. From now on we assume that #푈 ≥ 1.

75 2 The Lambrechts–Stanley Model of Configuration Spaces

Graphs Graphs′ Lemma 2.4.25. The morphism 퐴⟨푈⟩ → 퐴 ⊗ 푛⟨푈⟩ is surjective on cohomology. Proof. Choose some 푢 ∈ 푈. There is an explicit chain-level section of the morphism, sending 푥 ⊗ Γ to Γ푢,푥, the same graph with the vertex 푢 labeled by 푥 and all the other vertices labeled by 1푅. It is a well-defined chain map. It is clear that this is a section of the morphism in the lemma, hence the morphism of the lemma is surjective on cohomology. We now use a proof technique similar to the proof of [LV14, Lemma 8.3], ∗ Graphs′ e∨ Lie∨ working by induction. The dimension of 퐻 ( 푛⟨푈⟩) = 푛 ⟨푈⟩ = 푛 (푈) is well-known: ⎧(#푈 − 1)!, if 푖 = (푛 − 1)(#푈 − 1); 푖 Graphs′ { dim 퐻 ( 푛⟨푈⟩) = ⎨ (2.4.26) ⎩{0, otherwise. 푖 Graphs Lemma 2.4.27. For all sets 푈 with #푈 ≥ 1, the dimension of 퐻 ( 퐴⟨푈⟩) is the same as the dimension: 푖 Graphs′ 푖−(푛−1)(#푈−1) dim 퐻 (퐴 ⊗ 푛⟨푈⟩) = (#푈 − 1)! ⋅ dim 퐻 (퐴). The proof will be by induction on the cardinality of 푈. Graphs Lemma 2.4.28. The complex 퐴⟨1⟩ has the same cohomology as 퐴. Proof. Let ℐ be the subcomplex spanned by graphs with at least one internal ℐ Graphs ℐ vertex. We will show that is acyclic; as 퐴⟨1⟩/ ≅ 퐴, this will prove the lemma. There is an explicit homotopy ℎ that shows that ℐ is acyclic. Given a graph Γ with a single external vertex and at least one internal vertex, define ℎ(Γ) to be the same graph with the external vertex replaced by an internal vertex, a new external vertex labeled by 1퐴, and an edge connecting the external vertex to the new internal vertex (see Figure 2.4.1). Graphs The differential in 퐴⟨1⟩ only retains the internal differential of 퐴 and the contracting part of the differential. Contracting the new edge in ℎ(Γ) gives back Γ, and it is now straightforward to check that: 푑ℎ(Γ) = Γ ± ℎ(푑Γ). Now let 푈 be a set with at least two elements, and fix some element 푢 ∈ 푈. Let Graphs푢 Graphs 퐴⟨푈⟩ ⊂ 퐴⟨푈⟩ be the subcomplex spanned by graphs Γ such that 푢 has valence 1, is labeled by 1퐴, and is connected to another external vertex. We now get to the core of the proof. The idea (adapted from [LV14, Lemma 8.3]) is to “push” the labels of positive degree away from the chosen vertex 푢 through a homotopy. Roughly speaking, we use Figure 2.4.2 to move labels around up to homotopy.

76 2.4 From the model to forms via graphs

1 푥 퐴 푥 푢 ↦ 푢

Figure 2.4.1: The homotopy ℎ

푥 푥 푥 푑contr ( ) = −

Figure 2.4.2: Trick for moving labels around (gray vertices are either internal or external)

Graphs푢 Graphs Lemma 2.4.29. The inclusion 퐴⟨푈⟩ ⊂ 퐴⟨푈⟩ is a quasi-isomorphism.

Proof. Let 풬 be the quotient. We will prove that it is acyclic. The module 풬 further decomposes into a direct sum of modules (but the differential does not preserve the direct sum):

• The module 풬1 spanned by graphs where 푢 is of valence 1, labeled by 1퐴, and connected to an internal vertex;

• The module 풬2 spanned by graphs where 푢 is of valence ≥ 2 or has a label in 퐴>0.

We now filter 풬 as follows. For 푠 ∈ ℤ, let 퐹푠풬1 be the submodule of 풬1 spanned by graphs with at most 푠 + 1 edges, and let 퐹푠풬2 be the submodule spanned by graphs with at most 푠 edges. This filtration is preserved by the differential of 풬. Consider the E0 page of the spectral sequence associated to this filtration. Then 0 0 0 the differential 푑 is an morphism E 풬1 → E 풬2 by counting the number of edges (and using the crucial fact that dead ends are not contractible). It contracts the only edge incident to 푢. It is an isomorphism, with an inverse similar to the homotopy defined in Lemma 2.4.28, “blowing up” the point 푢 into a new edge connecting 푢 to a new internal vertex that replaces 푢. This shows that (E0풬, 푑0) is acyclic, hence E1풬 = 0. It follows that 풬 is acyclic.

77 2 The Lambrechts–Stanley Model of Configuration Spaces

Proof of Lemma 2.4.27. The case #푈 = 0 is obvious, and the case #푈 = 1 of the lemma was covered in Lemma 2.4.28. We now work by induction and assumes the claim proved for #푈 ≤ 푘, for some 푘 ≥ 1. Graphs푢 Let 푈 be of cardinality 푘 + 1. Choose some 푢 ∈ 푈 and define 퐴⟨푈⟩ as before. By Lemma 2.4.29 we only need to show that this complex has the right cohomology. It splits as:

Graphs푢 Graphs 퐴⟨푈⟩ ≅ ⨁ 푒푢푣 ⋅ 퐴⟨푈 − {푢}⟩ (2.4.30) 푣∈푈−{푢}

And therefore using the induction hypothesis:

푖 Graphs푢 푖−(푛−1) Graphs dim 퐻 ( 퐴⟨푈⟩) = 푘 ⋅ dim 퐻 ( 퐴⟨푈 − {푢}⟩) = 푘! ⋅ dim 퐻푖−푘(푛−1)(퐴).

Graphs Proof of Proposition 2.4.17. By Lemma 2.4.25, the morphism induced by 퐴 → 0 G퐴 on the E page is surjective on cohomology. By Lemma 2.4.27 and Equa- tion (2.4.26), both E0 pages have the same cohomology, and so the induced morphism is a quasi-isomorphism. Standard spectral arguments imply the proposition.

Graphs′ ∗ FM Proposition 2.4.31. The morphism 휔 ∶ 푅(푈) → ΩPA( 푀(푈)) is a quasi-isomorphism.

Proof. From Equation (2.1.20), Proposition 2.4.12, Lemma 2.4.16, and Proposi- tion 2.4.17, both CDGAs have the same cohomology of finite type, so it will suffice to show that the map is surjective on cohomology to prove thatitisa quasi-isomorphism. We work by induction. The case 푈 = ∅ is immediate, as we get

Graphs′ ∼ Graphs휑 ∗ FM 푅(∅) 푅(∅) = ΩPA( 푀(∅)) = ℝ and the last map is the identity. Suppose that 푈 = {푢} is a singleton. Since 휌 is a quasi-isomorphism, for every ∗ ∗ cocycle 훼 ∈ ΩPA(FM푀(푈)) = ΩPA(푀) there is some cocycle 푥 ∈ 푅 such that 휌(푥) is cohomologous to 훼. Then the graph 훾푥 with a single (external) vertex labeled Graphs′ by 푥 is a cocycle in 푅(푈), and 휔(훾푥) = 휌(푥) is cohomologous to 훼. This Graphs′ ∗ proves that 푅({푢}) → ΩPA(푀) is surjective on cohomology, and hence a quasi-isomorphism. Now assume that 푈 = {푢} ⊔ 푉, where #푉 ≥ 1, and assume that the claim is proved for sets of vertices of size at most #푉 = #푈 − 1. By Equation (2.1.20),

78 2.4 From the model to forms via graphs

we may represent any cohomology class of FM푀(푈) by an element 푧 ∈ G퐴(푈) satisfying 푑푧 = 0. Using the relations defining G퐴(푈), we may write 푧 as

′ 푧 = 푧 + ∑ 휔푢푣푧푣, 푣∈푉

′ where 푧 ∈ 퐴 ⊗ G퐴(푉) and 푧푣 ∈ G퐴(푉). The relation 푑푧 = 0 is equivalent to

′ ∗ 푑푧 + ∑ (푝푢 × 푝푣) (Δ퐴) ⋅ 푧푣 = 0, (2.4.32) 푣∈푉

and 푑푧푣 = 0 for all 푣. (2.4.33) Graphs′ By the induction hypothesis, for all 푣 ∈ 푉 there exists a cocycle 훾푣 ∈ 푅(푉) ∗ such that 휔(훾푣) represents the cohomology class of the cocycle 푧푣 in 퐻 (FM푀(푉)), and such that 휎∗(훾푣) is equal to 푧푣 up to a coboundary. By Equation (2.4.32), the cocycle

∗ Graphs′ ̃훾= ∑ (푝푢 × 푝푣) (Δ푅) ⋅ 훾푣 ∈ 푅 ⊗ 푅(푉) 푣∈푉 G Graphs′ is mapped to a coboundary in 퐴 ⊗ 퐴(푉). The map 휎∗ ∶ 푅 ⊗ 푅(푉) → 퐴 ⊗ G퐴(푉) is a quasi-isomorphism, hence ̃훾= 푑 ̃훾1 is a coboundary too. ′ It follows that 푧 − 휎∗( ̃훾1) ∈ 퐴 ⊗ G퐴(푉) is a cocycle. Thus by the induction Graphs′ hypothesis there exists some ̃훾2 ∈ 푅 ⊗ 푅(푉) whose cohomology class ′ ∗ ∗ represents the same cohomology class as 푧 −휎∗( ̃훾1) in 퐻 (퐴⊗G퐴(푉)) = 퐻 (푀× FM푀(푉)). ′ ′ ′ ′ We now let 훾 = − ̃훾1 + ̃훾2, hence 푑훾 = − ̃훾+ 0 = − ̃훾 and 휎∗(훾 ) is equal to 푧 up to a coboundary. By abuse of notation we still let 훾′ be the image of 훾′ under Graphs′ Graphs′ the obvious map 푅 ⊗ 푅(푉) → 푅(푈), 푥 ⊗ Γ ↦ 휄푢(푥) ⋅ Γ. Then

′ Graphs′ 훾 = 훾 + ∑ 푒푢푣 ⋅ 훾푣 ∈ 푅(푈) 푣∈푉 ∗ is a cocycle, and 휔(훾) represents the cohomology class of 푧 in ΩPA(FM푀(푈)). Graphs′ ∗ FM We’ve shown that the morphism 푅(푈) → ΩPA( 푀(푈)) is surjective on cohomology, and hence it is a quasi-isomorphism.

Proposition 2.4.34. If 퐴′ is another Poincaré duality model of 푀, then we have a weak equivalence of symmetric collections G퐴 ≃ G퐴′ . If moreover 휒(푀) = 0 then this weak ∨ equivalence is a weak equivalence of right Hopf e푛 -comodules. Proof. The CDGAs 퐴 and 퐴′ are models of the manifold, hence there exists some cofibrant CDGA 푆 and quasi-isomorphisms 푓 ∶ 푆 ∼ 퐴 and 푓 ′ ∶ 푆 ∼ 퐴′. This

79 2 The Lambrechts–Stanley Model of Configuration Spaces

′ ′ ′ yields two chain maps 휀, 휀 ∶ 푆 → ℝ[−푛] defined by 휀 = 휀퐴 ∘ 푓 and 휀 = 휀퐴′ ∘ 푓 . Mimicking the proof of Proposition 2.3.4, we can also find cocycles Δ,Δ′ ∈ 푆 ⊗ 푆 which satisfy Δ21 = (−1)푛Δ (and the same equation for Δ′) and mapped ′ respectively to Δ퐴 and Δ퐴′ under 푓 and 푓 . 휀,Δ 휀′,Δ′ We can then build symmetric collections Graphs푆 and Graphs푆 and quasi- 휀,Δ ′ 휀′,Δ′ isomorphisms 푓∗ ∶ Graphs푆 → G퐴 and 푓∗ ∶ Graphs푆 → G퐴′ like before. The 휀,Δ 휀′,Δ′ ′ differential of an edge 푒푢푣 in Graphs푆 (resp. Graphs푆 is 휄푢푣(Δ) (resp. 휄푢푣(Δ )), and an isolated internal vertex labeled by 푥 ∈ 푆 is identified with 휀(푥) (resp. 휀′(푥)). If moreover 휒(푀) = 0, we can choose Δ and Δ′ such that their product vanishes, Graphs Graphs thus both 푆 become right Hopf 푛-comodules and the projections ′ 푓∗, 푓∗ are compatible with the comodule structure. It thus suffices to prove that 휀,Δ 휀′,Δ′ we have a quasi-isomorphism between Graphs푆 and Graphs푆 to prove the proposition. 휀′,Δ′ 휀′,Δ We first have an isomorphism Graphs푆 ≅ Graphs푆 (with the obvious notations). Indeed, the two cocycles Δ and Δ′ are cohomologous, say Δ′ = Δ − 푑훼 for some 훼 ∈ 푆 ⊗ 푆 of degree 푛 − 1. If we replace 훼 by (훼 + (−1)푛훼21)/2, we can assume that 훼21 = (−1)푛훼. Moreover if 휒(푀) = 0, one can replace 훼 by 푛 훼 − (휇푆(훼) ⊗ 1 + (−1) 1 ⊗ 휇푆(훼))/2 to be able to assume that 휇푆(훼) = 0. We then obtain an isomorphism by mapping an edge 푒푢푣 to 푒푢푣 ± 휄푢푣(훼) (the sign depending on the direction of the isomorphism). This is clearly compatible with differentials, and if 휒(푀) = 0 with the comodule structure. The dg-module 푆 is cofibrant and ℝ[−푛] is fibrant (like all dg-modules). More- over we can assume that 휀 and 휀′ induce the same map on cohomology (it suffices ′ 휀′,Δ to rescale, say, 휀 , and there is an automorphism of Graphs푆 which takes care of this rescaling). Thus the two maps 휀, 휀′ ∶ 푆 → ℝ[−푛] are homotopic, and there exists some ℎ ∶ 푆[1] → ℝ[−푛] such that 휀(푥) − 휀′(푥) = ℎ(푑푥) for all 푥 ∈ 푆.

This homotopy induces a homotopy between the two morphisms Z휀, Z휀′ ∶ Δ Δ′ fGC푆[푛] → ℝ. Because Tw Gra푆 (푈) and Tw Gra푆 (푈) are cofibrant as modules over fGC푆[푛], we obtain quasi-isomorphisms

휀,Δ 휀′,Δ Graphs푆 ≃ Graphs푆 which are compatible with the comodule structure when 휒(푀) = 0 (see the proof of Proposition 2.4.12).

Proof of Theorem 2.4.14. The zigzag of the theorem becomes, after factorizing the

80 2.4 From the model to forms via graphs

Graphs first map through 퐴: Graphs Graphs휀 Graphs′ Graphs휀 퐴(푈) 푅(푈) 푅(푈) 푅(푈)

∗ G퐴(푈) ΩPA(FM푀(푈)) All these maps are quasi-isomorphisms by Lemma 2.4.16, Proposition 2.4.12, Proposition 2.4.17, and Proposition 2.4.31. Their compatibility with the comodule structures (under the relevant hypotheses) are due to Proposition 2.3.31, Proposition 2.4.12, and Proposition 2.4.13. The only thing missing from the theorem is that we must be able to choose any Poincaré duality model of the manifold, and not necessarily 퐴; this is the content of Proposition 2.4.34. We can thus settle the conjecture of Lambrechts–Stanley over ℝ for the class of manifolds that we consider:

Corollary 2.4.35. Let 푀 be a smooth simply connected closed manifold of dimension at least 4, and let 퐴 be any Poincaré duality model of 푀. Then the CDGA G퐴(푘) is a real model for Conf푘(푀). And this implies the real homotopy invariance of configuration spaces:

Corollary 2.4.36. The real homotopy type of the configuration space of a smooth simply connected closed manifold of dimension at least 4 only depends on the real homotopy type of 푀.

Proof. When dim 푀 ≥ 3, the Fadell–Neuwirth fibrations [FN62] Conf푘−1(푀 − ∗) ↪ Conf푘(푀) → 푀 show by induction that if 푀 is simply connected, then so is Conf푘(푀) for all 푘 ≥ 1. Hence the real model G퐴(푘) completely encodes the real homotopy type of Conf푘(푀). The degree-counting argument of Proposition 2.3.35 does not work in dimension 3, and so we cannot directly apply our results to 푆3 (the only simply connected 3-manifold). We however record the following partial result, communi- cated to us by Thomas Willwacher:

∗ 3 Proposition 2.4.37. The CDGA G퐴(푘), where 퐴 = 퐻 (푆 ; ℚ), is a rational model of 3 Conf푘(푆 ) for all 푘 ≥ 0. Proof. The claim is clear for 푘 = 0. Since 푆3 is a Lie group, the Fadell–Neuwirth fibration 3 3 3 Conf푘(ℝ ) ↪ Conf푘+1(푆 ) → 푆

81 2 The Lambrechts–Stanley Model of Configuration Spaces

3 is trivial [FN62, Theorem 4]. The space Conf푘+1(푆 ) is thus identified with 3 3 ∗ 3 ∨ 푆 × Conf푘(ℝ ), which is rationally formal with cohomology 퐻 (푆 ) ⊗ e3 (푘). It ∗ ∨ thus suffices to build a quasi-isomorphism between G퐴(푘 + 1) and 퐻 (푆3)⊗e푛 (푘) to prove the proposition. G (푘 ) 푘 = {0, … , 푘} To simplify notation, we consider 퐴 + (where + ), which is 3 3 3 obviously isomorphic to G퐴(푘 + 1). Let us denote by 휐 ∈ 퐻 (푆 ) = 퐴 the 3 volume form of 푆 , and recall that the diagonal class Δ퐴 is given by 1 ⊗ 휐 − 휐 ⊗ 1. We have an explicit map given on generators by:

푓 ∶ 퐻∗(푆3) ⊗ e∨(푘) → G (푘 ) 3 퐴 +

휐 ⊗ 1 ↦ 휄0(휐) 1 ⊗ 휔푖푗 ↦ 휔푖푗 + 휔0푖 − 휔0푗

The Arnold relations show that this is a well-defined algebra morphism. Let us prove that 푑 ∘ 푓 = 0 on the generator 휔푖푗 (the vanishing on 휐 ⊗ 1 is clear). We may assume that 푘 = 2 and (푖, 푗) = (1, 2), and then apply 휄푖푗 to get the general case. Then we have:

푑푓 (휔12) = (1 ⊗ 1 ⊗ 휐 − 1 ⊗ 휐 ⊗ 1) + (1 ⊗ 휐 ⊗ 1 − 휐 ⊗ 1 ⊗ 1) − (1 ⊗ 1 ⊗ 휐 − 휐 ⊗ 1 ⊗ 1) = 0

We know that both CDGAs have the same cohomology (Equation (2.1.20)), so to check that 푓 is a quasi-isomorphism it suffices to check that it is surjective 퐻∗(G (푘 )) ≅ 퐻∗(푆3) ⊗ e∨(푘) in cohomology. The cohomology 퐴 + 3 is generated in degrees 2 (by the 휔푖푗) and 3 (by the 휄푖(휐)), so it suffices to check surjectivity in these degrees. 3 휐 ⊗ 1 퐻3(G (푘 )) ≅ 퐻3(푆3) = In degree , the cocycle is sent to a generator of 퐴 + ℚ. Indeed, assume 휄0(휐) = 푑휔, where 휔 is a linear combination of the 휔푖푗 for degree reasons. In 푑휔, the sum of the coefficients of each 휄푖(휐) is zero, because they all come in pairs (푑휔푖푗 = 휄푗(휐) − 휄푖(휐)). We want the coefficient of 휄0(휐) to be 1, so at least one of the other coefficient must be nonzero to compensate, hence 푑휔 ≠ 휄0(휐). It remains to prove that 퐻2(푓 ) is surjective. We consider the quotient map 푝 ∶ G (푘 ) → e∨(푘) 휄 (휐) 휔 1 ≤ 푖 ≤ 푘 퐴 + 3 that maps 푖 and 0푖 to zero for all . We also ∗ 3 ∨ ∨ consider the quotient map 푞 ∶ 퐻 (푆 ) ⊗ e3 (푘) → e3 (푘) sending 휐 ⊗ 1 to zero. We

82 2.5 Factorization homology of universal enveloping 퐸푛-algebras get a morphism of short exact sequences:

∗ 3 ∨ 푞 ∨ 0 ker 푞 퐻 (푆 ) ⊗ e3 (푘) e3 (푘) 0

푓 =

푝 ∨ 0 ker 푝 G퐴(푘) e3 (푘) 0

We consider part of the long exact sequence in cohomology induced by these short exact sequences of complexes:

∨ 1 2 2 ∗ 3 ∨ ∨ 2 ∨ 2 e3 (푘) 퐻 (ker 푞) 퐻 (퐻 (푆 ) ⊗ e3 (푘)) = e3 (푘) e3 (푘)

= (1) 퐻2(푓 ) =

e∨(푘)1 퐻2( 푝) 퐻2(G (푘 )) e∨(푘)2 3 ker 퐴 + 3

For degree reasons, 퐻2(ker 푞) = 0 and so the map (1) is injective. By the four lemma, it follows that 퐻2(푓 ) is injective too. Both its domain and its codomain have the same finite dimension, thus 퐻2(푓 ) is an isomorphism.

2.5 Factorization homology of universal

enveloping 퐸푛-algebras 2.5.1 Factorization homology and formality Fix some dimension 푛. Let 푈 be a finite set and consider the space of framed embeddings of 푈 copies of ℝ푛 in itself, endowed with the compact open topology:

fr fr 푛 푛 푛 푛 Disk푛 (푈) ≔ Emb (ℝ × 푈, ℝ ) ⊂ Map(ℝ × 푈, ℝ ). (2.5.1) Using composition of embeddings, these spaces assemble to form a topo- fr logical operad Disk푛 . This operad is weakly equivalent to the operad of little fr 푛-disks [AF15, Remark 2.10], and the application that takes 푓 ∈ Disk푛 (푈) to 푛 {푓 (0 × 푢)}푢∈푈 ∈ Conf푈(ℝ ) is a homotopy equivalence. Similarly if 푀 is a framed manifold, then the spaces Embfr(ℝ푛 × −, 푀) assem- fr ble to form a topological right Disk푛 -module, again given by composition of fr fr embeddings. We call it Disk푀. If 퐵 is a Disk푛 -algebra, factorization homology is given by a derived composition product [AF15, Definition 3.2]:

fr 핃 fr fr fr ∫ 퐵 ≔ Disk푀 ∘ fr 퐵 = hocoeq(Disk푀 ∘ Disk푛 ∘ 퐵 ⇉ Disk푀 ∘ 퐵). (2.5.2) 푀 Disk푛

83 2 The Lambrechts–Stanley Model of Configuration Spaces

Thanks to [Tur13, Section 2], the pair (FM푀, FM푛) is weakly equivalent to the fr fr pair (Disk푀, Disk푛 ). So if 퐵 is an FM푛-algebra, its factorization homology can be computed as:

핃 ∫ 퐵 ≃ FM푀 ∘FM 퐵 = hocoeq(FM푀 ∘ FM푛 ∘ 퐵 ⇉ FM푀 ∘ 퐵). (2.5.3) 푀 푛 We now work in the category of chain complexes over ℝ. We use the formality theorem (Equation (2.1.15)) and the fact that weak equivalences of operads induce Quillen equivalence between categories of right modules (resp. categories of algebras) [Fre09, Theorems 16.A and 16.B]. Thus, to any homotopy class [퐵] of 퐸푛-algebras in the category of chain complexes, there corresponds a homotopy ̃ class [퐵] of e푛-algebras (which is generally not easy to describe). Using Theorem 2.4.14, a game of adjunctions [Fre09, Theorems 15.1.A and 15.2.A] shows that: ∨ 핃 ∫ 퐵 ≃ G ∘e 퐵,̃ (2.5.4) 푀 퐴 푛 ∨ where 퐴 is the Poincaré duality model of 푀 mentioned in the theorem, and G퐴 is the right e푛-module dual to G퐴. Note that we forget the Hopf structure of G퐴 in Equation 2.5.4.

2.5.2 Higher enveloping algebras

Consider the forgetful functor from nonunital 퐸푛-algebras to homotopy Lie algebras. Knudsen [Knu16, Theorem A] constructs a left adjoint 푈푛 to this forgetful functor, called the higher enveloping algebra functor. He also gives a way of computing factorization homology of higher enveloping algebras If 픤 is a Lie algebra, then so is 퐴 ⊗ 픤 for any CDGA 퐴. Then the factorization homology of 푈푛(픤) on 푀 is given by [Knu17, Theorem 3.16].:

CE −∗ ∫ 푈푛(픤) ≃ C∗ (A (푀) ⊗ 픤) (2.5.5) 푀 PL

CE −∗ where C∗ is the Chevalley–Eilenberg complex and APL (푀) is the CDGA of rational piecewise polynomial differential forms, with the usual grading reversed to obtain a homological grading. Let 푛 be at least 2. We use again the formality theorem and standard facts about Quillen equivalences induced by weak equivalences of operads. Consider the ∼ canonical resolution hoLie푛 Lie푛, the standard inclusion Lie푛 ⊂ e푛, and the Lie Graphs∨ maps of operads 푛 → 푛 which sends the generator (the bracket) to the graph with two external vertex and an edge between the two (see the part about twisting in Section 2.1.3). We can find a morphism of operads hoLie푛 → 퐶∗(FM푛)

84 2.5 Factorization homology of universal enveloping 퐸푛-algebras which makes the following diagram commute (the square on the right only up to homotopy): e ∼ Graphs∨ ∼ FM 푛 푛 퐶∗( 푛)

(ℎ) ∃ = ∼ Lie푛 Lie푛 hoLie푛 These morphisms of operads yield functors between the corresponding categories of algebras. The forgetful functor 푈푛 in Knudsen’s work corresponds (in an informal sense) to the right adjoint (obtained by a Kan extension) of the induced functor 퐶∗(FM푛)-Alg → hoLie-Alg which is the composite of the functor induced by the morphism of operads and a shift in degree. The inclusion Lie푛 ⊂ e푛 induces a functor e푛-Alg → Lie-Alg that maps an (푛 − 1)-Poisson algebra 퐵 to its underlying Lie algebra 퐵[1 − 푛]. Its left ̃ adjoint 푈푛 corresponds to 푈푛, thanks to the commutative diagram above; in other words, given a Lie algebra 픤, which we can also see as an hoLie-algebra, ̃ ̃ we have 푈푛(픤) ≃ 푈푛(픤). It is given by a symmetric algebra 푈푛(픤) = 푆(픤[푛 − 1]), with a shifted Lie bracket is defined using the Leibniz rule.

Proposition 2.5.6. Let 퐴 be a Poincaré duality CDGA. Then we have a quasi-isomorphism of chain complexes:

G∨ ∘핃 푆(픤[푛 − 1]) ∼ CCE(퐴−∗ ⊗ 픤). 퐴 e푛 ∗

∗ ∗ If 퐴 is a Poincaré duality model of 푀, we have 퐴 ≃ ΩPA(푀) ≃ APL(푀) ⊗ℚ ℝ [Har+11, Theorem 6.1]. It follows that the Chevalley–Eilenberg complex of the previous proposition is weakly equivalent to the Chevalley–Eilenberg complex of Equation (2.5.5). By Equation (2.5.3), the derived circle product over e푛 computes the factorization homology of 푈푛(픤) on 푀, and so we recover a variant of Knudsen’s theorem (over the reals) for closed framed simply connected manifolds. Let I be the unit of the composition product, defined by I(1) = ℝ and I(푈) = 0 for #푈 ≠ 1. Let Λ be the suspension of operads, satisfying

ΛP ∘ (푋[−1]) = (P ∘ 푋)[−1] = I[−1] ∘ (P ∘ 푋).

As as symmetric collection, ΛP is simply given by ΛP = I[−1] ∘ P ∘ I[1]. Recall 1−푛 that we let Lie푛 = Λ Lie. The symmetric collection

L푛 ≔ Lie ∘ I[1 − 푛] = I[1 − 푛] ∘ Lie푛 (2.5.7)

85 2 The Lambrechts–Stanley Model of Configuration Spaces

is a (Lie, Lie푛)-bimodule, i.e. a Lie-algebra in the category of Lie푛-right modules. We have L푛(푈) = (Lie푛(푈))[1 − 푛]. This bimodule satisfies, for any Lie algebra 픤, L ∘ 픤[푛 − 1] ≅ 픤 as Lie algebras. (2.5.8) 푛 Lie푛 We can view the CDGA 퐴−∗ as a symmetric collection concentrated in arity 0, and as such it is a commutative algebra in the category of symmetric collections. Thus the tensor product −∗ −∗ 퐴 ⊗ L푛 = {퐴 ⊗ L푛(푘)}푘≥0 becomes a Lie-algebra in right Lie푛-modules, where the right Lie푛-module structure comes from L푛 and the Lie algebra structure combines the Lie algebra −∗ structure of L푛 and the CDGA structure of 퐴 . Its Chevalley–Eilenberg complex CE −∗ CE C∗ (퐴 ⊗ L푛) is well-defined, and by functoriality of C∗ , it is a right Lie푛- module. The proof of the following lemma is essentially found (in a different language) in the work of Félix and Thomas [FT04, Section 2]. ∨ CE −∗ Lemma 2.5.9. The right Lie푛-modules G퐴 and C∗ (퐴 ⊗ L푛) are isomorphic. Proof. We will actually define a non-degenerate pairing CE −∗ ⟨−, −⟩ ∶ G퐴(푈) ⊗ C∗ (퐴 ⊗ L푛)(푈) → ℝ for each finite set 푈, compatible with differentials and the right Lie푛-(co)module structures. As both complexes are finite-dimensional in each degree, this is sufficient to prove that they are isomorphic. CE Recall that the Chevalley–Eilenberg complex C∗ (픤) is given by the cofree cocommutative conilpotent coalgebra 푆푐(픤[−1]), together with a differential induced by the Koszul duality morphism Λ−1Com∨ → Lie. It follows that as a CE −∗ module, C∗ (퐴 ⊗ L푛)(푈) is given by:

Σ푟 ⎛ −∗ −∗ ⎞ ⨁ ⎜ ⨁ 퐴 ⊗ L푛(푈1)[−1] ⊗ ⋯ ⊗ 퐴 ⊗ L푛(푈푟)[−1]⎟ 푟≥0 ⎝휋∈Part푟(푈) ⎠ Σ푟 ⎛ 푛−∗ ⊗푟 ⎞ = ⨁ ⎜ ⨁ (퐴 ) ⊗ Lie푛(푈1) ⊗ ⋯ ⊗ Lie푛(푈푟)⎟ (2.5.10) 푟≥0 ⎝휋∈Part푟(푈) ⎠ 푛−∗ where the sums run over all partitions 휋 = {푈1 ⊔ ⋯ ⊔ 푈푟} of 푈 and 퐴 = 퐴−∗[−푛] (which is a CDGA, Poincaré dual to 퐴). Fix some 푟 ≥ 0 and some partition 휋 = {푈1 ⊔⋯⊔푈푟}. We define a first pairing: ⊗푈 ∨ 푛−∗ ⊗푟 (퐴 ⊗ e푛 (푈))⊗((퐴 ) ⊗ Lie푛(푈1) ⊗ ⋯ ⊗ Lie푛(푈푟)) → ℝ (2.5.11) as follows:

86 2.5 Factorization homology of universal enveloping 퐸푛-algebras

• On the 퐴 factors, the pairing uses the Poincaré duality pairing 휀퐴. It is given by:

(푎 ) ⊗ (푎′ ⊗ ⋯ ⊗ 푎′ ) ↦ ±휀 (푎 ⋅ 푎′ ) … 휀 (푎 ⋅ 푎′ ), 푢 푢∈푈 1 푟 퐴 푈1 1 퐴 푈푟 푟

where 푎푈 = ∏ 푎푢. 푖 푢∈푈푖 e∨ 푟 Lie e∨ • On the factor 푛 (푈)⊗⨂푖=1 푛(푈푖), it uses the duality pairing on 푛 (푈)⊗ e e Com Lie 푟 Lie 푛(푈) (recalling that 푛 = ∘ 푛 so we can view ⨂푖=1 푛(푈푖) as a submodule of e푛(푈)).

The pairing in Equation (2.5.11) is the product of the two pairings we just ⊗푈 ∨ CE −∗ defined. It is extended linearly on allof (퐴 ⊗ e푛 (푈)) ⊗ C∗ (퐴 ⊗ L푛)(푈), ⊗푈 ∨ and it factors through the quotient defining G퐴(푈) from 퐴 ⊗ e푛 (푈). To check the non-degeneracy of this pairing, recall the vector subspaces G퐴⟨휋⟩ of Lemma 2.4.22, which are well-defined even though they are not preserved by 0 the differential if we do not consider the graded space E G퐴. Fix some partition 휋 = {푈1, … , 푈푟} of 푈, then we have an isomorphism of vector spaces:

⊗푟 ∨ ∨ G퐴⟨휋⟩ ≅ 퐴 ⊗ Lie푛 (푈1) ⊗ ⋯ ⊗ 퐴 ⊗ Lie푛 (푈푟).

It is clear that G퐴⟨휋⟩ is paired with the factor corresponding to 휋 in Equa- tion (2.5.10), using the Poincaré duality pairing of 퐴 and the pairing between Lie푛 and its dual; and if two elements correspond to different partitions, then their pairing is equal to zero. Since both 휀퐴 and the pairing between Lie푛 and its dual are non-degenerate, the total pairing is non-degenerate. The pairing is compatible with the Lie푛-(co)module structures, i.e. the following diagram commutes (a relatively easy but notationally tedious check):

CE −∗ 1⊗∘ G퐴(푈) ⊗ C∗ (퐴 ⊗ L푛)(푈/푊) 푊 CE −∗ G퐴(푈) ⊗ C∗ (퐴 ⊗ L푛)(푈) ⊗ Lie푛(푊)

∨ ∘푊⊗1 ⟨−,−⟩ ⟨−,−⟩ CE −∗ ⟨−,−⟩ G퐴(푈/푊) ⊗ C∗ (퐴 ⊗ L푛)(푈/푊) Lie푛 ∨ ℝ Lie푛 (푊) ⊗ Lie푛(푊)

Finally, it is a straightforward but long computation that the pairing commutes with differentials (i.e. ⟨푑(−), −⟩ = ±⟨−, 푑(−)⟩). It follows directly from the ′ ′ ′ ″ fact that 휀퐴(푎푎 ) = ∑ ±휀퐴(푎Δ퐴)휀퐴(푎 Δ퐴), which in turns stems from the (Δ퐴) definition of Δ퐴.

87 2 The Lambrechts–Stanley Model of Configuration Spaces

Proof of Proposition 2.5.6. The operad e푛 is given by the composition product Com∘ Lie푛 equipped with a distributive law that encodes the Leibniz rule. We get the following isomorphism (natural in 픤): G∨ ∘ 푆(픤[푛 − 1]) = G∨ ∘ (Com ∘ 픤[푛 − 1]) 퐴 e푛 퐴 e푛 ≅ G∨ ∘ (e ∘ 픤[푛 − 1]) 퐴 e푛 푛 Lie푛 ≅ G∨ ∘ 픤[푛 − 1]. 퐴 Lie푛 ∨ CE −∗ According to Lemma 2.5.9, the right Lie푛-module G퐴 is isomorphic to C∗ (퐴 ⊗ −∗ CE L푛). The functoriality of 퐴 ⊗ − and C∗ (−), as well as Equation (2.5.8), imply that we have the following isomorphism (natural in 픤): G∨ ∘ 픤[푛 − 1] ≅ CCE(퐴−∗ ⊗ L ) ∘ 픤[푛 − 1] 퐴 Lie푛 ∗ 푛 Lie푛 ≅ CCE(퐴−∗ ⊗ ((L ) ∘ 픤[푛 − 1])) ∗ 푛 Lie푛 CE −∗ ≅ C∗ (퐴 ⊗ 픤). The derived circle product is computed by taking a cofibrant resolution of ∼ 푆(픤[푛 − 1]). Let 푄픤 픤 be a cofibrant resolution of the Lie algebra 픤. Then 푆(푄픤[푛 − 1]) is a cofibrant e푛-algebra, and by Künneth’s formula 푆(푄픤[푛 − 1]) → 푆(픤[푛 − 1]) is a quasi-isomorphism. It follows that: G∨ ∘핃 푆(픤[푛 − 1]) = G∨ ∘ 푆(푄 [푛 − 1]). 퐴 e푛 퐴 e푛 픤 We therefore have a commutative diagram:

G∨ ∘핃 푆(픤[푛 − 1]) G∨ ∘ 푆(픤[푛 − 1]) 퐴 e푛 퐴 e푛

≅ ≅

CE −∗ CE −∗ C∗ (퐴 ⊗ 푄픤) C∗ (퐴 ⊗ 픤)

The Chevalley–Eilenberg functor preserves quasi-isomorphisms of Lie algebras, and therefore the bottom map is a quasi-isomorphism. The proposition follows.

2.6 Outlook: The case of the 2-sphere and oriented manifolds

We provide a generalization of the previous work for the 2-sphere, and we formu- late a conjecture for higher dimensional closed manifolds that are not necessarily framed.

88 2.6 Outlook: The case of the 2-sphere and oriented manifolds

2.6.1 Framed little disks and framed configurations Following Salvatore–Wahl [SW03, Definition 2.1], we describe the framed little disks operad as a semi-direct product. If 퐺 is a topological group and P is an operad in 퐺-spaces, the semi-direct product P⋊퐺 is the topological operad defined by (P ⋊ 퐺)(푛) = P(푛) × 퐺푛 and explicit formulas for the composition. Similarly if 퐻 is a commutative Hopf algebra and C is a Hopf cooperad in 퐻-comodules, then the semi-direct product C ⋊ 퐻 is defined by formally dual formulas. The operad FM푛 is an operad in SO(푛)-spaces, the action rotating configurations. There is thus an operad fFM푛 = FM푛 ⋊ SO(푛), the framed Fulton–MacPherson operad, weakly equivalent to the standard framed little disks operad. Given an oriented 푛-manifold 푀, there is a corresponding right module over fFM푛, which we call fFM푀 [Tur13, Section 2]. The space fFM푀(푈) is a principal ×푈 SO(푛) -bundle over FM푀(푈). Since SO(푛) is an algebraic group, fFM푛 and fFM푀(푈) are respectively an operad and a module in semi-algebraic spaces.

2.6.2 Cohomology of fFM푛 and potential model The cohomology of SO(푛) is classically given by Pontryagin classes and Euler classes:

∗ 퐻 (SO(2푛); ℚ) = 푆(훽1, … , 훽푛−1, 훼2푛−1)(deg 훼2푛−1 = 2푛 − 1) ∗ 퐻 (SO(2푛 + 1)) = 푆(훽1, … , 훽푛)(deg 훽푖 = 4푖 − 1)

∨ ∨ ∗ ⊗푈 By the Künneth formula, fe푛 (푈) = e푛 (푈) ⊗ 퐻 (SO(푛)) . We now provide explicit formulas for the cocomposition [SW03]. If 푥 ∈ 퐻∗(SO(푛)) and 푢 ∈ 푈, ∗ ⊗푈 denote as before 휄푢(푥) ∈ 퐻 (SO(푛)) . Let 푊 ⊂ 푈, then if 푥 is either 훽푖 or 훼2푛−1 in the even case, we have: ⎧ ∨ {휄∗(푥) ⊗ 1 + 1 ⊗ 휄푢(푥), if 푢 ∈ 푊; ∘푊 (휄푢(푥)) = ⎨ (2.6.1) ⎩{휄푢(푥) ⊗ 1, otherwise.

∨ ∨ The formula for ∘푊(휔푢푣) depends on the parity of 푛. If 푛 is odd, then ∘푊(휔푢푣) ∨ is still given by Equation (2.1.10). Otherwise, in fe2푛 we have: ⎧ {휄∗(훼2푛−1) ⊗ 1 + 1 ⊗ 휔푢푣, if 푢, 푣 ∈ 푊; ∨ { ∘푊 (휔푢푣) = ⎨휔푢푣 ⊗ 1, if 푢, 푣 ∉ 푊; (2.6.2) { ⎩휔푢∗ ⊗ 1, if 푣 ∈ 푊, 푢 ∉ 푊.

From now on, we focus on the case of oriented surfaces. Let 푀 = 푆2 be the 2-sphere, the only simply connected compact surface. We can choose 퐴 =

89 2 The Lambrechts–Stanley Model of Configuration Spaces

퐻∗(푆2) = 푆(휐)/(휐2) as our Poincaré duality model for 푆2. The Euler class of 퐴 is 2 푒퐴 = 휒(푆 )vol퐴 = 2휐, and the diagonal class is given by Δ퐴 = 휐 ⊗ 1 + 1 ⊗ 휐. It 1 1 is a standard fact about diagonal classes that 휇퐴(Δ퐴) = 푒퐴. Let also 훼 ∈ 퐻 (푆 ) be the generator (the “Euler class”).

Definition 2.6.3. The framed LS CDGA fG퐴(푈) is given by: ⊗푈 ∨ fG퐴(푈) = (퐴 ⊗ fe2 (푈)/(휄푢(푎) ⋅ 휔푢푣 = 휄푣(푎) ⋅ 휔푢푣), 푑), where the differential is given by 푑휔푢푣 = 휄푢푣(Δ퐴) and 푑휄푢(훼) = 휄푢(푒퐴). ∨ Proposition 2.6.4. The collection {fG퐴(푈)}푈 is a Hopf right fe2 -comodule, with cocomposition given by the same formula as Equation (2.2.2). Proof. The proofs that the cocomposition is compatible with the cooperad struc- ∨ ture of fe2 , and that this is compatible with the quotient, is the same as in the proof of Proposition 2.2.1. It remains to check compatibility with differentials. We check this compatibility on generators. The internal differential of 퐴 = ∗ 2 ∨ ∨ 퐻 (푆 ) is zero, so it is easy to check that ∘푊(푑(휄푢(푎))) = 푑(∘푊(휄푢(푎))) = 0. Similarly, using Equation (2.6.1), checking the equality on 훼 is immediate. As before there several are cases to check for 휔푢푣. If 푢, 푣 ∈ 푊, then by Equa- tion (2.6.2), ∨ 푑(∘푊(휔푢푣)) = 푑(휄∗(훼) ⊗ 1 + 1 ⊗ 휔푢푣) = 휄∗(푒퐴) ⊗ 1 ∨ = 휄∗(휇퐴(Δ퐴)) ⊗ 1 = ∘푊(푑휔푢푣), and otherwise the proof is identical to the proof of Proposition 2.2.1.

∗ 2.6.3 Connecting fG퐴 to ΩPA(fFM푆2 ) The framed little 2-disks operad is known to be formal [GS10; Šev10]. We will focus on the proof of Giansiracusa–Salvatore [GS10], which goes along the same line as the proof of Kontsevich of the formality of FM푛. To simplify notations, let 퐻 = 퐻∗(푆1), which is a Hopf algebra. Graphs The operad 2 is an operad in 퐻-comodules, so we may consider the Graphs semi-direct product 2 ⋊ 퐻. Giansiracusa and Salvatore construct a zigzag: fe∨ ∼ Graphs ∼ fFM 2 2 ⋊ 퐻 ΩPA( 2). (2.6.5) Graphs ∼ e∨ The first map is the tensor product of 2 2 and the identity of 퐻. Graphs The second map is given by the Kontsevich integral on 2 and by sending ∗ 1 the generator 훼 ∈ 퐻 to the volume form of ΩPA(푆 ) (pulled back by the relevant projection). They check that both maps are maps of Hopf (almost) cooperads, and they are quasi-isomorphisms by the Künneth formula.

90 2.6 Outlook: The case of the 2-sphere and oriented manifolds

∨ ∗ 2 Theorem 2.6.6. The Hopf right comodule (fG퐴, fe2 ), where 퐴 = 퐻 (푆 ; ℝ), is quasi- ∗ ∗ isomorphic to the Hopf right comodule (ΩPA(fFM푆2 ), ΩPA(fFM2)). Proof. It is now straightforward to adapt the proof of Theorem C to this setting, reusing the proof of Giansiracusa–Salvatore [GS10]. We build the zigzag:

fG Graphs ∗ fFM 퐴 ← 퐴 ⋊ 퐻 → ΩPA( 푆2 ).

We simply choose 푅 = 퐴 = 퐻∗(푆2), mapping 휐 ∈ 퐻2(푆2) to the volume form of 푆2. Note that the propagator can be made completely explicit on 푆2, and it can be checked that 푍휑 vanishes on all connected graphs with more than one Graphs vertex [CW16, Proposition 80]. The middle term is a Hopf right ( 2 ⋊ 퐻)- Graphs comodule built out of 퐴 and 퐻, using formulas similar to the formulas Graphs Graphs defining 2 ⋊ 퐻 out of 2 and 퐻. The first map is given by the tensor Graphs G product of 푅 → 퐴 and the identity of 퐻. Graphs The second map is given by the morphism of Proposition 2.3.31 on the 퐴 factor, composed with the pullback along the projection fFM푆2 → FM푆2 . The generator 훼 ∈ 퐻 is sent to a pullback of a global angular form 휓 of the principal 2 2 SO(2)-bundle fFM푆2 (1) → FM푆2 (1) = 푆 induced by the orientation of 푆 . This 2 form satisfies 푑휓 = 휒(푆 )vol푆2 . The proof of Giansiracusa–Salvatore [GS10] then adapts itself to prove that these two maps are maps of Hopf right comodules. The Künneth formula implies that the first map is a quasi-isomorphism, and the second map induces an isomorphism on the E2-page of the Serre spectral sequence associated to the bundle fFM푆2 → FM푆2 and hence is itself a quasi-isomorphism. or 2 Corollary 2.6.7. The CDGA fG퐻∗(푆2)(푘) is a real model for Conf푘 (푆 ), the principal ×푘 2 2 SO(2) -bundle over Conf푘(푆 ) induced by the orientation of 푆 .

A conjecture in higher dimensions If 푀 is an oriented 푛-manifold, Defini- tion 2.6.3 readily adapts to define fG퐻∗(푀), by setting 푑훼 to be the Euler class of 푀 (when 푛 is even), and 푑훽푖 to be the 푖th Pontryagin class of 푀. The proof of Proposition 2.6.4 adapts easily to this new setting, and fG퐻∗(푀) becomes a Hopf ∨ right fe푛 -comodule. It was recently proved that the framed little disks operads is formal for even 푛 and not formal for odd 푛 ≥ 3 [Mor16; KW17].

Conjecture 2.6.8. If 푀 is a formal, simply connected, oriented closed 2푛-manifold, then ∨ ∗ ∗ the pair (fG퐻∗(푀), fe2푛) is quasi-isomorphic to the pair (ΩPA(fFM푀), ΩPA(fFM2푛)). To adapt our proof directly to this conjecture, the difficulty would be the same as encountered by Giansiracusa–Salvatore [GS10], namely finding forms in

91 2 The Lambrechts–Stanley Model of Configuration Spaces

∗ ∗ ΩPA(fFM푛) corresponding to the generators of 퐻 (SO(푛)) and compatible with the Kontsevich integral. The proof of the formality of fFM2푛 for 푛 > 1 [KW17] is rather more involved than the proof for 푛 = 1, and it would be an interesting question to try and adapt it for the conjecture. If 푀 itself is not formal, it is also not clear how to define Pontryagin classes in some Poincaré duality model of 푀 – the Euler class was simply given the Euler characteristic. Nevertheless, for any oriented manifold 푀 we get invariants of fe -algebras by considering the functor fG∨ ∘핃 (−). Despite not necessarily 푛 퐴 fe푛 computing factorization homology, these invariants could prove to be interesting.

92 3 Configuration Spaces of Manifolds with Boundary

We now extend the results of the previous chapter to compact manifolds with boundary which admit a “surjective pretty model” [CLS15a]. For concreteness, let (푀, 휕푀) be a compact manifold with boundary. Given some integer 푘 ≥ 0, its 푘th (ordered) configuration space is:

×푘 Conf푘(푀) ≔ {(푥1, … , 푥푘) ∈ 푀 ∣ 푥푖 ≠ 푥푗 ∀푖 ≠ 푗}.

Suppose that the pair (푀, 휕푀) has a surjective pretty model induced by 휓 ∶ 푃 → 푄, where 푃 is a Poincaré duality CDGA and 휓 is surjective (see Section 3.1.2 for the definitions). In short, this means that 푀 has been obtained by removing from a closed manifold 푁 (of which 푃 is a model) a sub-polyhedron 푋 (of which 푄 is a model). Roughly, the pretty model construction is a way of encoding Poincaré–Lefschetz duality to obtain a model for 푀 = 푁 − 푋. This situation occurs the case if 푀 is closed (in which case 푄 = 0), if both 푀 and 휕푀 are 2-connected and the boundary retracts rationally onto its half-skeleton, if 푀 is a disk bundle over a closed manifold, or if 푀 is obtained by removing the thickening of a high-codimensional subpolyhedron from a closed manifold (see Theorem 3.1.12). Then a model of 푀 is given by 퐴 = 푃/퐼, where 퐼 = im(휓!) is an ideal of the Poincaré duality CDGA 푃. We use the same construction as in Chapter 2 to get a CDGA G퐴̃ (푘) (Equa- tion (3.3.19)), which is somehow a “perturbation” of G퐴(푘). We show that there is ∗ ∗ an isomorphism of graded vector spaces between 퐻 (G퐴(푘)) and 퐻 (Conf푘(푀)) over ℚ (see Theorem 3.3.16), which generalizes the result of [LS08a]. If 푀 is closed, then 퐼 = 0 and we recover the model considered in Chapter 2. If 푀 is simply connected with simply connected boundary but does not admit a surjective pretty model, we have analogous results as soon as dim 푀 ≥ 7 using what we dub “Poincaré–Lefschetz duality models”. Just like in the previous chapter, we consider a colored version of the Fulton– MacPherson compactification SFM푀(∅, 푘). This is a stratified space which contains ̊ Conf푘(푀) as its interior, and the inclusion is a weak equivalence. If 푀 is framed,

Based on a work-in-progress in collaboration with P. Lambrechts.

93 3 Configuration Spaces of Manifolds with Boundary

then the collection SFM푀(∅, −) = {SFM푀(∅, 푘)}푘≥0 is equipped with an action of the Fulton–MacPherson operad, just like in the previous chapter. The first result of this chapter can be summarized as:

Theorem D (See Theorem 3.5.37). Let 푀 be a smooth, simply connected connected compact 푛-manifold with simply connected boundary of dimension at least 5. Assume that either 푀 admits a surjective pretty model, or that 푛 ≥ 7 so that 푀 admits a Poincaré– Lefschetz duality model. Let 퐴 be the model built either out of the surjective pretty model or the Poincaré–Lefschetz duality model. ̃ ∗ Then for all 푘 ≥ 0, the CDGA G퐴(푘) is weakly equivalent to ΩPA(SFM푀(∅, 푘)), and the equivalence is compatible with the action of the symmetric group Σ푘. ∨ ∗ The collection G퐴̃ is a Hopf right comodule over the cooperad e푛 = 퐻 (FM푛). More- ∨ over, if 푀 is framed, then the right Hopf comodule (G퐴, e푛 ) is weakly equivalent to ∗ ∗ (ΩPA(SFM푀(∅, −)), ΩPA(FM푛)).

Corollary E. Let 푀 be a manifold with boundary which satisfies the hypotheses of Theorem D. Then the real homotopy type of Conf푘(푀) only depends on the real homotopy type of the pair (푀, 휕푀).

One can also consider colored configuration in the manifold 푀 and study the space, for 푘, 푙 ≥ 0:

̊ Conf푘,푙(푀) ≔ {(푥1, … , 푥푘, 푦1, … , 푦푙) ∈ Conf푘+푙(푀) ∣ 푥푖 ∈ 휕푀, 푦푗 ∈ 푀}.

Once again, we can consider its (colored) Fulton–MacPherson compactification SFM푀(푘, 푙). This is a stratified space which contains Conf푘,푙(푀) as its interior, the inclusion being a homotopy equivalence. When 푀 is framed, the collection SFM푀 is equipped with an action of SFM푛, an operad weakly equivalent to the Swiss- SGraphs Cheese operad [Vor99]. Willwacher [Wil15] has built a model 푛 for SFM Graphs 푛 which is similar in spirit to Kontsevich’s graph model 푛 (due to the non-formality of the Swiss-Cheese operad [Liv15], it is not possible to go all the ∗ way to the cohomology 퐻 (SFM푛)). We then have:

Theorem F (See Theorem 3.5.42). Let 푀 be a smooth, simply connected manifold with a simply connected boundary of dimension at least 5. Assume either that 푀 admits a SGraphs surjective pretty model or that 푛 ≥ 7. Then we have an explicit model 푅(푘, 푙) of Conf푘,푙(푀) which is built out of the pretty (resp. PLD) model. It is compatible with the action of the symmetric groups Σ푘 × Σ푙. SGraphs The collection 푅 is a Hopf right comodule over Willwacher’s [Wil15] model SGraphs 푛 for the Swiss-Cheese operad. Moreover, if 푀 is framed, then the Hopf right SGraphs SGraphs ∗ SFM ∗ SFM comodule ( 푅, 푛) is weakly equivalent to (ΩPA( 푀), ΩPA( 푛)).

94 3.1 Background and recollections

Outline In Section 3.1, we recall some background on relative cooperads and comodules over them, pretty models, colored Fulton–MacPherson compactifications, and graph complexes. In Section 3.2, we define Poincaré–Lefschetz duality models, a generalization of surjective pretty models, and we prove that any simply connected manifold with simply connected boundary of dimension at least 7 admits a Poincaré–Lefschetz duality model. In Section 3.3, we describe the CDGA G퐴̃ (푘), and we prove that it computes the Betti numbers of Conf푘(푀). In Section 3.4, we compute the cohomology of the Swiss-cheese version of the graph complex twisted by a Maurer–Cartan element corresponding to the HKR isomorphism. In Section 3.5, we show that G퐴̃ (푘) is actually a real model of Conf푘(푀), and we prove that it is compatible with the action of the Fulton–MacPherson 푆 푐푀,z휑 operad. We also build a graph complex SGraphs푅 (푘, 푙) and we show that it is a real model for Conf푘,푙(푀), compatible with the action of the compactified colored Fulton–MacPherson operad.

3.1 Background and recollections

We mostly reuse the conventions of Section 2.1.1.

3.1.1 Colored (co)operads and (co)modules We deal with special types of two-colored operads, called relative operads [Vor99], or Swiss-Cheese type operads. Definition 3.1.1. Given an operad P, a relative operad over P is an operad in the category of right P-modules (see also Definition 1.1.1). This is equivalent to an operad with two colors (traditionally called the “closed” color 픠 and the “open” color 픬) such that operations with a closed output may only have closed inputs and are given by P. We can encode the part of the operad with an open output as a bisymmetric collection Q, i.e. as a functor from the category of pairs of sets and pairs of bijections to dg-modules. The first set in the pair corresponds to open inputs, and the second to closed inputs. There is an identity 휂픬 ∈ Q({∗}, ∅), and the operadic composition structure maps are given by:

∘푇 ∶ Q(푈, 푉/푇) ⊗ P(푇) → Q(푈, 푉) 푇 ⊂ 푉; ∘푊,푇 ∶ Q(푈/푊, 푉) ⊗ Q(푊, 푇) → Q(푈, 푉 ⊔ 푇) 푊 ⊂ 푈. As mentioned in the definition, we can equivalently view Q as an operad in the category of right P-modules. The P-module in arity 푈 is given by Q(푈, −), and one

95 3 Configuration Spaces of Manifolds with Boundary

checks that the operad structure maps ∘푊,− ∶ Q(푈/푊, −) ⊗ Q(푊, −) → Q(푈, −) are morphisms of right P-modules. Given a (one-colored) cooperad C, a relative cooperad over C is defined dually as a bisymmetric collection D equipped with structure maps:

∨ ∘푇 ∶ D(푈, 푉) → D(푈, 푉/푇) ⊗ C(푊) 푇 ⊂ 푉; (3.1.2) ∨ ∘푊,푇 ∶ D(푈, 푉 ⊔ 푇) → D(푈/푊, 푉) ⊗ D(푊, 푇) 푊 ⊂ 푈. (3.1.3) Finally, a comodule over a relative C-cooperad D is given by a bisymmetric collection N equipped with structure maps:

∨ ∘푇 ∶ N(푈, 푉) → N(푈, 푉/푇) ⊗ C(푇) 푇 ⊂ 푉; (3.1.4) ∨ ∘푊,푇 ∶ N(푈, 푉 ⊔ 푇) → N(푈/푊, 푉) ⊗ D(푊, 푇) 푊 ⊂ 푈. (3.1.5) We can also define relative Hopf cooperads as relative cooperads in the category of CDGAs.

3.1.2 Pretty models for compact manifolds with boundary The cohomology of a compact manifold with boundary does not satisfy Poincaré duality; it satisfies Poincaré–Lefschetz duality instead. Cordova Bulens, Lam- brechts, and Stanley [CLS15a; CLS15b] use this idea to define “pretty rational models” for Poincaré duality pairs such as (푀, 휕푀). The rough idea is that one can see 푀 as the complement in some closed manifold 푁 of the thickening of some subpolyhedron 퐾 ⊂ 푁. One then takes a Poincaré duality model 푃 of 푁 and a model 푄 of 퐾, and formally “kills” the forms that are dual to homology classes on 퐾 to obtain a model of 푀. To be explicit, the starting data in the definition of a pretty model is: • a Poincaré duality CDGA 푃 of formal dimension 푛 (in the rough picture above, it would be a model for 푁);

• a connected CDGA 푄 satisfying 푄≥푛/2−1 = 0 (representing a model for 퐾 ⊂ 푁);

• a morphism 휓 ∶ 푃 → 푄 (representing the restriction Ω∗(푁) → Ω∗(퐾)). The morphism 휓 makes 푄 into a 푃-module. Moreover, the dual 푃∨ (resp. 푄∨) is a 푃-module (resp. 푄-module) by letting (푥 ⋅ 푓 )(푦) = ±푓 (푥푦), and so is the desuspension 푃∨[−푛] (resp. 푄∨[−푛]). The Poincaré duality structure on 푃 induces an isomorphism of 푃-modules:

≅ ∨ 휃푃 ∶ 푃 푃 [−푛] (3.1.6)

96 3.1 Background and recollections

! given by 푎 ↦ 휀푃(푎 ⋅ −). The “shriek map” 휓 is the morphism of 푃-modules defined by the composite map:

∨ −1 ! ∨ 휓 [−푛] ∨ 휃푃 휓 ∶ 푄 [−푛] 푃 [−푛] ≅ 푃. (3.1.7)

More concretely, given 훼 ∈ 푄∨[−푛], 휓!(훼) is uniquely determined by:

! ∀푥 ∈ 푃, 휀푃(휓 (훼)푥) = 훼(휓(푥)). (3.1.8)

One must further assume that 휓휓! is “balanced”, i.e. that for all 푓 , 푔 ∈ 푄∨[−푛], one has 휓휓!(푓 ) ⋅ 푔 = 푓 ⋅ 휓휓!(푔). Note that, under our assumptions, this is automatically the case by degree reasons (in fact, we have 휓휓! = 0 because of the hypothesis that 푄≥푛/2−1 = 0).

Definition 3.1.9. The mapping cone of a chain map 푓 ∶ 푋 → 푌, denoted either by cone(푓 ) or 푌 ⊕푓 푋[1], is given as a graded vector space by 푌 ⊕ 푋[1], and the differential is given by 푑(푦, 푥) = (푑푌푦 + 푓 (푥), 푑푋푥). ! ! ∨ ∨ Note that 휓휓 = 0 implies that cone(휓휓 ) = 푄⊕휓휓! 푄 [1−푛] = 푄⊕푄 [1−푛] is simply given by the direct sum of 푄 and 푄∨[1−푛]. The balancedness assumption ∨ ∨ ensures that one can make the mapping cones 푃 ⊕휓! 푄 [1−푛] and 푄⊕푄 [1−푛] into CDGAs, using the product of 푃 (resp. 푄) and the 푃-module (resp. 푄-module) structure of 푄∨[1 − 푛], and by letting the product of two elements of 푄∨[1 − 푛] be zero. In other words, the product is defined by:

(푥, 훼) ⋅ (푦, 훽) ≔ (푥푦, 푥 ⋅ 훽 + 훼 ⋅ 푦).

Definition 3.1.10. The pretty model associated to (푃, 푄, 휓) is the CDGA morphism: ∨ ∨ 휆 ≔ 휓 ⊕ id ∶ 푃 ⊕휓! 푄 [1 − 푛] → 푄 ⊕ 푄 [1 − 푛]. The pair (푀, 휕푀) admits a surjective pretty model if there exists a surjective morphism 휓 ∶ 푃 → 푄 as above such that there is a commutative diagram of CDGAs:

∨ ∼ ∼ ∗ 퐵 ≔ 푃 ⊕ ! 푄 [1 − 푛] 푅 Ω (푀) 휓 푓 푔 휆≔휓⊕id 휌 res (3.1.11)

∨ ∼ ∼ ∗ 퐵휕 ≔ 푄 ⊕ 푄 [1 − 푛] 푅휕 푔 Ω (휕푀) 푓휕 휕 with each row a zigzag of quasi-isomorphisms.

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Theorem 3.1.12 (Cordova Bulens, Lambrechts, and Stanley [CLS15a; CLS15b]). The pair (푀, 휕푀) admits a surjective pretty model in the following cases: • If both 푀 and 휕푀 are 2-connected and the boundary 휕푀 retracts rationally on its half-skeleton [CLS15a, Definition 6.1]; • If 푀 (resp. 휕푀) is the associated disk bundle (resp. sphere bundle) of a vector bundle of even rank over a simply connected Poincaré duality space; • If 푀 is obtained by removing a tubular neighborhood of a 2-connected subpolyhedron 퐾 from a 2-connected closed manifold 푁 satisfying 2 dim 퐾 + 3 ≤ dim 푁 = dim 푀. By the results from Section 2.1.4, when 푀 is simply connected and 휕푀 = ∅, one can take a Poincaré duality model 푃 of 푀 and 푄 = 0 to obtain a surjective pretty model of (푀, ∅). The CDGA 퐵휕 is a Poincaré duality CDGA of formal dimension 푛 − 1, with augmentation 휀 ∶ 퐵 → 한[1 − 푛] given on 퐵휕 휕 (푄 ⊕ 푄∨[1 − 푛])푛−1 = (푄0)∨ 0 ∨ by evaluation on 1푄 ∈ 푄 . In other words the volume form is given by 1푄. We will also write 휀퐵 ∶ 퐵 → 한[−푛] for the linear map given by 휀퐵(푥, 푦) = 휀푃(푥).

Remark 3.1.13. The map 휀퐵 is not a chain map unless 푄 = 0. Indeed if 푄 ≠ 0, ∨ ! ∨ then we have 푑(1푄) = 휓 (1푄) = vol푃, and thus 휀퐵 ∘ 푑 ≠ 0. Instead, there is a Stokes-like formula: 휀 (푑(푥, 훼)) = 휀 (휓(푥), 훼), ∀푥 ∈ 푃, 훼 ∈ 푄∨[1 − 푛], (3.1.14) 퐵 퐵휕 which means that 휀 = (휀 , 휀 ) defines a chain map 휀 ∶ cone(휓⊕id) → ℝ[−푛+1]. 퐵 퐵휕 Let 퐼 ⊂ 푃 be the image of 휓!, which is an ideal. Then when 휓 is surjective, the ∨ projection 푃 ⊕휓! 푄 [1 − 푛] → 퐴 ≔ 푃/퐼 is a quasi-isomorphism. Thus if (푀, 휕푀) admits a surjective pretty model, then 푀 admits a model which is a quotient of a Poincaré duality CDGA. It is an open conjecture whether this is the case for all compact manifolds with boundary. There is an interpretation of Poincaré–Lefschetz duality in this case. The kernel 퐾 ≔ ker 휓 ⊂ 푃 is equal to ker(휓 ⊕ id), which is a model for Ω∗(푀, 휕푀) ≔ hoker(Ω∗(푀) → Ω∗(휕푀)). ∨ Recall the isomorphism 휃푃 ∶ 푃 → 푃 [−푛] from Equation (3.1.6). It induces dual isomorphisms:

≅ ∨ ≅ ∨ 휃푃 ∶ 푃/퐼 퐾 [−푛], 휗푃 ∶ 퐾 (푃/퐼) [−푛], (3.1.15)

98 3.1 Background and recollections that are both induced by the non-degenerate pairing (푃/퐼) ⊗ (ker 휓) → ℝ[−푛], 푥 ⊗ 푦 ↦ 휀푃(푥푦). Example 3.1.16. Consider the manifold 푀 = 퐷푛 with boundary 휕푀 = 푆푛−1. Intuitively, we can see 푀 as a sphere with a (thick) point removed. Thus consider ∗ 푛 2 the Poincaré duality CDGA 푃 = 퐻 (푆 ) = 푆(vol푛)/(vol푛) (with a generator of degree 푛), and let 푄 = 퐻∗(pt) = ℝ. Let also 휓 ∶ 푃 → 푄 be the augmentation. 푛 ∨ We get as a model for 퐷 the mapping cone 퐵 = 푃 ⊕휓! 푄 [1 − 푛], which is ∨ spanned by three generators 1푃, 1푄, and vol푛. All the nontrivial products vanish, ∨ ∗ 푛 and 푑(1푄) = vol푛. The quotient 퐴 = 푃/퐼 is isomorphic to 퐻 (퐷 ) = ℝ, and ∨ ∗ 푛 ∗ 푛−1 퐵휕 = 푄 ⊕ 푄 [1 − 푛] is isomorphic to 퐻 (휕퐷 ) = 퐻 (푆 ) – both manifolds being formal.

3.1.3 Compactified configuration spaces

Recall the Fulton–MacPherson operad FM푛 from Section 2.1.2, defined using compactified configuration spaces. Recall also its right modules FM푀, defined when 푀 is a closed framed manifold. These constructions generalize to manifolds with boundary. Given a manifold with boundary (푀, 휕푀) and finite sets 푈, 푉, define the colored configuration spaces: ̊ Conf푈,푉(푀) ≔ {푐 ∈ Conf푈⊔푉(푀) ∣ 푐(푈) ⊂ 휕푀, 푐(푉) ⊂ 푀}. (3.1.17) If 푀 is a compact manifold, then this configuration space can be compactified into SFM푀(푈, 푉). This is a stratified manifold of dimension (푛 − 1) ⋅ #푈 + 푛 ⋅ #푉. Note that SFM푀(∅, {∗}) ≅ 푀, as a point may be infinitesimally close to the boundary of 푀. 푛 푛 Similarly if 푀 = ℍ ⊂ ℝ is the upper half-space, then Conf푈,푉(푀), after modding it out by the group of translations preserving ℍ푛 and positive dilata- tions, can be compactified into a space SFM푛(푈, 푉) [Vor99]. It is a stratified manifold of dimension 푛 ⋅ #푉 + (푛 − 1) ⋅ #푈 − 푛 as soon as #푈 + 2#푉 ≥ 2, and it is reduced to a point otherwise. One has homeomorphisms SFM푛(푈, ∅) ≅ FM푛−1(푈) 푛−1 and SFM푛(1, 1) ≅ 퐷 , and homotopy equivalences SFM푛(∅, 푉) ≃ FM푛(푉). In this way, one obtains a relative operad SFM푛 over FM푛, weakly equivalent to the Swiss-Cheese operad [Vor99] where operations with no open inputs are allowed (see [HL12, Section 3] or Remark 1.1.2 for an explanation of the difference). If 푀 is framed, then the collection SFM푀 also assemble into a right SFM푛-module using insertion of configurations:

∘푇 ∶ SFM푀(푈, 푉/푇) × FM푛(푇) → SFM푀(푈, 푉) 푇 ⊂ 푉; (3.1.18) ∘푊,푇 ∶ SFM푀(푈/푊, 푉) × SFM푛(푊, 푇) → SFM푀(푈, 푉 ⊔ 푇) 푊 ⊂ 푈. (3.1.19)

99 3 Configuration Spaces of Manifolds with Boundary

The spaces SFM푛(푈, 푉) are clearly semi-algebraic sets. Moreover, 푀 itself can be endowed with a semi-algebraic structure. Indeed, it can be triangulated and is 퐶1-diffeomorphic to some simplicial complex in some ℝ푁. Each simplex is a semi-algebraic set, and there is a finite number of them (because 푀 is compact), hence 푀 itself is semi-algebraic. It follows that the spaces SFM푀(푈, 푉) are semi- algebraic too. We can thus apply the theory of PA forms (see Section 2.1.2) and ∗ ∗ obtain relative “almost” cooperads and comodules ΩPA(SFM푛) and ΩPA(SFM푀). Moreover, the projections

푝푈,푉 ∶ SFM푀(푈 ⊔ 퐼, 푉 ⊔ 퐽) → SFM푀(푈, 푉) (3.1.20) which forget some of the points are semi-algebraic bundles. Just like in Chap- ter 2, we follow the proof from [LV14, Section 5.9] step by step, and we work by induction on the number of points forgotten (because SA bundles can be composed, see [Har+11, Proposition 8.5]). If we forget a terrestrial point, then this essentially follows from the fact that the projections FM휕푀(푈 ⊔ 퐼) → FM휕푀(푈) are SA bundles. For aerial points, let us give an idea of the key point of the proof for the projection 푝 ∶ SFM푀(∅, 2) → SFM푀(∅, 1) which forgets the point 2. Recall that SFM푀(∅, 1) is homeomorphic to 푀. The fiber of 푝 over a point 푥 of the interior of 푀 is given by 푀 with an open disk centered around 푥. The fiber over a point 푥 of the boundary has three types of points: either point 2 is far away from point 1, or it is infinitesimally close to point 1. The set of configurations from the first case is given by 푀 with a half-disk around point 1 removed. The set of configurations from the second case is given by the space SFM푛(∅, 2) which specify how the two points are infinitesimally arranged. The intersection of these two sets is givenby SFM푛(∅, 1), i.e. point 2 is infinitesimally close to 1 but “almost” leaves it. In other words, the fiber is:

푝−1(푥) = 푀 − {half-disk around 푥} ∪ SFM (∅, 2), (3.1.21) SFM푛(∅,1) 푛 which is also homeomorphic to 푀 with an open disk removed, see Figure 3.1.1.

3.1.4 Graphs and the Swiss-Cheese operad The Swiss-Cheese operad is not formal [Liv15; Wil17], thus there can be no ∗ ∗ quasi-isomorphism between 퐻 (SFM푛) and ΩPA(SFM푛). Nevertheless, there is a SGraphs SFM model 푛 of 푛, due to Willwacher [Wil15], which is similar in spirit Graphs ∗ SFM ∗ FM ∗ FM to the cooperad 푛. The cohomology 퐻 ( 푛) ≅ 퐻 ( 푛) ⊗ 퐻 ( 푛−1) SGraphs splits as a Voronov product (see [Vor99] and Section 1.4.3), and 푛 is a Graphs Graphs way of intertwining 푛 and 푛−1 in a way that corrects the lack of formality.

100 3.1 Background and recollections

SFM푛(0, 1)

푀 minus open half-disk

SFM푛(0, 2)

Figure 3.1.1: The fiber of SFM푀(∅, 2) → SFM푀(∅, 1) = 푀 above a point of the boundary: (푀 − {half-disk}) ∪ SFM (∅, 2) SFM푛(∅,1) 푛

Remark 3.1.22. Our notations differ slightly from the notations of[Wil15]. We call SGraphs Graphs1 푛 what is called 푛 there, i.e. the space of operations with output of “type 1” (which corresponds to “open”). This is a relative Hopf cooperad over Graphs Graphs2 푛 (which would be 푛 in Willwacher’s paper, the space of operations with output of “type 2”, i.e. closed). Let us assume from the start that 푛 ≥ 3 to avoid some difficulties that arise when 푛 = 2. Note that in the 푛 = 2 case we have found a different model (Chapter 1) for SFM2 which is instead inspired by Tamarkin’s proof of the formality of the little 2- disks operad, obtained by intertwining the operad of parenthesized permutations and the operad of parenthesized chord diagrams. The idea is to construct a relative Hopf cooperad SGra푛 over Gra푛, with two types of vertices: aerial ones, corresponding to closed inputs, and terrestrial ones, corresponding to open inputs. Edges are oriented, and the source of an edge may only be an aerial vertex. More concretely, one defines:

SGra푛(푈, 푉) ≔ 푆(푒푣푢)푢∈푈,푣∈푉 ⊗ 푆(푒푣푣′ )푣,푣′∈푉 (3.1.23) where the generators all have degree 푛 − 1 and the cooperad structure is given by formulas similar to Equation (2.1.10). A monomial in SGra푛(푈, 푉) can be seen as a directed graph with two kinds of vertices: aerial and terrestrial. The set 푈 is the set of terrestrial vertices, and the set 푉 is the set of aerial vertices. Note that unlike graphs in Gra푛, double edges 2 2 (of the type 푒푣푢 or 푒푣푣′ ) and loops (also known as tadpoles, of the type 푒푣푣) are allowed. ″ ∗ This allows us to produce a first morphism 휔 ∶ SGra푛 → ΩPA(SFM푛). One can

101 3 Configuration Spaces of Manifolds with Boundary

′ ′ ∗ define, for 푣, 푣 ∈ 푉, 휔 (푒푣푣′ ) ≔ 푝푣푣′ (vol푛−1), where

푛−1 ′ 푛−1 ″ 푛−1 푛−1 vol푛−1 ∈ ΩPA (SFM푛(∅, {푣, 푣 })) ≃ ΩPA (FM푛({푣, 푣 })) ≃ ΩPA (푆 ).

푛−1 ℎ Recall that SFM푛({푢}, {푣}) is homeomorphic to 퐷 , and we write vol푛−1 for the (푛 − 1)-form on SFM푛({푢}, {푣}) obtained by pulling back the volume form of 푆푛−1 along the map 퐷푛−1 → 푆푛−1 given by the hyperbolic geodesic (see [Wil15, ′ ∗ ℎ Equation (8)]). Then for 푢 ∈ 푈 and 푣 ∈ 푉, define 휔 (푒푣푢) ≔ 푝푣푢(vol푛−1). If Γ ∈ SGra푛(푈, 푉) is a graph, let

푐(Γ) ≔ ∫ 휔′(Γ). (3.1.24) SFM푛(푈,푉) Note that these are analogous to the coefficients that appear in Kontsevich’s universal 퐿∞ formality morphism 푇poly → 퐷poly, defined for 푛 = 2 [Kon03]. The cooperad SGra푛 is then twisted with respect to the sum of the Maurer– Cartan element 휇 ∈ Def(hoLie푛 → Gra푛) with the Maurer–Cartan element defined by 푐, to obtain a relative Hopf cooperad Tw SGra푛 over Tw Gra푛. Concretely, Tw SGra푛(푈, 푉) is spanned by graphs with four types of vertices: they can be either aerial or terrestrial, and either internal or external. Internal terrestrial vertices are of degree 1 − 푛 and indistinguishable among themselves, and internal aerial vertices are of degree −푛 and indistinguishable among themselves. Edges remain oriented and of degree 푛 − 1. External terrestrial vertices are in bijection with 푈, and external aerial vertices are in bijection with 푉. The cooperad structure maps collapse subgraphs, and the product glues graphs along external vertices.

푣1 푣2

푢

Figure 3.1.2: A colored graph in Tw SGra푛({푢}, {푣1, 푣2}). Terrestrial vertices are drawn on a dotted line to distinguish them, even though they are not ordered. This graph is of degree 5(푛 − 1) − 푛 − (푛 − 1).

The differential has several summands: • A first summand (corresponding to 휇) contracts edges between two aerial vertices, with at least one being internal.

• A second summand is given by contracting subgraphs Γ′ ⊂ Γ with at most one external vertex, which must be terrestrial, to obtain Γ/Γ′, with

102 3.1 Background and recollections

coefficient 푐(Γ′). One should note that in Γ/Γ′, the new vertex representing the collapsed subgraph is terrestrial, even if Γ′ is fully aerial.

• Finally, a third summand is given by forgetting some internal vertices and keeping a subgraph Γ′ ⊂ Γ, with coefficient 푐(Γ/Γ′).

In the second and third cases, if the graph Γ/Γ′ contains an edge whose source is terrestrial, then the summand is defined to be zero (seeKon03 [ , §6.4.2.2]). ∗ One then checks that there is an extension 휔 ∶ Tw SGra푛 → ΩPA(SFM푛), given by a formula analogous to Equation (2.1.14). It remains to mod out by the bi-ideal SGraphs of graphs with internal components to obtain a Hopf cooperad 푛, and to check that 휔 factors through the quotient. SGraphs ∗ SFM Theorem 3.1.25 (Willwacher [Wil15]). The morphism 휔 ∶ 푛 → ΩPA( 푛) is a quasi-isomorphism of relative Hopf cooperads.

3.1.5 Graph complexes There is a dg-module of particular interested appearing in Kontsevich’s proof of the formality of FM푛. Define

fGC푛 ≔ Tw Gra푛(∅)[−푛] (3.1.26) to be the full graph complex. It is spanned by graphs containing only internal vertices, with a degree shift: if 훾 ∈ fGC푛 is a graph with 푘 edges and 푙 vertices, then its degree 푘(푛 − 1) − 푙푛 + 푛. The differential is given by contracting edges. Its suspension fGC푛[푛] is a CDGA, and the product is the disjoint union of graphs. Since the differential cannot create new connected components, this a free CDGA, and we have: fGC푛 = 푆(GC푛[푛])[−푛] (3.1.27) 2 where GC푛 is the submodule of connected graph. The variants GC푛 (where bivalent vertices are allowed but univalent vertices forbidden) and GC푛 (where loops are allowed) are defined similarly. Remark 3.1.28. Our notations are slightly nonstandard. It is often the dual complex ∨ GC푛 which is called the graph complex, with a differential that is dually given by creating internal vertices. This dual is a Lie algebra using insertion of graphs. The homology of this dg-module is particularly hard to compute. It is known 0 ∨ that 퐻 (GC2 ) is isomorphic to the Grothendieck–Teichmüller Lie algebra 픤픯픱1 <0 ∨ 1 and that 퐻 (GC2 ) = 0 [Wil14]; it is a conjecture that 퐻 (GC2) = 0. The ∗ 2,∨ vanishing of 퐻 (GC푛 ) in some degrees shows that the operad of little 푛-disks is intrinsically formal as a Hopf cooperad [FW15]. And this homology computes

103 3 Configuration Spaces of Manifolds with Boundary the homotopy groups of the space of rational automorphisms of the little 푛-disks operad [FTW17]. There is also a Swiss-Cheese version of that graph complex appearing in the description of Willwacher’s model for the Swiss-Cheese operad:

fSGC푛 ≔ Tw SGra푛(∅, ∅)[−푛] = 푆(SGC푛[푛])[−푛] (3.1.29) is the full Swiss-Cheese graph complex, a symmetric algebra on its subcomplex ∨ of connected graphs. Its dual SGC푛 is a Lie algebra using insertion of graphs ∨ (and the Lie algebra GC푛 acts on SGC푛), and is sometimes also denoted KGC푛. ∨ The Kontsevich integrals 푐 ∈ SGC푛 define a Maurer–Cartan element in this Lie algebra.

3.2 Poincaré–Lefschetz duality models 3.2.1 Motivation In Section 3.1.2, we recalled the definition of pretty models. If 푀 admits a surjective pretty model (see Theorem 3.1.12) induced by 휓 ∶ 푃 → 푄, then we get a diagram as in Equation (3.1.11):

non degen. 퐾 ≔ ker 휆 pairing

∼ ∨ ∼ ∼ ∗ 퐴 ≔ 푃/퐼 퐵 ≔ 푃 ⊕ ! 푄 [1 − 푛] 푅 Ω (푀) (3.2.1) 휋 휓 푓 푔 PA 휆=휓⊕id 휌 res

∨ ∼ ∼ ∗ 퐵휕 ≔ 푄 ⊕ 푄 [1 − 푛] 푅휕 푔 ΩPA(휕푀) 푓휕 휕

Recall that 퐼 ⊂ 푃 is the image of 휓! ∶ 푄∨[1 − 푛] → 푃. We consider the diagonal ⊗2 cocycle Δ푃 ∈ 푃 as in Equation (2.1.18), and we implicitly view it as an element ⊗2 ⊗2 of 퐵 using the inclusion 퐵 ⊂ 푃. We let Δ퐴 ∈ 퐴 be the image of Δ푃 under the projection 휋 ∶ 퐵 → 퐴. Let 퐾 be the kernel of 휓, which is also the kernel of 휆 = 휓 ⊕ id. While they are quite useful, we do not know whether all manifolds with boundary admit a surjective pretty model. We will deal with a more general notion, which we dub “Poincaré–Lefschetz duality models” (or “PLD models” for short), and which are available as soon as dim 푀 ≥ 7 and 푀 and 휕푀 are simply connected. The goal is to obtain a diagram similar to Equation (3.2.1) without ∨ ∨ assuming that 퐵 (resp. 퐵휕) is of the form 푃 ⊕휓! 푄 [1 − 푛] (resp. 푄 ⊕ 푄 [1 − 푛]).

104 3.2 Poincaré–Lefschetz duality models

Let us first motivate our definition. It is always possible to find a surjective model 휌 for 푀:1 푓 ∗ 푅 ∼ ΩPA(푀) 휌 res (3.2.2) 푓휕 ∗ 푅휕 ∼ ΩPA(휕푀)

We can still define a chain map (휀, 휀휕) ∶ cone(휌) → ℝ[−푛 + 1] by 휀(푥) = ∫ 푓 (푥) and 휀 (푦) = ∫ 푓 (푦). If we let 퐾 = ker(휌), this yields dual maps 푀 휕 휕푀 휕 푅 ∨ ∨ 휃푅 ∶ 푅 → 퐾푅 [−푛] and 휗푅 ∶ 퐾푅 → 푅 [−푛], which are quasi-isomorphisms by Poincaré–Lefschetz duality. However, they are not necessarily isomorphisms, contrary to the maps 휃퐵, 휗퐵 defined in the setting of surjective pretty models. This prevents us from carrying out the arguments of Section 3.3. The idea of this section is to adapt the proofs of [LS08b] in order to obtain a surjective model of (푀, 휕푀) for which 휃 is surjective and 휗 is injective. If the morphism 휃푅 were surjective (which is equivalent to 휗푅 being injective), then we would obtain an induced isomorphism 푅/ ker 휃 ≅ 퐾∨[−푛], and the quotient map 푅 → 푅/ ker 휃푅 would be a quasi-isomorphism by the 2-out-of-3 property. We would thus get an actual isomorphism between a model of 푀 and a model for the homology of 푀.

3.2.2 Definition and existence Definition 3.2.3. A Poincaré–Lefschetz duality pair (or “PLD pair” for short) of formal dimension 푛 is a CDGA morphism 휆 ∶ 퐵 → 퐵휕 between two connected CDGAs, equipped with a chain map2 휀 ∶ cone(휆) → ℝ[1 − 푛], and such that:

• The pair (퐵 , 휀 ) is a Poincaré duality CDGA of formal dimension 푛 − 1; 휕 퐵휕 ∨ • Let 퐾 ≔ ker 휆, and let 휃퐵 ∶ 퐵 → 퐾 [−푛] be defined by 휃퐵(푏)(푘) = 휀(푏푘); then we require 휃퐵 to be surjective and to be a quasi-isomorphism.

휆 If (퐵 −→퐵휕, 휀) is a PLD pair, then

• the CDGA 퐵 is quasi-isomorphic to its quotient 퐴 ≔ 퐵/ ker 휃퐵; ∨ ⊗2 • the map 휃 ∶ 퐴 → 퐾 [−푛] induced by 휃퐵 is an isomorphism of 퐵 -modules;

1 We can also assume that the maps 푓, 푓휕 factor through the sub-CDGAs of trivial forms in order to integrate along the fibers of the canonical projections,CW16 see[ , Appendix C]. 2Recall that this is equivalent to the data of two linear maps 휀 ∶ 퐵푛 → ℝ and 휀 ∶ 퐵푛−1 → ℝ 퐵 퐵휕 휕 satisfying the “Stokes formula” 휀 (푑푥) = 휀 (휆(푥)) and 휀 (푑푦) = 0. 퐵 퐵휕 퐵휕

105 3 Configuration Spaces of Manifolds with Boundary

푖 푛−푖 • Equivalently, the pairing 퐴 ⊗ 퐾 → ℝ, given by 푎 ⊗ 푘 ↦ 휀퐵(푎푘), is non-degenerate for all 푖 ∈ ℤ. Example 3.2.4. A surjective pretty model is an example of a PLD pair. Definition 3.2.5. A Poincaré–Lefschetz duality model of (푀, 휕푀) is a PLD pair 휆 ∶ 퐵 → 퐵휕 which is a model of the inclusion 휕푀 ⊂ 푀, in the sense that we can fill out the following diagram:

∼ ∼ ∗ 퐵 푅 ΩPA(푀) 휆 휌 res . ∼ ∼ ∗ 퐵휕 푅휕 ΩPA(휕푀)

Remark 3.2.6. In that case, 퐻∗(퐴) ≅ 퐻∗(푀), 퐻∗(퐾) ≅ 퐻∗(푀, 휕푀), and the ∗ isomorphism 휃 is given by Poincaré–Lefschetz duality 퐻 (푀) ≅ 퐻푛−∗(푀, 휕푀). Proposition 3.2.7. Let 푀 be a simply connected 푛-manifold with simply connected boundary, and assume that 푛 ≥ 7. Then (푀, 휕푀) admits a PLD model.

Proof. We start with some surjective model 휌 ∶ 푅 → 푅휕 as in Equation (3.2.2), and we will build a PLD model out of it. We keep the terminology of [LS08b]. We can find a surjective quasi-isomorphism 푔휕 ∶ 푅휕 → 퐵휕, where 퐵휕 is a Poincaré duality CDGA, by [LS08b]. We let ‴ ′ 퐾푅 be the kernel of 휌 ≔ 푔휕 ∘ 휌 and we define the chain map 휀푅 ∶ cone(휌 ) → ℝ[−푛 + 1]. Let 풪 ≔ ker 휗푅 ⊂ 퐾푅 be the ideal of “orphans”, i.e.

풪 ≔ ker 휗푅 = {푦 ∈ 퐾푅 ∣ ∀푥 ∈ 푅, 휗푅(푦)(푥) = 휀푅(푥푦) = 0}. (3.2.8) We could consider (for a moment) the new short exact sequence:

휆 0 → (퐾 ≔ 퐾푅/풪) → (퐵 ≔ 푅/풪) 퐵휕 → 0. (3.2.9)

∨ There is an induced chain map 휀퐵 ∶ cone(휆) → ℝ[−푛 + 1], and 휗퐵 ∶ 퐾 → 퐵 [−푛] is injective because we killed all the orphans. Thus we do obtain an isomorphism ∨ 퐵/ ker 휃퐵 ≅ 퐾 [−푛] induced by 휃퐵. The problem is that the ideal 풪 is not necessarily acyclic, thus 휆 is not necessarily a model of (푀, 휕푀) anymore. Indeed, by Poincaré–Lefschetz duality, all we know is that a cycle 표 ∈ 풪 is always the boundary of some element 푧 ∈ 퐾; but one may not always choose 푧 ∈ 풪. The idea, just like in [LS08b], is to extend the CDGA 푅 by acyclic cofibrations (over 퐵휕) in order to make the ideal of orphans acyclic. Thanks to our connectivity assumptions on the manifold and its boundary, we can assume that the model 휌 ∶ 푅 → 푅휕 of (푀, 휕푀) and the chain map ′ 휀푅 ∶ cone(휌 ) → ℝ[−푛 + 1] satisfy:

106 3.2 Poincaré–Lefschetz duality models

′ ′ • 휌 is surjective, hence so is 휌 ≔ 푔휕 ∘ 휌, and we let 퐾푅 = ker(휌 );

• both 푅 and 푅휕 are of finite type;

0 0 1 1 • both 푅 and 푅휕 are simply connected, i.e. 푅 = 푅휕 = ℝ and 푅 = 푅휕 = 0;

2 2 3 • we have 푅휕 ⊂ ker 푑 and 퐾 ⊂ ker 푑;

• the morphisms 휃푅, 휗푅 induced by 휀푅 are quasi-isomorphisms.

We say that the pair (휌, 휀푅) is a “good” pair if it satisfies all these assumptions. Let us say that its orphans are 푘-half-acyclic if 퐻푖(풪) = 0 for 푛/2 + 1 ≤ 푖 ≤ 푘. This condition is void when 푘 = 푛/2, and if 푘 = 푛 + 1 then Poincaré–Lefschetz duality implies that 풪 is acyclic (see [LS08b, Proposition 3.6]). We will now work by induction (starting at 푘 = 푛/2) and we assume that we are given a good pair (휌, 휀푅) whose orphans are (푘 − 1)-half-acyclic. We will build an extension:

휌 0 퐾푅 푅 푅휕 0

∼ ∼ = (3.2.10) ̂ ̂ ̂휌 0 퐾푅 푅 푅휕 0

′̂ and an extension 푅̂휀 ∶ cone(휌 ) → ℝ[−푛 + 1] of 휀푅 such that ( ̂휌, ̂휀) is good and its orphans are 푘-half-acyclic. We follow closely [LS08b, Sections 4 and 5], adapting the proof where needed. Let 푙 = dim(풪푘 ∩ker 푑)−dim(푑(풪푘−1)) and choose 푙 linearly independent cycles 푘 훼1, … , 훼푙 ∈ 풪 such that

푘 푘−1 풪 ∩ ker 푑 = 푑(풪 ) ⊕ ⟨훼1, … , 훼푙⟩. (3.2.11)

These are the obstructions to 풪 being 푘-half acyclic. Because 휗푅 is a quasi- ′ 푘−1 ′ isomorphism and 휗푅(훼푖) = 0, there exists 훾푖 ∈ 퐾 such that 푑훾푖 = 훼푖. ∗ ∗ Let 푚 ≔ dim 퐻 (푅) = dim 퐻 (퐾푅), and choose cycles ℎ1, … , ℎ푚 ∈ 푅 such that ∗ ([ℎ1], … , [ℎ푚]) is a basis of 퐻 (푅). By Poincaré–Lefschetz duality there exists ′ ′ ′ ∗ cycles ℎ1, … , ℎ푚 in 퐾푅 such that 휀(ℎ푖ℎ푗) = 훿푖푗 (and these form a basis for 퐻 (퐾)). Let ′ ′ ′ 푘−1 훾푖 ≔ 훾푖 − ∑ 휀(훾푖 ℎ푗)ℎ푗 ∈ 퐾 , (3.2.12) 푗

3Since we require 휌 to be surjective, it is possible that not all elements of 푅2 are cycles, because some of the classes from 퐻2(휕푀) may need to be killed. Check Example 3.1.16 when 푀 = 퐷3.

107 3 Configuration Spaces of Manifolds with Boundary

푘−1 and let Γ be the subspace of 퐾 generated by the 훾푖. Then a proof similar to the proof of [LS08b, Lemma 4.1] shows that 푑훾푖 = 훼푖, and if 푦 ∈ 푅 is a cycle, then 휀(훾푖푦) = 휗(훾푖)(푦) = 0. Now let

̂ 푅 ≔ (푅 ⊗ 푆(푐1, … , 푐푙, 푤1, … , 푤푙), 푑푐푖 = 훼푖, 푑푤푖 = 푐푖 − 훾푖), (3.2.13) where the 푐푖 and the 푤푖 are new variables of degrees 푘 − 1 and 푘 − 2. ̂ Extend 휌 to ̂휌∶ 푅 → 푅휕 by declaring that ̂휌(푐푖) = ̂휌(푤푖) = 0. It follows that ̂ 퐾푅 ≔ ker ̂휌= (퐾푅 ⊗ 푆(푐푖, 푤푗)1≤푖,푗≤푙, 푑). (3.2.14)

It is not hard to see that 푅 ↪ 푅̂ is an acyclic cofibration (compare with [LS08b, ̂ Lemma 4.2]), and so is 퐾 ↪ 퐾. Finally, since all the 훾푖 and 훼푖 are in ker ,̂휌 one can extend 휀푅 to 푅̂휀 ∶ cone( ̂휌)→ ℝ[−푛 + 1] by formulas identical to [LS08b, Equation 4.5], which works because 푛 ≥ 7. It is clear that ( ̂휌,푅 ̂휀 ) is still a good ̂ ̂ pair. We let 휃푅, 휗푅 be the quasi-isomorphisms induced by the pairing, and we ̂ ̂ finally let 풪 ≔ ker 휗푅. It remains to check that 풪̂ is 푘-half-acyclic, knowing that 풪 is (푘−1)-half-acyclic. First, we can reuse the proofs of [LS08b, Lemmas 5.2 and 5.3] to check that if 푖 > 푛 − 푘 + 2 then 풪푖 ⊆ 풪푖̂ , and if 푖 ∈ {푘 − 2, 푘 − 1} then 풪푖̂ ∩ ker 푑 ⊂ 풪푖 ∩ ker 푑. The only difference is that instead of the hypothesis 푑(푅2) = 0, we instead have 2 푑(퐾푅) = 0. Now, just like [LS08b], we have several cases to check. If we have 푘 ≥ 푛/2 + 2, or if we have 푛 odd and 푘 = (푛+1)/2+1, then the proof goes through unchanged up to slight changes of notation. However, if 푛 is even and 푘 = 푛/2 + 1, more significant adaptations are needed, even if the idea is the same. We’d like tocheck that 풪푘̂ ∩ ker 푑 ⊂ 푑(풪푘−1̂ ). We already know that

푘̂ 푘 푘−1 풪 ∩ ker 푑 ⊂ 풪 ∩ ker 푑 = 푑(풪 ) ⊕ ⟨훼푖⟩. (3.2.15)

Since 풪푘−1 ⊂ 풪푘−1̂ (the proof is identical to [LS08b, Lemma 5.7]), it is sufficient 푘−1̂ ̄ to check that 풪 ∩ ⟨훼푖⟩ = 0. Let us write 푗 ∶ 퐾푅 → 푅 for the inclusion, 푅 ≔ 푅/풪, ̄ 퐾푅 ≔ 퐾푅/풪, 푛 = 2푚 and 푘 = 푛/2+1 = 푚+1. Then we have long exact sequences in cohomology and morphisms between them:

0 퐻푚(푅) 휋 퐻푚(푅)̄ 훿 퐻푚+1(풪) 0

푗 ̄횥 = (3.2.16) 푚 휋 푚 ̄ 훿 푚+1 0 퐻 (퐾) 퐻 (퐾푅) 퐻 (풪) 0

108 3.2 Poincaré–Lefschetz duality models

푚+1 The space 퐻 (풪) is generated by the classes of the 훼푖. We obtain a section 푚+1 푚 ̄ 휎 ∶ 퐻 (풪) → 퐻 (퐾푅) of 훿 by letting 휎([훼푖]) ≔ [훾푖]. Suppose that we have some nonzero element 훼 ∈ ⟨훼푖⟩; we’d like to show 푘̂ ̂ 푚 ̄ that it is not in 풪 , i.e. that it is not an orphan in 퐾푅. The pairing 퐻 (퐾푅) ⊗ 푚 ̄ 퐻 (푅) → ℝ induced by 푅̄휀 is non-degenerate, and 휎(훼) ≠ 0, hence there exists some 훽 ∈ 퐻푚(푅)̄ such that 휀(휎(훼)훽) ≠ 0. But as we saw before, for any 푚 [ℎ] = ∑ 푥푗[ℎ푗] ∈ 퐻 (푅), 휀푅(휎(훼)ℎ) = 0, because 휀푅(훾푖ℎ푗) = 0. It follows that 휀 (휎(훼)휎(훿(훽))) = 휀 (휎(훼)훽) ≠ 0 훿(훽) = ∑ 훽 훾 ∈ 퐻푚+1(풪) 푅 푅 . If we write 푗 푗 푗 , we 푤 = ∑ 훽 푤 ∈ 푅̂ ̂휀 (훼푤) = 휀 (휎(훼)휎(훿(훽))) ≠ 0 can let 푗 푗 푗 , and then by definition 푅 푅 , 푘−1̂ thus 훼 is not an orphan. This completes the proof that 풪 ∩ ⟨훼푖⟩ = 0, and thus that 풪푘̂ ∩ ker 푑 ⊂ 푑(풪푘−1̂ ). We have thus covered all the cases to prove that if 풪 is (푘 − 1)-half-acyclic, then 풪̂ is 푘-half-acyclic. ̂ By induction, we obtain a good pair ( ̂휌∶ 푅 → 푅휕,푅 ̂휀 ) whose ideal of orphans is (푛 + 1)-half acyclic (and hence actually acyclic), obtained by a sequence of ̂ ̂ acyclic cofibrations from 푅. It then remains to define 퐵 ≔ 푅/풪 and 휀퐵 to be the map induced by 푅̂휀 on the quotient to prove Proposition 3.2.7. Given a PLD model of (푀, 휕푀), we obtain a diagram (similar to (3.2.1)):

non degen. 퐾 ≔ ker 휆 pairing

퐴 ≔ 퐵/ ker 휃 ∼ 퐵 ∼ 푅 ∼ Ω∗ (푀) (3.2.17) 퐵 휋 푔 푓 PA 휆 휌 res ∼ ∼ ∗ 퐵휕 푔 푅휕 ΩPA(휕푀) 휕 푓휕

We also see in the proof that 휀 satisfies by construction 휀 푔 = ∫ 푓 (−). We 퐵 퐵 푀 also have 휀 푔 = ∫ 푓 (−) by construction (see Remark 2.1.17). 퐵휕 휕 휕푀 휕

3.2.3 Diagonal classes

휆 Let (퐵 −→퐵휕, 휀) be a Poincaré–Lefschetz duality pair, with 퐾 = ker 휆. Recall that ∨ ⊗2 we write 휃 ∶ (퐴 = 퐵/ ker 휃퐵) → 퐾 [−푛] for the isomorphism of 퐵 -modules ∨ induced by the surjective 휃퐵 ∶ 퐵 → 퐾 [−푛]. ⊗2 The multiplication 휇퐾 ∶ 퐾 → 퐾 can then be dualized into a morphism of 퐵⊗2-modules 훿 ∶ 퐴 → 퐴⊗2[−푛], and we let

푛 Δ퐴 ≔ 훿(1퐴) ∈ (퐴 ⊗ 퐴) (3.2.18)

109 3 Configuration Spaces of Manifolds with Boundary

be a representative of the diagonal class. Graded commutativity of 휇퐾 implies 21 푛 ⊗2 that Δ퐴 = (−1) Δ퐴. The fact that 훿 is a morphism of 퐵 -modules imply that Δ퐴 satisfies ∀푎 ∈ 퐴, 훿(푎) = (푎 ⊗ 1)Δ퐴 = Δ퐴(1 ⊗ 푎). (3.2.19) Remark 3.2.20. If the PLD model comes from a surjective pretty model, then we ⊗2 ⊗2 check that this diagonal class is indeed the image of Δ푃 ∈ 푃 ⊂ 퐵 under the projection 휋 ∶ 퐵 → 퐴.

The duality between 퐴 and 퐾 can also be turned into a cocycle Δ퐴퐾 ∈ 퐴⊗퐾. Let ∗ ∗ {푎푖} be some basis of 퐴, and let {푎푖 } be the dual basis of 퐾, satisfying 휀퐵(푎푖푎푖 ) = 훿푖푗. Then we define: ∗ 푛 Δ퐴퐾 ≔ ∑ ±푎푖 ⊗ 푎푖 ∈ (퐴 ⊗ 퐾) . (3.2.21)

It is not hard to check that Δ퐴 is the image of Δ퐴퐾 under the composite map 1⊗휋∘휄∶퐴⊗퐾→퐴⊗퐵→퐴⊗퐴, and by definition:

″ ′ ∀푎 ∈ 퐴, ∑ 휀퐵(푎Δ퐴퐾)Δ퐴퐾 = 푎. (3.2.22) (Δ퐴퐾)

Note also that, under the multiplication map 휇퐴 ∶ 퐴 ⊗ 퐴 → 퐴, we have ∗ 휇퐴(Δ퐴) = ∑ ±휋(푎푖푎푖 ) = 휒(퐴)휋(vol퐾) is equal to the Euler characteristic of 퐵 multiplied by the volume form vol퐾 ∈ 퐾 (the only element of 퐵 satisfying 휀퐵(vol퐾) = 1, representing the fundamental class of (푀, 휕푀)). But by degree ∨ reasons, vol퐾 ∈ 퐾 ⊂ 퐵 is in the kernel of 휃퐵 ∶ 퐵 → 퐾 [−푛] when 휕푀 ≠ ∅, hence 휋(vol퐾) = 0 by definition. In other words,

휇퐴(Δ퐴) = 0 if 휕푀 ≠ ∅. (3.2.23)

3.3 The model and its cohomology

From now on, let us assume that 푀 is a smooth, simply connected manifold with nonempty boundary. We also fix a PLD model of 푀 as in Equation (3.2.17).

3.3.1 The dg-module model of Conf푘(푀)

Given a finite set 푉 and an element 푣 ∈ 푉, define the canonical injection 휄푣 ∶ 퐴 → 퐴⊗푉 by 휄푣(푎) ≔ 1 ⊗ ⋯ ⊗ 1 ⊗⏟ 푎 ⊗1 ⊗ ⋯ ⊗ 1. (3.3.1) 푣

Definition 3.3.2. Define a symmetric collection of CDGAs G퐴 by: ⊗푉 ∨ G퐴(푉) ≔ (퐴 ⊗ e푛 (푉)/(휄푣(푎) ⋅ 휔푣푣″ = 휄푣′ (푎) ⋅ 휔푣푣′ ), 푑(휔푣푣′ ) = (휄푣 ⋅ 휄푣′ )(Δ퐴)) with the obvious actions of the symmetric groups.

110 3.3 The model and its cohomology

This definition also makes sense when 휕푀 = ∅, and it yields the symmetric collection of CDGAs considered in Section 2.2. When 푀 is a closed manifold, as soon as its Euler characteristic 휒(푀) van- ∨ ishes, then there is a structure of Hopf right e푛 -comodule on G퐴 (Proposi- tion 2.2.1), with cocomposition structure maps characterized by (compare with Equation (2.1.10)):

∨ ′ ∘푇 (휔푣푣′ ) = 1 ⊗ 휔푣푣′ , if {푣, 푣 } ⊂ 푇; ∨ ′ ∘푇 (휔푣푣′ ) = 휔[푣][푣′] ⊗ 1, if {푣, 푣 } ⊄ 푇; (3.3.3) ∨ ∘푇 (휄푣(푎)) = 휄[푣](푎) for 푎 ∈ 퐴, 푣 ∈ 푉; where [푣] ∈ 푉/푇 is the class of 푣 in the quotient.

Proposition 3.3.4. If 휕푀 ≠ ∅, then the symmetric collection of CDGAs G퐴 forms a ∨ right Hopf e푛 -comodule, with the same formulas. Proof. Comparing with the proof of Proposition 2.2.1, we see that almost all the arguments are the same. The only difficulty is to check that the cocomposition is compatible with the differential, which required that the Euler characteristic vanished in the boundaryless case. ∨ ∨ It is immediate to check that 푑(∘푇 (휄푣(푎))) = ∘푇 (푑(휄푣(푎))) = 휄[푣](푑푎), thus it suffices to check that the same equality holds on the generators 휔푣푣′ . If either 푣 ∉ 푇 or 푣′ ∉ 푇, this is again immediate; hence it suffices to check that

∨ ∨ ′ 푑(∘푇 (휔푣푣′ )) = ∘푇 (푑(휔푣푣′ )) for 푣, 푣 ∈ 푇 The LHS of that equation always vanishes. On the other hand, the RHS is equal to 휄∗(휇퐴(Δ퐴)), where 휇퐴 ∶ 퐴 ⊗ 퐴 → 퐴 is the product. But by Equation (3.2.23), 휇퐴(Δ퐴) = 0. Example 3.3.5. Recall from Example 3.1.16 the model for (퐷푛, 푆푛−1), with 퐴 = ℝ, ∨ and Δ퐴 = 0. It follows that in this case, G퐴 is isomorphic to e푛 seen as a Hopf right comodule over itself. This is not surprising, given that FM푛 is formal as an operad, and hence as a module over itself, and that SFM퐷푛 (∅, −) is weakly equivalent to FM푛 as a right FM푛-module.

3.3.2 Computing the homology

We now prove that G퐴 has the right cohomology, in the spirit of [LS08a] and using the methods of [CLS15b] to deal with manifolds with boundary. From then on and until the end of this section, we fix some integer 푘 ≥ 0. We can work over ℚ in this section.

111 3 Configuration Spaces of Manifolds with Boundary

The general idea goes as follows. If 푊 is a manifold with boundary, and 푋 ⊂ 푊 is a sub-polyhedron, then by [CLS15b], it suffices to know a CDGA model ofthe square of inclusions 휕푊 푊 (3.3.6)

휕푊푋 ≔ 푋 ∩ 휕푊 푋 to obtain a complex computing the cohomology of 푊 − 푋. Therefore, to compute the cohomology of Conf푘(푀), we need to find such 푘 ⋃ 푘 models for 푊 = 푀 and 푋 = Δ(푘) ≔ 1≤푖,푗≤푘 Δ푖푗, where Δ푖푗 ≔ {푥 ∈ 푀 ∣ 푥푖 = 푥푗}. Since the sub-polyhedron Δ(푘) can be decomposed into the sub-polyhedra Δ푖푗, we can use the techniques of [LS08a] to further simplify the description of the dg- module model as a “total cofiber” indexed by graphs, which will be isomorphic to G퐴. Let us now give the details. Let 퐸 = {(푖, 푗) ∣ 1 ≤ 푖 < 푗 ≤ 푘} be a set of pairs, and let Γ be the poset of subsets of 퐸 ordered by reverse inclusion. We can see an element 훾 ∈ Γ as a graph on 푘 vertices, with an edge between 푖 and 푗 iff (푖, 푗) ∈ 훾. In particular ∅ ∈ Γ is the “empty” graph with no edges (but 푘 vertices). Using this point of view, we can define the “zeroth homotopy group” 휋0(훾) of a graph 훾 ∈ Γ, which is a partition of {1, … , 푘}. We obtain a functor ∇ from Γ to the category of topological spaces defined by

푘 훾 ↦ ∇(훾) ≔ ⋂ Δ푒 ⊂ 푀 , (3.3.7) 푒∈퐸훾

푘 where Δ(푖,푗) is simply the small diagonal Δ푖푗. Note that ∇(∅) = 푀 , and that if 훾′ ⊃ 훾 then there is an inclusion ∇(훾′) ⊂ ∇(훾). The space ∇(훾) is homeomorphic to the product 푀휋0(훾), and under these homeomorphisms, the inclusion ∇(훾′) ⊂ ∇(훾) is the cofibration induced by iterations of the diagonal map 푀 → 푀 × 푀. We thus obtain that:

휋0(훾) Δ(푘) = ⋃ Δ푖푗 = colim훾∈Γ ∇(훾) = colim훾∈Γ 푀 , (3.3.8) 1≤푖<푗≤푘 and this is in fact a homotopy colimit. We will first aim to build a CDGA model for the square (3.3.6) (with 푊 = 푀푘 and 푋 = Δ(푘)) out of the diagram (3.2.1). The previous description of Δ(푘) as ⊗휋0(훾) a homotopy colimit tells us that a model for Δ(푘) is given by lim훾∈Γop 퐵 , where the maps in the diagram are induced by iterations the multiplication 휇퐵 ⊗푘 ⊗휋0(∅) of 퐵. The inclusion Δ(푘) is modeled by the canonical map from 퐵 = 퐵 to the colimit.

112 3.3 The model and its cohomology

푘 It remains to find a model for 휕푊푋 = Δ(푘) ∩휕(푀 ) and models for the inclusion maps. The morphism 퐵 → 퐵휕 is surjective, hence 퐵휕 is isomorphic to 퐵/퐾 where 퐾 ≔ ker(퐵 → 퐵휕). We get: Lemma 3.3.9. For all 푖 ≥ 0, the left-hand side square is a CDGA model for the right-hand side square, where the horizontal maps are the diagonal maps:

(푖) 휇퐵 훿 퐵 퐵⊗푖 푀 푀푖 is a model for (푖) 휇퐵 훿 퐵/퐾 퐵⊗푖/퐾⊗푖 휕푀 휕(푀푖)

Proof. The idea is the same as in [CLS15b, Proposition 5.1]. We work by induction. The case 푖 = 1 is obvious (as 퐵휕 ≅ 퐵/퐾), and they prove the case 푖 = 2. Now let use assume that the proposition is true for a given 푖 ≥ 2. There is a diagram, where all the inclusions are either induced by diagonal maps or induced by 휕푀 ⊂ 푀: 푀 × 푀푖 푀

휕(푀 × 푀푖) 푀 × 휕(푀푖) .

(ho. pushout)

(휕푀) × 푀푖 (휕푀) × (휕(푀푖)) 휕푀 The diagram of the proposition is the “outer” diagram, and the bottom left square is a (homotopy) pushout. Let 푃 be the (homotopy) pullback in CDGAs of

⊗푖 ⊗푖 ⊗푖 ⊗푖 ⊗푖 퐵휕 ⊗ 퐵 → 퐵휕 ⊗ (퐵 /퐾 ) ← 퐵 ⊗ (퐵 /퐾 ).

Then the induction hypothesis and the fact that homotopy pushouts of spaces become homotopy pullbacks of models imply that the following diagram is a CDGA model of the previous one (where the maps are either induced by ⊗2 휇퐵 ∶ 퐵 → 퐵 or 휆 ∶ 퐵 → 퐵휕):

퐵 ⊗ 퐵⊗푖 퐵

푃 퐵 ⊗ (퐵⊗푖/퐾⊗푖)

(ho. pullback)

⊗푖 ⊗푖 ⊗푖 퐵휕 ⊗ 퐵 퐵휕 ⊗ (퐵 /퐾 ) 퐵휕

113 3 Configuration Spaces of Manifolds with Boundary

Now, as in the proof [CLS15b, Lemma 5.3], it is clear that the natural map 퐵⊗(푖+1) → 푃 is surjective and that its kernel is 퐾⊗(푖+1), in other words that 푃 ≅ 퐵⊗(푖+1)/퐾⊗(푖+1). The proposition then follows immediately.

푘 The space Δ(푘) ∩ 휕(푀 ) admits a description as a colimit similar to the one Equation 3.3.8. Indeed, a point of 푀푘 is in the boundary iff one of the coordinates is in the boundary of 푀. Now if a point is in both ∇(훾) and 휕(푀푘), then at least one of the coordinates (say 푥푖) is in the boundary, and thus all the points indexed by some 푗 in the same connected component as 푖 in 훾 is also in the boundary. We then obtain:

Δ ∩ 휕(푀푘) = colim colim (휕푀)푆 × 푀휋0(훾)−푆. (3.3.10) (푘) 훾∈Γ ∅⊊푆⊂휋0(훾)

For a fixed 훾, the inner colimit is precisely the image of 휕(푀휋0(훾)) under the diagonal embedding 푀휋0(훾) ↪ 푀푘. Combining this with the previous lemma, we then obtain:

푘 Proposition 3.3.11. A model for the square (3.3.6), with 푊 = 푀 and 푋 = Δ(푘), is given by:

훼 퐵⊗푘 푘 퐵⊗푘/퐾⊗푘

휉푘

훽푘 ⊗휋0(훾) ⊗휋0(훾) ⊗휋0(훾) lim훾∈Γop 푅 lim훾∈Γop 푅 /퐾

The map 훼푘 is surjective, thus we have a canonical quasi-isomorphism ker 훼푘 = ⊗푘 ∼ 퐾 hoker 훼푘. Therefore, by [CLS15b, Proposition 3.1]:

Corollary 3.3.12. The cohomology of the cone

∨ 휉 ̄ ⊗푘 ∨ cone((hoker 훽푘) [−푛푘] 푘 (퐾 ) [−푛푘]) of the map induced by 휉푘 on the kernels is isomorphic, as a graded vector space, to the 푘 cohomology of Conf푘(푀) = 푀 − Δ(푘).

Armed with this corollary, we can now define a cubical diagram 퐶• (see [LS08a, Section 7]) whose total cofiber computes the cohomology of Conf푘(푀) as a graded vector space. Given 훾 ∈ Γ, the chain complex 퐶훾 is defined to be ⊗휋0(훾) ∨ ⊗푘 ∨ ′ (퐾 ) ; note in particular that 퐶∅ = (퐾 ) . If 훾 ⊃ 훾, then the map 퐶훾′⊃훾 ∶ 퐶훾′ → 퐶훾 is induced by the dual of the multiplication of 퐾.

114 3.3 The model and its cohomology

Recall that the total cofiber of 퐶• is a representative of the homotopy colimit of 퐶, given by the chain complex [LS08a, Definition 7.2]:

TotCof 퐶• ≔ (⨁ 퐶훾 ⋅ 푦훾, 퐷), (3.3.13) 훾∈Γ where 푦훾 is some variable of degree −#퐸훾, deg(푥 ⋅ 푦훾) = deg(푥) + deg(푦훾), and

퐷(푥 ⋅ 푦훾) = ±(푑푥) ⋅ 푦훾 + ∑ ±(퐶훾′⊃훾푥) ⋅ 푦훾−푒. (3.3.14) 푒∈퐸훾

Proposition 3.3.15. The total cofiber of 퐶• computes the cohomology of Conf푘(푀) as a graded vector space, up to suspension by 푛푘.

Proof. The proof is almost identical to the proof of [LS08a, Theorem 9.2]. First ′ define an auxiliary cubical diagram 퐶• just like 퐶•, except for 퐶∅ which we set ∨ ′ ⊗휋0(훾) equal to (hoker 훽푘) . The maps 퐶훾⊃∅ are induced by the inclusions 퐾 → ⊗휋0(훾) 퐵 , which go through the definition of hoker 훽푘. By Poincaré–Lefschetz ∨ ⋃ duality, (hoker 훽푘) is a model for Δ(푘) = 훾∈Γ ∇(훾), each 퐶훾 is a model for ′ ∇(훾), and the maps between the 퐶훾 are models for the inclusions by Lemma 3.3.9. ′ We thus obtain that TotCof 퐶• is acyclic by [LS08a, Proposition 9.1] and the homotopy invariance of total cofibers. ′ ∨̄ The morphism 퐶훾 → 퐶훾 is given by the identity if 훾 ≠ ∅, and it is given by 휉푘 ′ ″ if 훾 = ∅. This yields a morphism of cubical diagrams 퐶• → 퐶•. Define 퐶• to be ′ the object-wise mapping cone of 퐶• → 퐶•, so that there is a short exact sequence

′ ″ 0 → 퐶• → 퐶• → 퐶• → 0.

′ ″ For 훾 ≠ ∅, the map 퐶훾 → 퐶훾 is the identity, hence 퐶훾 is acyclic. It follows that ″ ∨̄ ′ TotCof 퐶• is quasi-isomorphic to the cone of 휉푘 ∶ 퐶∅ → 퐶∅, which computes the cohomology of Conf푘(푀) by Corollary 3.3.12. There is a long exact sequence between the homologies of the total cofibers ′ ″ 퐶•, 퐶•, and 퐶•: total cofibers commute with mapping cones up to homotopy, because both are types of homotopy colimits. The proposition then follows from ′ the fact that 퐻∗(TotCof 퐶•) = 0.

Theorem 3.3.16. Let 푀 be a simply connected manifold with simply connected boundary and which admits a Poincaré–Lefschetz duality model. Let 퐴 be the model of 푀 obtained from this PLD model (see Section 3.2). Then there is an isomorphism of graded vector ∗ ∗ spaces between 퐻 (G퐴(푘); ℚ) and 퐻 (Conf푘(푀); ℚ).

115 3 Configuration Spaces of Manifolds with Boundary

Proof. There is in fact an isomorphism of dg-modules

G퐴(푘) ≅ (TotCof 퐶•)[−푛푘]. It is induced by the Poincaré–Lefschetz isomorphism 휃 ∶ 퐴 → 퐾∨[−푛]. We use the crucial fact that the multiplication of 퐾 is dual, under this isomorphism, to the ⊗2 map 퐴 → 퐴 defined by 푎 ↦ (푎 ⊗ 1)Δ퐴 = (1 ⊗ 푎)Δ퐴 (which is true by definition of Δ퐴 in our setting). The proof is then similar to the proof of Lemma 2.5.9.

In other words, G퐴(푘) is a dg-module model of Conf푘(푀). Unfortunately, in general if 휕푀 ≠ ∅ then G퐴(푘) is not an actual model of Conf푘(푀): the algebra structure is not the correct one.

3.3.3 The perturbed model

We define in the next section a “perturbed” version of G퐴(푉), which will be the actual model for Conf푘(푀), and we will prove that it is isomorphic to G퐴(푉) as a dg-module. Choose some section 푠 ∶ 퐵휕 → 퐵 of 휌. Using the Poincaré duality of 퐵휕, this section 푠 corresponds to some element 휎퐵 ∈ 퐵휕 ⊗ 퐵 of degree 푛 − 1. Note that, ⊗2 by definition, the image of 휎퐵 in 퐵휕 is the diagonal class of 퐵휕. The map 푠 does not commute, in general, with the differential, though for any 푥 ∈ 퐵휕, one has 푑푠(푥) − 푠(푑푥) ∈ 퐾 = ker 휌. It follows that in general 푑휎퐵 ≠ 0, but 푑휎퐵 ∈ 퐵휕 ⊗ 퐾. We in fact have that 푑휎퐵 is a representative of the image of Δ퐴퐾 under the map 퐵 ⊗ 퐾 → 퐵휕 ⊗ 퐾. Let ′ ″ 휎퐴 = (1 ⊗ 휋)(휎퐵) ≔ ∑ 휎 ⊗ 휎 ∈ 퐵휕 ⊗ 퐴. (3.3.17) (휎퐴) Remark 3.3.18. We are going to use Sweedler’s notation as above extensively. If the element 휎 (or any other tensor) appears multiple times in an equation, we are going to write it as follows:

′ ″ ′ ″ 휎 ⊗ 휎 = ∑ 휎1 ⊗ 휎1 ⊗ 휎2 ⊗ 휎2 and so on if it appears more than twice. We define the perturbed model to be: ⊗푉 G퐴̃ (푉) ≔ (퐴 ⊗ 푆( ̃휔푣푣′ )푣,푣′∈푉/퐽, 푑 ̃휔푣푣′ = (휄푣 ⋅ 휄푣′ )(Δ퐴)) (3.3.19) 2 where the ideal of relations is generated by ̃휔푣푣′ = 0 and, for all 푏 ∈ 퐵 and all subsets 푇 ⊂ 푉 of cardinality at least two:

′ ″ ∑ ±(휄푣(휋(푏)) ⋅ ∏ ̃휔푣푣′ ) + ∑ ±휀휕(휌(푏) ∏ 휎푣) ∏ 휄푣(휎푣 ). (3.3.20) 푣∈푇 푣≠푣′∈푇 푣∈푇 푣∈푇

116 3.3 The model and its cohomology

Note in particular that we have

푛 ′ ′ ″ ″ ̃휔12 − (−1) ̃휔21 + ∑ ±휀휕(휎1휎2)휎1 ⊗ 휎2 = 0 ∈ G퐴̃ (2), (3.3.21) with the last summand being a cycle. The relations for #푇 = 2 are perturbations of the symmetry relation of G퐴(푉), and the relations for #푇 = 3 are perturbations ∨ of the Arnold relations in e푛 (푉) (where, by “perturbation”, we informally mean that the perturbed relation is the sum of the standard relation and terms which have a strictly lower number of generators ̃휔푣푣′ ). Example 3.3.22. Let us consider 푀 = 푆1 × [0, 1] (even though it doesn’t satisfy our assumptions about connectivity). We can find a PLD model for 푀 where:

∗ • 퐵휕 = 퐻 (휕푀) is four-dimensional, generated by 1, 푡, 푑휑, and 푡푑휑 with 푡 being idempotent;

• 퐴 = 퐻∗(푀) = 퐻∗(푆1) is two-dimensional, generated by 1 and 푑휑;

• 퐾 = 퐻∗(푀, 휕푀) is two-dimensional, generated by 푑푡 and 푑푡 ∧ 푑휑.

Then one of the nontrivial relation in G퐴̃ (2) is given by (푑휑 ⊗ 1) ̃휔21 + (1 ⊗ 푑휑) ̃휔12 + (푑휑 ⊗ 푑휑) = 0. To obtain some intuition about this relation, consider 2 2 that Conf2(푀) ≃ Conf2(ℝ − {0}) is homotopy equivalent to Conf3(ℝ ). The element 푑휑 correspond to the two points are rotating around the origin. Thus ∗ 2 we can identify ̃휔12 with 휔12 ∈ 퐻 (Conf3(ℝ )), 푑휑 ⊗ 1 with 휔13, and 1 ⊗ 푑휑 with 휔23. The perturbed relation in G퐴̃ (2) is then nothing but the usual Arnold ∨ ∗ 2 relation in e2 (3) = 퐻 (Conf3(ℝ )).

Proposition 3.3.23. There is an isomorphism of dg-modules between G퐴(푉) and G퐴̃ (푉). Proof. Let us fix 푉 = {1, … , 푘} for some 푘 ≥ 0. Consider the standard basis of ∨ e푛 (푘) given by monomials of the type: 휔 … 휔 , 푖1푗1 푖푟푗푟 with 1 ≤ 푖1 < ⋯ < 푖푟 ≤ 푟 and 푖푙 < 푗푙 for all 푙. By choosing some basis {푎1, … , 푎푚} of 퐴, we obtain a basis of G퐴(푘) by labeling the last element of each connected component of such a monomial by some 푎푖. We claim that if we replace all the 휔푖푗 by ̃휔푖푗 in this basis, then we obtain a basis of G퐴̃ (푉). Using the perturbed Arnold relations, it’s clear that any element of G퐴̃ (푉) can be written as a linear combination of these elements. Moreover, using the same argument that proves that there is no nontrivial relation between the ∨ elements of the standard basis of e푛 (푘), we can prove that there is non nontrivial relation between the elements of our claimed basis.

117 3 Configuration Spaces of Manifolds with Boundary

There is thus a linear isomorphism G퐴(푉) → G퐴̃ (푉) which is defined on the basis by replacing all the 휔푖푗 by ̃휔푖푗. It’s then clear that this map preserves the internal differential of 퐴 and the part of the differential which splits an 휔 푖푙푗푙 (which can be written down explicitly in the basis: it merely splits a connected component into two).

∗ Corollary 3.3.24. There is an isomorphism of graded vector spaces 퐻 (Conf푘(푀)) ≅ ∗ 퐻 (G퐴̃ (푘)). Proof. This follows immediately from Theorem 3.3.16

Remark 3.3.25. It is often the case that G퐴 and G퐴̃ are actually equal. For example, if ∗ 푛−1 푀 is obtained by removing a point from a closed manifold, then 퐵휕 = 퐻 (푆 ) = 2 푆(푣)/(푣 ), and 휎퐴 = 푣 ⊗ 1. Then in all the “corrective terms” in the definition of G퐴̃ (푉), the power of 푣 is at least two, hence the term vanishes. Example 3.3.26. If 푀 = 퐷푛 and we use the PLD model of Example 3.1.16, then G퐴̃ = G퐴 (see Example 3.3.5).

3.4 Computation of Swiss-Cheese graph cohomology

In this section, we prove statements related to the vanishing of the homology of a ∨ twist of SGC푛 (see Section 3.1.5). We assume that 푛 ≥ 3 throughout the section. Elements of SGC푛 are given by linear combinations of graphs; we identify an element in the dual with a (possibly infinite) sum of graphs using the dual basis. ∨ Let 훿 be the differential in SGC푛 : it is given by “splitting” an aerial vertex (in all ∨ possible ways) in two vertices connected by an edge. Then SGC푛 is a (complete) dg-Lie algebra, with the bracket given by [훾, 훾′] = 훾 ∘ 훾″ − ±훾′ ∘ 훾, where 훾 ∘ 훾′ is defined by the sum over the terrestrial vertices of 훾 of the insertion of 훾′ into that terrestrial vertex and reconnecting the incident edges in all possible ways. ∨ Recall the Kontsevich integrals 푐 ∈ SGC푛 (see Equation (3.1.24)). It is well- known (see e.g. [Wil15, Section 6.1]) that 푐 starts with a linear part 푐0 corresponding to the Hochschild–Kostant–Rosenberg map, which we now describe. Let 훾푗, for 푗 ≥ 0, be the graph with one aerial vertex, 푗 terrestrial vertex, and an edge from the aerial vertex to each terrestrial vertex.

훾푗 ≔ (3.4.1) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 푗 vertices

118 3.4 Computation of Swiss-Cheese graph cohomology

∨ 1 Definition 3.4.2. The linear part 푐0 ∈ SGC푛 of 푐 is given by 푐0(훾푗) = 푗! and 푐0(훾) = 0 if 훾 does not have exactly one aerial vertex.

∨ The element 푐0 itself is a Maurer–Cartan element in 푆퐺퐶푛 . In other words, 푐−푐0, which vanishes on all graphs with exactly one aerial vertex, is a Maurer–Cartan in the twisted Lie algebra

∨,푐0 ∨ SGC푛 ≔ (SGC푛 , 훿 + [푐0, −]). (3.4.3)

We’d like to prove that 푐 is gauge equivalent to 푐0, or equivalently that 푐 − 푐0 is gauge trivial in the twisted Lie algebra.

푖 ∨,푐0 Proposition 3.4.4. For each 푖 ∈ ℤ, the dimension of 퐻 (SGC푛 ) is at most equal to 푖 ∨ 푖 ∨ the dimension of 퐻 (GC푛 ) ⊕ 퐻 (GC푛−1). This is to be compared with a similar result for 푛 = 2 at the beginning of [Wil15, ∗ ∨,푐0 ∗ ∨ Section 5], stating that 퐻 (SGC2 ) ≅ 퐻 (GC2 ). This result relies in turn on [Wil16, Appendix F], and we will use similar proof techniques.

Proof of Proposition 3.4.4. Call an aerial vertex a “wedge” if it is bivalent and connected to two terrestrial vertices. For example, the only aerial vertex of 훾2 is a ∨,푐0 wedge. We define a complete decreasing filtration on SGC푛 by the number of aerial vertices that are not wedges. On the E0 page, the differential is given by the bracket 훾 ↦ [훾2, −], which can be represented as follows:

↦

Graphs∨ This roughly corresponds to the differential in 푛−1. Note that the creations of “dead ends” in 훾 ∘ 훾2 (i.e. when all the incoming edges at a vertex are put on one side of the wedge) are canceled with the summands coming from 훾2 ∘ 훾. Our goal is to prove that on E1 = 퐻(E0, 푑0), there only remains univalent terrestrial vertices (“hairs”) and no wedges. The computation is conceptually similar to the one right after [Wil16, Claim 1], except that instead of computing the Hochschild homology of a symmetric algebra, we compute the e푛−1-cohomology of a symmetric algebra with trivial bracket. The E0 page splits in a direct sum as follows. Given some 훾 ∈ E0, we consider the “index” idx(훾) obtained by removing all wedges and cutting off the terrestrial vertices, keeping a half-edge. We obtain in this way a possibly disconnected graph with only aerial vertices, each one having a certain number of half-edges attached to it. The only restriction is that a zero-valent vertex may not have two half-edges

119 3 Configuration Spaces of Manifolds with Boundary

0 0 attached to it (otherwise it would be a wedge). Then EΓ ≔ {훾 ∈ E ∣ idx(훾) = Γ} 0 0 is a subcomplex, and that E = ⨁Γ EΓ. ∗ 0 0 We’d like to compute 퐻 (EΓ, 푑 ). There are two cases to consider. If Γ = ∅, 0 0 ∨ ∨ then (E∅, 푑 ) is isomorphic to GC푛−1 in the following way. If 훾 ∈ GC푛−1, then 0 we can view it as an element of E∅ by viewing all its vertices as terrestrial, and by replacing all its edges by wedges. The differential 푑0 is then the differential of ∨ GC푛−1. Now if Γ ≠ ∅, then the actual shape of Γ doesn’t matter, only the number of half-edges does. If we number the half-edges as ℎ1, … , ℎ푘, then we see that Graphs∨ we are actually computing the 푛−1-cohomology of the symmetric algebra 푆(ℎ1, … , ℎ푘) and taking the summand where each half-edge appears exactly once. Since Graphs∨ ≃ e , we have 퐻∗(E0 ) = 퐻∗ (푆(ℎ , … , ℎ ); ℝ)(1,…,1). 푛−1 푛−1 Γ e푛−1 1 푘 Applying the higher HKR theorem, this cohomology is given by a symmetric 0 algebra on a shift of the generators. Going back to EΓ, under this identification, ∗ 0 0 퐻 (EΓ, 푑 ) is given by graphs where the code is Γ,there are no wedges, and each terrestrial vertex is univalent. 1 ∗ ∨ In other words, we can identify E as a direct sum of 퐻 (GC푛−1) and a version of the hairy graph complex (as a vector space) where each vertex may have multiple hairs. The differential 푑1 is given by the part of the differential of ∨,푐0 1 SGC푛 which raises the number of non-wedge vertices by exactly one, i.e. 푑 = 훿 + [훾 , −] + [훾 , −] + ∑ [훾 , −] 0 1 푗≥3 푗 . We can now apply the same technique which appears in the proof of [Wil16, 1 1,−1 Claim 2]. The differential 푑 splits into three parts: 푑 = [훾0, −] which 1,0 decreases the number of terrestrial vertices by 1, 푑 = 훿 + [훾1, −] which keeps ′ 푑1,≥1 = ∑ [훾 , −] E1 it constant, and 푗≥3 푗 which increases it. Let be the quotient of E1 by graphs with terrestrial vertices as well as graphs in the image of 푑1,−1 (i.e. graphs with univalent aerial vertices connected to another aerial vertex). The map ′ 푑1,0 induces a differential on E1 . We can then reuse the proof of [Wil16, Claim 2] 1 1 ∗ ∨ to prove that (E , 푑 ) is quasi-isomorphism to the direct sum of 퐻 (GC푛−1) and ′ the quotient (E1 , 푑1,0). 1′ 1,0 ∨ Then (E , 푑 = 훿) has the same cohomology as GC푛 , using an argument identical to the one in [Wil16, Appendix F.2.1]. Thus, the E2 page is a direct sum ∗ ∨ ∗ ∨ of 퐻 (GC푛 ) and 퐻 (GC푛−1). It follows that the dimension of the cohomology ∨,푐0 ∨ ∨ of SGC푛 is at most the dimension of the cohomology of GC푛 ⊕ GC푛−1. ∗ ∨ ∗ ∨ Recall that 퐻 (GC푛 ) and 퐻 (GC푛−1) contain some special classes, the circular graphs (or loops, but this terminology clashes with our terminology for edges between a vertex and itself), see e.g. [Wil14, Proposition 3.4]. We depict these circular graphs in Figure 3.4.1.

120 3.4 Computation of Swiss-Cheese graph cohomology

Figure 3.4.1: Circular graphs in GC푛 and GC푛−1

∨ Corollary 3.4.5. The Maurer–Cartan element 푐 is gauge equivalent to 푐0 in SGC푛 . ∨ Proof. According to [FW15, Proposition 2.2.3], the cohomology of GC푛 is given by a sum of the circular graphs and a part which vanishes in degrees ∗ > −푛 (we work with cohomological conventions, so the degree is reversed). The integral defining 푐 vanishes on all the circular graphs for degree reasons: the total degree of a circular graph with 푘 “edges” (either an edge between two aerial vertices or a wedge between two terrestrial vertices) is −푘 < 0. It then follows ∨,푐0 from Proposition 3.4.4 that 푐 − 푐0 lives in a Lie subalgebra of SGC푛 whose cohomology vanishes in degrees ∗ > −(푛 − 1). Hence, by obstruction theory,4 the Maurer–Cartan element 푐 − 푐0 is gauge trivial. SGraphs The definition of the cooperad 푛 (cf. Section 3.1.4) depends on the choice of this Maurer–Cartan element, 푐 or 푐0.

SGraphs푐0 SGraphs Definition 3.4.6. The cooperad 푛 is defined similarly to 푛, but the Maurer–Cartan element 푐0 is used in place of 푐 in the definition of the differential. ∨ By choosing a path object for SGC푛 , we can turn the gauge equivalence between 푐 and 푐0 into: Corollary 3.4.7. There is a zigzag of quasi-isomorphism of cooperads:

∼ ∼ SGraphs SGraphs푐0 푛 ⋅ 푛 .

SGraphs푐0 Let us also record the following observation. The comodule 푛 (∅, −) is Graphs not isomorphic to 푛 seen as a comodule over itself, because its graphs can contain internal terrestrial vertices. However, SFM푛(∅, {∗}) is a point, and there is a weak equivalence of FM푛-modules given by:

∘푈 FM푛(푈) ≅ SFM푛(∅, {∗}) × FM푛(푈) SFM푛(∅, 푈). (3.4.8)

This is modeled by the following proposition. Define the map

SGraphs푐0 Graphs 휈 ∶ 푛 (∅, 푉) → 푛(푉) (3.4.9) 4The E1 page of the spectral sequence computing 휋 Map ((푆(SGC [1]), 훿 ), ℝ) = ∗ CDGA 푛 푐0 ∨,푐0 휋∗MC•(SGC푛 ) vanishes in the right degrees.

121 3 Configuration Spaces of Manifolds with Boundary

SGraphs푐0 as follows. Given some Γ ∈ 푛 (∅, 푉), if Γ only has univalent terrestrial vertices, then 휈(Γ) is the graph with these univalent vertices and their incident edges removed. Otherwise, if Γ has terrestrial vertices of valence greater than one, then 휈(Γ) = 0.

SGraphs푐0 Graphs Proposition 3.4.10. The maps 휈 ∶ 푛 (∅, 푉) → 푛(푉) defines a quasi- Graphs isomorphism of Hopf right 푛-modules. Proof. For a given 푉, the map is the composite of two quasi-isomorphisms of CDGAs (hence it is itself a CDGA map and a quasi-isomorphism):

• The comodule structure map

∨ SGraphs푐0 SGraphs푐0 Graphs ∘푉 ∶ 푛 (∅, 푉) → 푛 (∅, {∗}) ⊗ 푛(푉), which is a quasi-isomorphism because it models the weak equivalence FM푛(∅, {∗}) × FM푛(푉) → SFM푛(∅, 푉);

SGraphs푐0 • The map 푛 (∅, {∗}) → ℝ given by 푐0, tensored with the identity of Graphs SGraphs푐0 푛(푉). This is a quasi-isomorphism because 푛 (∅, {∗}) is a model for SFM푛(∅, {∗}), which is a point, and 푐0 is a nontrivial cocycle.

SGraphs푐0 Graphs The fact that 푛 is a relative operad over 푛 also shows that this Graphs is a morphism of 푛-comodules.

3.5 A model of right Hopf comodules 3.5.1 The propagator The proof of Theorem C in the case of closed manifolds relied on the existence of a propagator, see Section 2.3.3. Let us briefly recall its construction in this case, before going back to the case 휕푀 ≠ ∅. The propagator 휑 is an (푛 − 1)-form on FM푀(2) whose differential was the pullback of the diagonal class. Moreover, 푛−1 recall that the projection 휕FM푀(2) → 푀 is an 푆 -bundle, and the restriction of the propagator to the boundary is a global angular form for this bundle. If 푛−1 푀 is framed then this bundle is trivial (with ∘1 ∶ 푀 × 푆 → FM푀(2) being an isomorphism of bundles), and one can further assume that 휑 is equal to 1×vol푛−1. This propagator was constructed by considering a global angular form of the previous bundle, pulling it back to a tubular neighborhood of the boundary inside FM푛(2), multiplying it by a bump function, and then extending it by zero outside of the tubular neighborhood (see [CW16, Proposition 7]). Moreover, it ∗ ∗ can be chosen so that it belongs to the subalgebra Ωtriv(FM푀(2)) ⊂ ΩPA(FM푀(2))

122 3.5 A model of right Hopf comodules of “trivial” PA forms (see [CW16, Appendix C]), which implies that it can be integrated along the fibers of the canonical projections 푝푉. ̄ Let us now return to the case 휕푀 ≠ ∅. Consider the double 퐷 = 푀 ∪휕푀 푀, a closed manifold. One can construct a propagator 휓 ∈ Ωtriv(FM퐷(2)). There is an “inclusion” map 푖 ∶ SFM푀(∅, 2) → FM퐷(2) (which is not actually injective on the set of configurations 푥 ∈ SFM푀(2) where 푝1(푥) = 푝2(푥) ∈ 휕푀, just like the standard map SFM푛(∅, 2) → FM푛(2) is not injective). Moreover, there is a “mirror” map 휏1 ∶ SFM푀(∅, 2) → FM퐷(2) which is defined similarly, except that the first point of the configuration is sent to its mirror in 푀̄ ⊂ 퐷. We then define the propagator on 푀 to be: 1 휑 ≔ (푖∗휓 − 휏∗휓). (3.5.1) 2 1 One can write down the cohomology of 퐻∗(퐷) in terms of the cohomology of 푀 and 휕푀 and check on a basis that [푑휑] ∈ 퐻∗(푀 × 푀) is indeed the diagonal class of 푀. Note that this propagator is not symmetrical: if the point 1 (the “origin” of the propagator) becomes infinitesimally close to 휕푀, then the two contributions cancel out and 휑 become zero; this is not the case for the point 2 (the “target”). Consider the space:

퐸 = {푥 ∈ SFM푀(∅, 2) ∣ 푝1(푥) = 푝2(푥)} (3.5.2) which is an 푆푛−1-bundle 퐸̊ when restricted over the interior of 푀. Given that 휓 is a global angular form on 휕FM퐷(2), we check that 휑 is a global angular form on this 푆푛−1-bundle. Remark 3.5.3. We go back to the construction of the diagram of Equation (3.2.17) ∗ ∗ and replace ΩPA(−) with its quasi-isomorphic subalgebra Ωtriv(−), and then ∗ build a PLD model from that. We can then compose with the inclusion Ωtriv(−) → ∗ ΩPA(−) and we obtain a diagram:

푅 ∼ Ω∗ (푀) ∼ Ω∗ (푀) 푓 triv PA 휌 res res ∼ ∗ ∼ ∗ 푅휕 Ωtriv(푀) ΩPA(푀) 푓휕

This allows to assume that if 푥 ∈ 푅 (resp. 푦 ∈ 푅휕), then 푔(푥) (resp. 푔휕(푦)) is a trivial form on 푀 (resp. 휕푀) – compare with Remark 2.3.18. Let us fix the differential of the resulting propagator. Recall the diagonal class Δ퐴퐾 ∈ 퐴 ⊗ 퐾 from Equation (3.2.21), which gets sent to Δ퐴 ∈ 퐴 ⊗ 퐴 under 휋 퐾 ⊂ 퐵 −→퐴. Then we have:

123 3 Configuration Spaces of Manifolds with Boundary

⊗2 Lemma 3.5.4. There exists an element Δ푅 ∈ 푅 ⊗ 푅 such that (휋 ⊗ id)푔 (Δ푅) = Δ퐴퐾 and 휇푅(Δ푅) = 0. Proof. The proof is identical to the proof of Proposition 2.3.4, recalling that 푅 → 퐵 and 퐵 → 퐴 are both surjective, and that 휇퐴(Δ퐴) = 0 when 휕푀 ≠ 0 (Equa- tion (3.2.23)).

⊗2 Definition 3.5.5. For convenience, define a cocycle Δ퐵 in 퐵 by:

⊗2 Δ퐵 ≔ 푔 (Δ푅).

Finally, we’d like to fix the value of the propagator on the subspace SFM푀({2}, {1}) of SFM푀(∅, {1, 2}). In order to define the inclusion

푗 ∶ SFM푀({2}, {1}) ↪ SFM푀(∅, {1, 2}), (3.5.6) we choose some inward pointing vector field on 휕푀 and we use it to infinitesimally push the second point into the interior of 푀. Recall the element 휎퐵 ∈ 퐵휕 ⊗ 퐵 defined in Section 3.3.3, with 푑휎퐵 ∈ 퐵휕 ⊗ 퐾 being cohomologous to the diagonal class of 푀 with the first factor restricted to 휕푀. Using the fact that we have a surjective quasi-isomorphism (thanks to the five lemma): cone(푅휕 ⊗ ker 휌) → cone(퐵 ⊗ 퐾), (3.5.7) we can find some element 휎푅 ∈ 푅휕 ⊗ 푅 with (푔휕 ⊗ 푔)(휎푅) = 휎퐵 and 푑휎푅 ∈ ∗ 푅휕 ⊗ ker 휌. Then 푗 휑 − (푓휕 ⊗ 푓 )(휎푅) is an exact form, by checking on cohomology. 푛−1 Proposition 3.5.8. There exists a form 휑 ∈ ΩPA (SFM푀(∅, 2)) such that:

• 푑휑 is the pullback of (푓 ⊗ 푓 )(Δ푅) along the product of the projections SFM푀(2) → 푀 × 푀;

• its restriction to the bundle 퐸̊ from Equation (3.5.2) is a global angular form;

• if moreover 푀 is framed we choose 휑 so that 휑|퐸̊ = 1 × vol푛−1; • 푗∗휑 = 휑| = (푓 ⊗ 푓 )(휎 ); SFM푀({2},{1}) 휕 푅 ∗ • for any 푥 ∈ 푅, one has (푝2)∗(푝1(푥)휑) = 0. Proof. We’ve already seen how to construct a propagator satisfying the first three ̄ properties, using the mirror 퐷 = 푀 ∪휕푀 푀. ∗ The difference 푗 휑 − (푓휕 ⊗ 푓 )(휎푅) is an exact form, say 푑휉 where 휉 is a trivial ∗ form on SFM푀({1}, {2}). The application 푗 is surjective (because SFM푀({1}, {2}) ̂ is a submanifold with corners of SFM푀(∅, {1, 2})), hence there exists some 휉 such

124 3.5 A model of right Hopf comodules that 푗∗휉̂ = 휉. It then suffices to replace 휑 by 휑 − 푑휉,̂ which still satisfies the first three properties. Finally, for the last property, we reuse the proof technique of [CM10, Lemma 3]. Namely, we consider the chain map:

∗ 푟 ∶ Ωtriv(푀) → 푅 ′ ″ ′ ″ 푥 ↦ ∑ (∫ 푥Δ )Δ + ∑ (∫ 푥|휕푀휎 )휎 , 푀 휕푀 (Δ푅) (휎푅) satisfying 푓 = 푓 ∘ 푟 ∘ 푓, and the homotopy:

∗ ∗−1 ℎ ∶ Ωtriv(푀) → Ωtriv (푀) ∗ 푥 ↦ (푝1)∗(푝2(푥) ∧ 휑). Then we replace ℎ by ℎ′ ≔ (id −푓 푟) ∘ ℎ ∘ (id −푓 푟) in order to obtain a new homotopy satisfying ℎ′′′′′ ∘ 푓 = 0. Let us write down the integral kernel of ℎ′. In order to simplify notations, we write ∫ (−) to indicate that we push forward 푖,푗,… ∗ along the projection which forgets the points 푖, 푗, … , we write Δ푖푗 = 푝푖푗(Δ푅) and ∗ 휎푖푗 = 푝푖푗(휎푅). Then the new propagator is:

12̃휑 ≔ 휑12 − ∫ 휑23Δ13 − ∫ 휑23휎13 − ∫ 휑13Δ23 − ∫ 휑13휎23 3 3 3 3

+ ∫ Δ24휑34Δ13 + ∫ 휎24휑23Δ13 + ∫ Δ24휑34휎13 + ∫ 휎42휑34휎13. 3,4 3,4 3,4 3,4 By construction, this new propagator ̃휑 satisfies the last property. To check that the corrective term ̃휑− 휑 is a cycle, one needs to apply the Stokes formula and check that all the terms cancel out, which follows from the general properties ∗ of the diagonal class, the fact that 휑|퐸̊ is a global angular form, and that 푗 휑 = 휎. ∗ Similarly to check that 푗 ( ̃휑− 휑) = 0 and that ( ̃휑− 휑)|퐸̊ , one need to use the functoriality property of integral along fibers [Har+11, Proposition 8.9] and check that all the terms cancel out.

3.5.2 Colored labeled graphs Graphs We now introduce a colored version of the graph comodule 푅 considered in Section 2.3. Recall the cocycle Δ푅 from Lemma 3.5.4. We also define Δ = (id ⊗휌)(Δ ) ∈ 푅 ⊗ 푅 and Δ ∗ ≔ (id ⊗푔 )(Δ ) for notational 푅,푅휕 푅 휕 푅,Ω (휕푀) 휕 푅,푅휕 convenience. ∗ Definition 3.5.9. The CDGA of (푅, Ωtriv(휕푀))-labeled graphs on the finite sets 푈, 푉 is:

∗ ⊗푈 ⊗푉 SGra푅,Ω∗(휕푀)(푈, 푉) ≔ ((Ωtriv(휕푀)) ⊗ 푅 ⊗ SGra푛(푈, 푉), 푑)

125 3 Configuration Spaces of Manifolds with Boundary

′ with differential defined by 푑푒푣푣⁗ = 휄푣푣′ (Δ푅) for 푣, 푣 ∈ 푉, and 푑푒푣푢 = 휄푣푢(Δ푅,Ω∗(휕푀)) for 푢 ∈ 푈 and 푣 ∈ 푉.

Proposition 3.5.10. The bisymmetric collection SGra푅,Ω∗(휕푀) is a Hopf right comodule over SGra푛.

Proof. The proof is identical to the proof of Proposition 2.3.7. Recall that we do not need additional assumptions on the Euler characteristic as soon as we assume 휕푀 ≠ ∅ (see Lemma 3.5.4).

This comodule has a graphical description similar to SGra푛. The difference is that aerial points (corresponding to the interior of 푀) are labeled by 푅, while terrestrial points (corresponding to the boundary) are labeled by 푅휕. Given a graph Γ, the differential 푑Γ is a sum over the set of edges of Γ. For each summand, one removes the edge from the graph and multiplies the endpoints of the edge by either Δ푅 (if both endpoints are aerial) or Δ푅,휕 (if one endpoint is aerial and the other terrestrial). We call this “splitting” the edge, see Figure 3.5.1. ′ ″ ⊗2 Recall that we write Δ푅 = ∑ Δ푅 ⊗ Δ푅 ∈ 푅 . (Δ푅)

′ ″ 푥 푦 푥Δ푅 푦Δ푅 ↦ ∑

(Δ푅)

Figure 3.5.1: The splitting differential of SGra푅,Ω∗(휕푀) (a gray vertex can be of any kind).

The product glues graphs along their vertices (multiplying the labels), and the comodule structure collapses subgraphs, multiplying the labels and applying 휌 to them if necessary.

′ Proposition 3.5.11. There is a morphism of bisymmetric collections 휔 ∶ SGra푅,Ω∗(휕푀) → ∗ ΩPA(SFM푀) characterized by

′ ∗ 휔 (휄푣(푥)) = 푝푣(푔(푥)) 푣 ∈ 푉, 푥 ∈ 푅 ′ ∗ ∗ 휔 (휄푢(푥)) = 푝푢(푥) 푢 ∈ 푈, 푥 ∈ Ωtriv(휕푀) ′ ∗ ′ 휔 (푒푣푣′ ) = 푝푣푣′ (휑) 푣, 푣 ∈ 푉 ′ ∗ ∗ 휔 (푒푣푢) = 푝푣푢(푗 (휑)) 푢 ∈ 푈, 푣 ∈ 푉

126 3.5 A model of right Hopf comodules

5 where 푗 ∶ SFM푀(1, 1) → SFM푀(∅, 2) is the inclusion. Moreover if 푀 is framed, then together with Willwacher’s morphism [Wil15], this defines a Hopf right comodule morphism

′ ∗ ∗ 휔 ∶ (SGra푅,Ω∗(휕푀), SGra푛) → (ΩPA(SFM푀), ΩPA(SFM푛)).

Proof. The proof is identical to the proof of Proposition 2.3.11.

∨ ∗ Let us define a Maurer–Cartan element 푐휑 ∈ SGC푛 ⊗Ωtriv(휕푀). Let 훾 ∈ SGC푛 be a connected graph, let 퐼 be the set of its terrestrial vertices and 퐽 of its aerial ′ vertices. Then 훾 defines a form 휔 (훾) on SFM푀(퐼, 퐽), which we can pull back along the composition map ∘퐼,퐽 ∶ 휕푀 × SFM푛(퐼, 퐽) → SFM푀(퐼, 퐽) and then integrate on the SFM푛(퐼, 퐽) factor, in other words:

∗ ′ 푐휑(훾) ≔ (푝휕푀)∗(∘퐼,퐽(휔 (훾))). (3.5.12)

Remark 3.5.13. For all 훾, 휔′(훾) is a trivial form (in the sense of [CW16, Ap- pendix C]), hence 푐휑(훾) is obtained by pushing forward a trivial form along a trivial bundle and is therefore trivial itself.

Definition 3.5.14. The twisted colored labeled graph comodule Tw SGra푅,Ω∗(휕푀) is the Hopf right (Tw SGra푛)-comodule obtained by twisting SGra푅,Ω∗(휕푀) with respect to the Maurer–Cartan element 푐휑.

Let us give a graphical description of the module Tw SGra푅,Ω∗(휕푀)(푈, 푉). It is spanned by graphs with four types of vertices: on the one hand, either aerial or terrestrial, and on the other hand either external or internal. Recall that aerial vertices are labeled by 푅 while terrestrial ones are labeled by 푅휕. External aerial vertices are in bijection with 푉, while external terrestrial vertices are in bijection with 푈. Internal aerial vertices are of degree −푛, while internal terrestrial vertices are of degree −(푛 − 1). Both kinds of internal vertices are indistinguishable among themselves. Finally, edges are of degree 푛 − 1, and the source of an edge may only be aerial. The product glues graphs along external vertices (multiplying the labels). The comodule structure maps collapse subgraphs, and the label of the collapsed subgraph is the product of all the labels inside that subgraph (applying 휌 ∶ 푅 → 푅휕 as needed). Finally, the differential has several parts:

• A first part comes from SGra푅,Ω∗(휕푀): it splits edges between vertices of any type, and then multiplies the endpoints of the removed edge by either Δ푅 or Δ푅,휕.

5Recall that one must choose a collar 휕푀 × [0, 1) ⊂ 푀 in order to define this inclusion.

127 3 Configuration Spaces of Manifolds with Boundary

푦1 푦2 푦3 푣1 푣2

(푥 , 푥 ∈ 푅 and 푦 , 푦 , 푦 ∈ 푅) 푥1 푥2 1 2 휕 1 2 3 푢

Figure 3.5.2: A colored labeled graph in Tw SGra푅,Ω∗(휕푀)({푢}, {푣1, 푣2}).

• A second part contracts edges between two aerial vertices, one of them being internal, and multiplies the labels of the endpoints in the process. This includes dead ends, i.e. edges connected to a univalent internal vertex.

• A third part contracts a subgraph Γ′ with at most one external vertex. The label of the contracted subgraph is the product of all the labels inside it ̄′ ̄′ multiplied by 푐휑(Γ ), where Γ is the subgraph with the labels removed. If the result contains a “bad edge” (whose source is terrestrial), then the summand vanishes.

Remark 3.5.15. Note that there are several differences compared to the description of the differential from Section 3.1.4: dead ends are contractible, and the last part of the differential which forgets some internal vertices is not present. This comes from the fact that in the definition of the twisting of a right comodule overa cooperad, the Maurer–Cartan element of the deformation complex can only act “from the right” on the comodule. Dead ends aren’t contractible in Tw SGra푛 because the contraction of a dead end appears twice in the differential, one “from the left” and one “from the right”, and they cancel each other (see [Wil14, Appendix I.3]). This cancellation does not occur in Tw SGra푅,Ω∗(휕푀). The last part of the differential on Tw SGra푛 came exclusively from the left action of the deformation complex on SGra푛, and so cannot appear in Tw SGra푅,Ω∗(휕푀). Remark 3.5.16. One should not forget that when a subgraph Γ′ ⊂ Γ with only internal vertices is contracted (the third part), the result may be a terrestrial vertex even though the subgraph contains only aerial vertices, see Figure 3.5.3.

푥

↦ 휌(푥)

Figure 3.5.3: Collapsing an aerial vertex into a terrestrial vertex

128 3.5 A model of right Hopf comodules

′ ∗ Proposition 3.5.17. The morphism 휔 ∶ SGra푅,Ω∗(휕푀) → ΩPA(SFM푀) of Proposi- tion 3.5.11 extends to a morphism of bisymmetric collections 휔 ∶ Tw SGra푅,Ω∗(휕푀) → ∗ ΩPA(SFM푀), given on a graph Γ ∈ SGra푅,Ω∗(휕푀)(푈⊔퐼, 푉⊔퐽) ⊂ Tw SGra푅,Ω∗(휕푀)(푈, 푉) by: ′ ′ 휔(Γ) ≔ (푝푈,푉)∗(휔 (Γ)) = ∫ 휔 (Γ). SFM푀(푈⊔퐼,푉⊔퐽)→SFM푀(푈,푉) Moreover if 푀 is framed then, together with Willwacher’s morphism [Wil15], this as- signment defines a morphism of Hopf right comodules

∗ ∗ 휔 ∶ (Tw SGra푅,Ω∗(휕푀), Tw SGra푛) → (ΩPA(SFM푀), ΩPA(SFM푛)). Proof. First note that we have chosen the propagator 휑 so that it is a trivial form, ∗ ∗ and we have assumed that the morphisms 푅 → ΩPA(푀) and 푅휕 → ΩPA(휕푀) factor through the sub-CDGAs of trivial forms. Hence for any graph Γ, 휔′(Γ) is a trivial form and can be integrated along the fiber of 푝푈,푉. We can now reuse the proof of Proposition 2.3.17. The difference is the description of the decomposition of the fiberwise boundary of 푝푈,푉 used to show that 휔 is a chain map through the application of Stokes’ formula. This description is very similar to the one implicitly used by [Wil15] (see also [Kon03, Section 5.2.1]) with some variations accounting from the fact that no normalization is done to compactify 푀 (see the discussion before the proof of Lemma 2.3.22). More concretely, the boundary of SFM푀(푈, 푉) is given by:

휕SFM푀(푈, 푉) = ⋃ im(∘푇) ∪ ⋃ im(∘푊,푇), 푇∈ℬℱ′(푉) (푊,푇)∈ℬℱ″(푈;푉) where: ℬℱ′(푉) ≔ {푇 ⊂ 푉 ∣ #푇 ≥ 2}, ℬℱ″(푈; 푉) ≔ {(푊, 푇) ∣ 푊 ⊂ 푈, 푇 ⊂ 푉, 2 ⋅ #푇 + #푊 ≥ 2}.

Note that in the description of 휕SFM푛(푈, 푉), there is an additional condition 푊 ∪ 푇 ⊊ 푈 ∪ 푉. Indeed in SFM푛 the normalization by the affine group prevents the points from becoming infinitesimally close all at once; in SFM푀, no such normalization occurs. Then the fiberwise boundary of the canonical projection 푝푈,푉 is given by:

휕 SFM푀(푈, 푉) = ⋃ im(∘푇) ∪ ⋃ im(∘푊,푇) ⊂ SFM푀(푈 ⊔ 퐼, 푉 ⊔ 퐽), 푇∈ℬℱ′(푉,퐽) (푊,푇)∈ℬℱ″(푈,퐼;푉,퐽) where 푇 ∈ ℬℱ′(푉, 퐽) ⊂ ℬℱ′(푉 ⊔ 퐽) ⟺ #(푇 ∩ 퐽) ≤ 1, (푊, 푇) ∈ ℬℱ″(푈, 퐼; 푉, 퐽) ⊂ ℬℱ″(푈 ⊔ 퐼, 푉 ⊔ 퐽) ⟺ 푉 ∩ 푇 = ∅, #(푈 ∩ 푊) ≤ 1.

129 3 Configuration Spaces of Manifolds with Boundary

One can then check that the boundary faces of that decomposition correspond to the summands of the differential.

Definition 3.5.18. Define the full colored graph complex

fSGC푅 ≔ Tw SGra푅,Ω∗(휕푀)(∅, ∅)[−푛].

This is the (shifted) CDGA of colored, labeled graphs with only internal vertices. The product is the disjoint union of graphs, thus fSGC푅[푛] is free as an algebra, generated by the graded module SGC푅[푛] of connected graphs. Each Tw SGra푅(푈, 푉) is a module over the CDGA fSGC푅[푛] by adding connected components.

푆 Definition 3.5.19. The colored partition function Z휑 ∶ fSGC푅[푛] → ℝ is the CDGA morphism given by the restriction in empty arity

푆 ∗ ∗ Z휑 ≔ 휔|(∅,∅) ∶ Tw SGra푅,Ω∗(휕푀)(∅, ∅) → ΩPA(SFM푀(∅, ∅)) = ΩPA({∗}) = ℝ.

푆 푆 Let ℝ휑 be the one-dimensional fSGC푅[푛]-module induced by Z휑.

∨ Remark 3.5.20. Since fSGC푅[푛] is a CDGA, the dual module SGC푅 is naturally an hoLie푛-algebra. The differential of fSGC푅 cannot create more than two connected ∨ components, thus SGC푅 is actually a Lie푛-algebra. The differential blows up ∨ vertices (like in GC푛 ) and joins pairs of vertices by an edge, while the Lie bracket 푆 joins two graphs by an edge. The algebra morphism Z휑 is uniquely determined 푆 by its restriction z휑 to SGC푅, which can be seen as a Maurer–Cartan element in ∨ the Lie푛-algebra SGC푅.

푆 푐휑,z휑 Definition 3.5.21. The reduced colored labeled graph comodule SGraphs푅,Ω∗(휕푀) is the bisymmetric collection given in each arity by:

푆 푐휑,z휑 푆 SGraphs ∗ (푈, 푉) ≔ ℝ ⊗ Tw SGra ∗ (푈, 푉). 푅,Ω (휕푀) 휑 fSGC푅[푛] 푅,Ω (휕푀)

푆 푐휑,z휑 Proposition 3.5.22. The bisymmetric collection SGraphs푅,Ω∗(휕푀) forms a Hopf right SGraphs SGra ∗ SFM comodule over 푛, and the map 휔 ∶ Tw 푅,Ω∗(휕푀) → ΩPA( ) factors through a Hopf right comodule morphism:

푐 ,z푆 SGraphs 휑 휑 SGraphs ∗ SFM ∗ SFM 휔 ∶ ( 푅,Ω∗(휕푀), 푛) → (ΩPA( 푀), ΩPA( 푛)). Proof. Identical to the proof of Proposition 2.3.31.

130 3.5 A model of right Hopf comodules

We now show that our comodule is quasi-isomorphic to a simpler one, in the spirit of Corollary 3.4.7. Recall the circular graph from Figure 3.4.1. We will need an explicit name for the circular graph of length 1 in GC푛−1, say:

ℓ1 ≔ (3.5.23)

∨ ∗ Lemma 3.5.24. The Maurer–Cartan element 푐휑 ∈ SGC푛 ⊗ Ωtriv(휕푀) is gauge equivalent to ∨ 푐푀 ≔ 푐0 ⊗ 1 + ℓ1 ⊗ 푒휕푀, where:

∨ • 푐0 ∈ SGC푛 is the linear part of 푐 (Definition 3.4.2);

• ℓ1 ∈ SGC푛 is the circular graph of length 1 in GC푛−1 (Equation (3.5.23)); • 푒 ≔ 휒(휕푀) ⋅ vol is the Euler class of 휕푀 (and vol = 푔 (vol ) where 휕푀 휕푀 휕푀 휕 푅휕 vol is a fixed cocycle satisfying 푓 (vol ) = vol ). 푅휕 휕 푅휕 퐵휕

Proof. Let us first show that 푐휑 agrees with 푐푀 on circular graphs, i.e. that 푐휑 vanishes on circular graphs except for ℓ1, on which it is equal to 푒푀. The aerial circular graphs which produce nontrivial classes have odd length,6 hence they contain a vertex with in and out valences of 1. One can construct the propagator ̄ 휓 on 퐷 = 푀 ∪휕푀 푀 (see Section 3.5.1) so that 휑 exhibits symmetry properties that make 푐휑 vanish on these graphs, by an argument similar to Kontsevich’s trick [Kon94, Lemma 2.1]. Moreover, using a similar argument, we see that 푐휑 vanishes on the circular graphs from GC푛−1, because apart from the first one, they all contain a copy of the following graph (which represent a “bivalent” terrestrial vertex, the “outside” vertices may be the same one):

Thus when computing the integral defining 푐, one can consider the automorphism of SFM푛({푢1, 푢2, 푣}, ∅) which maps 푣 to its symmetric with respect to the 푛−1 barycenter of 푢1 and 푢2 inside ℝ × {0}, which changes the sign of the integral. The only exception is ℓ1, for which we can explicitly compute 푐푀(ℓ1) = 푒푀 using the fourth property of 휑 in Proposition 3.5.8 and the standard fact that the Euler class 푒휕푀 is the image of the diagonal class of 휕푀 under the multiplication map.

6Depending on the parity of 푛, they are congruent to either 1 or 3 modulo 4.

131 3 Configuration Spaces of Manifolds with Boundary

It then follows that 푐휑 − 푐푀 is a Maurer–Cartan element in the twisted Lie ∨,푐0 ∗ algebra SGC푛 ⊗ Ω (휕푀). According to Proposition 3.4.4, if we forget about ∨,푐0 circular graphs the cohomology of SGC푛 is concentrated below degree −(푛−1), and the cohomology of 휕푀 below degree 푛 − 1. It follows that 푐휑 − 푐푀 lives in ∨,푐0 ∗ a Lie subalgebra of SGC푛 ⊗ Ω (휕푀) whose cohomology vanishes in positive degrees. Thus, by obstruction theory, 푐휑 − 푐푀 is gauge trivial and we have proved the proposition.

SGraphs푐0 Recall the cooperad 푛 from Definition 3.4.6. We can define a Hopf 푆 푐푀,z휑 right comodule SGraphs푅,Ω∗(휕푀) over this cooperad, by twisting SGra푅,Ω∗(휕푀) with respect to 푐푀 instead of 푐휑. Corollary 3.5.25. There is zigzag of quasi-isomorphisms of Hopf right comodules:

푆 푆 푐휑,z휑 ∼ ∼ 푐푀,z휑 SGraphs SGraphs SGraphs SGraphs푐0 ( 푅,Ω∗(휕푀), 푛) ⋅ ( 푅,Ω∗(휕푀), 푛 ). Moreover, we can now get rid of the Ω∗(휕푀) and label the terrestrial vertices by elements of 푅휕 instead. This was not possible before vecause 푐휑 could produce ∼ ∗ forms that were not in the image of 푔휕 ∶ 푅휕 Ω (휕푀), but this is not the case 푆 푐푀,z휑 for 푐 ⊗ 1. We thus define a Hopf right comodule (SGraphs , SGraphs푐0 ), 푀 푅,푅휕 푛 and we obtain: Lemma 3.5.26. There is a quasi-isomorphism of Hopf right comodules induced by ∼ ∗ 푔휕 ∶ 푅휕 Ω (휕푀):

푆 푆 푐푀,z휑 푐0,z휑 푐0 ∼ 푐0 (SGraphs , SGraphs ) (SGraphs ∗ , SGraphs ). 푅,푅휕 푛 푅,Ω (휕푀) 푛

3.5.3 Labeled graphs and proof of Theorem D

z휑 We now introduce a variant Graphs푅 of the graph comodule from Chapter 2. Recall that 푗∗휑 = 휑| is equal to (푔 ⊗ 푔)(휎 ), where 휎 = ∑ 휎′ ⊗ 휎″ ∈ SFM푀({2},{1}) 휕 푅 푅 푅휕 ⊗ 푅 was defined in Proposition 3.5.8.

z휑 Definition 3.5.27. The CDGA of reduced 푅-labeled graphs Graphs푅 (푉) on the finite set 푉 is spanned by directed graphs with external (corresponding to 푉) vertices and internal vertices, with all the vertices labeled by 푅. The algebra structure glues graphs along external vertices, and the comodule structure collapses subgraphs. A connected component with only internal vertices is identified with 푆 the real number given by the partition function Z휑, by viewing the component as having no terrestrial vertices. The differential has three parts:

132 3.5 A model of right Hopf comodules

• Splitting any edge and multiplying the labels of the endpoints by Δ푅; • Contracting an edge between an internal vertex and another vertex, multiplying the labels;

• Removing one internal vertex 푢. Let 푥 ∈ 푅 be the label of 푢, and let 푣1, … , 푣푘 be the vertices connected to 푢, with respective labels 푦1, … , 푦푘 ∈ 푅. Then in the differential, the vertex 푢 and its incident edges are removed, the label of ″ ′ ′ 푣푖 becomes 푦푖휎푖 , and the new graph is multiplied by 휀휕(휌(푥)휎1 … 휎푘) ∈ ℝ. See Figure 3.5.4 for an example.

푥 ″ ″ ″ 푦 휎 푦 휎 푦3휎 푦 푦 푦 ↦ 휀 (휌(푥)휎′ 휎′ 휎′ ) ⋅ 1 1 2 2 3 1 2 3 휕 1 2 3 1 2 3 1 2 3

휑 Figure 3.5.4: The third part of the differential in Graphs푅

Roughly speaking, in this third part of the differential, the removed vertex becomes infinitesimally close to the boundary of 푀. Each edge thus becomes 푗∗휑 = 휑| = (푔 ⊗ 푔)(휎 ), and we multiply the labels accordingly. The SFM푀({1},{2}) 휕 푅 vertex 푢 is then alone in its connected component and is thus identified with the integral of its label. 푐 ,z푆 For each finite set of external vertices 푉, there is a map 휈 ∶ SGraphs 푀 휑 (∅, 푉) → 푅 푅,푅휕 z 푐 ,z푆 Graphs 휑 (푉) defined as follows. Given some graph Γ ∈ SGraphs 푀 휑 (∅, 푉), we 푅 푅,푅휕 remove all the edges from an aerial vertex to a terrestrial vertex and we multiply their endpoints by 휎푅 ∈ 푅휕 ⊗ 푅. We then obtain a graph where all the terrestrial vertices are zero-valent, and we identify those with the real number given by the z휑 integral over 휕푀 of their labels. In this way, we obtain an element of Graphs푅 (푉). This is a chain map, the part of the differential described in Figure 3.5.4 being induced by Figure 3.5.3. 휑 Proposition 3.5.28. The symmetric collection Graphs푅 is a Hopf right comodule over Graphs 푛. The maps

푆 푐푀,z휑 ∼ z휑 휈푅 ∶ SGraphs푅 (∅, −) Graphs푅 Graphs define a quasi-isomorphism of Hopf right 푛-comodules.

133 3 Configuration Spaces of Manifolds with Boundary

Proof. We use an argument similar to the one of Section 2.4.2. Filter both complexes by the number of edges minus the number of vertices. The only part of the differential which remains on the E0 page is the contraction of aerial edges and the contraction of a subgraph into a terrestrial vertex induced by 푐푀, which is nonzero on trees only (see Definition 3.4.2 and Lemma 3.5.24, the summand induced by ℓ1 ⊗ 푒휕푀 vanishes because it strictly decreases the filtration). It follows that both complexes split as a direct sum in terms of connected components, just like in Lemma 2.4.21. We can now reuse the same “trick” we used for the proof of Lemma 2.4.27, to show that in cohomology, there is only one label for each connected component. We thus reduce to the morphism SGraphs푐0 Graphs 휈 ∶ 푛 (∅, 푉) → 푛(푉) from Proposition 3.4.10 (tensored with the identity of 푅 for each connected component). Since that morphism is a quasi-isomorphism, we obtain that the induced morphism on the E1 page is an isomorphism. It follows by standard spectral sequence arguments (the filtration is bounded below for a fixed 푉) that the morphism of the proposition is a quasi-isomorphism.

z 휑 ̃ We would now like to connect Graphs푅 (푉) with G퐴(푉).

Definition 3.5.29. The full labeled graph complex fGC푅 is the CDGA of aerial graphs with only internal vertices, the differential being defined in a manner similar to Definition 3.5.27. The partition function Z휑 ∶ fGC푅[푛] → ℝ is induced 푆 by Z휑. Let ℝ휑 be the one-dimensional fGC푅[푛]-module induced by Z휑.

The (shifted) commutative algebra fGC푅 is free, generated by the submodule ∨ GC푅 of connected graphs. The differential of fGC푅 turns GC푅 into a Lie푛-algebra up to homotopy (see Remark 3.5.20, and note that the part of the differential that removes one vertex may produce more than two connected components, so this ∨ is not a Lie푛-algebra), and Z휑 defines a Maurer–Cartan element z휑 ∈ GC푅. We can reuse the counting argument from Proposition 2.3.35 (and the last property of the propagator in Proposition 3.5.8) to show that the partition function vanishes on any graph with no bivalent vertex labeled by 1푅. We can then define Z휀 ∶ fGC푅 → ℝ to be the algebra morphism which vanishes on all connected graphs 훾 except those with a single vertex. If that vertex is labeled by 푥 ∈ 푅 we set Z (훾) = ∫ 푔(푥). 휀 푀 Unfortunately, we still cannot prove directly that Z휑 = Z휀; however, we can follow the discussion at the beginning of Section 2.4. Let Ω∗(Δ1) = 푆(푡, 푑푡) be the algebra of polynomials forms on Δ1, a path object in the category of CDGAs. We then obtain:

134 3.5 A model of right Hopf comodules

∗ 1 Proposition 3.5.30. There exists a homotopy ℎ ∶ fGC푅[푛] → Ω (Δ ) such that the following diagram commutes:

fGC푅[푛] Z휑 Z ℎ 휀 푑 푑1 ∗ 1 0 ℝ ∼ Ω (Δ ) ∼ ℝ

The reduced labeled graph comodule can be viewed as a tensor product ℝ ⊗ Tw Gra (푉). We can then define Graphs′ ≔ Ω∗(Δ1) ⊗ 휑 fGC푅[푛] 푅 푅 ℎ fGC푅[푛] z Tw Gra and Graphs 휀 ≔ ℝ ⊗ Tw Gra . Both symmetric collections de- 푅 푅 휀 fGC푅[푛] 푅 Graphs fine Hopf right 푛-comodules. Proposition 3.5.31. We have a zigzag of quasi-isomorphisms of comodules:

z Graphsz휀 ∼ Graphs′ ∼ Graphs 휑 푅 푅 푅 .

Proof. The fGC푅[푛]-module Tw Gra푅(푉) is quasi-free and is equipped with a good filtration, hence Tw Gra (푉) ⊗ (−) preserves quasi-isomorphisms. 푅 fGC푅[푛] The claim now follows from the previous proposition.

z휀 We can now do the same construction as Graphs푅 but with 퐵 replacing 푅, Δ퐵 (from Definition 3.5.5) replacing Δ푅, 휎퐵 replacing 휎푅, and taking as “partition function” the map which sends a graph with a single vertex labeled by 푏 to Graphs 휀퐵(푏) and all other connected graphs to zero. We obtain a Hopf right 푛- Graphs comodule 퐵, where graphs with internal components containing at least two vertices vanish, and an isolated internal vertex labeled by 푏 ∈ 퐵 is identified with 휀퐵(푏). Graphs Graphs Finally, we can consider a quotient 퐴 of 퐵, where we apply the projection 휋 ∶ 퐵 → 퐴 to all the remaining labels. It is not necessarily the case that the two morphisms 휀, 휀′ ∶ cone(휌) → ℝ[−푛 + 1], defined respectively by the composites (∫ 푔(−), ∫ 푔 (−)) and (휀 푓 , 휀 푓 ), 푀 휕푀 휕 퐵 퐵휕 휕 are equal. Nevertheless, up to rescaling 휀 (which induces an automorphism of z휀 푛−1 Graphs푅 ), they induce the same map up to quasi-isomorphism, 퐻 (cone(휌)) being one-dimensional. Since all cochain complexes are fibrant and cofibrant, we obtain a diagram:

cone(휌) 휀 휀′ 휀ℎ (3.5.32)

푑1 푑0 ℝ[−푛 + 1] ∼ 푃[−푛 + 1] ∼ ℝ[−푛 + 1]

135 3 Configuration Spaces of Manifolds with Boundary where 푃 is the standard path object for the base field in the category of cochain complexes. The chain map 휀ℎ is uniquely determined by a graded map ℎ ∶ cone(휌) → ℝ[−푛 + 1] satisfying 휀(푥) − 휀′(푥) = ℎ(푑푥). Since the product of fGC푅 is merely given by the disjoint union of graphs, this yield a homotopy between the two morphisms fGC푅[푛] → ℝ induced. Similarly to Proposition 3.5.31, we then obtain a zigzag of quasi-isomorphisms:

′ Graphsz휀 ∼ Graphs″ ∼ Graphsz휀 푅 푅 푅 . (3.5.33)

Now 휀′ is compatible with (휀 , 휀 ), we obtain the following proposition: 퐵 퐵휕 Graphs Proposition 3.5.34. We have quasi-isomorphisms of Hopf right 푛-comodules

′ z휀 휋 Graphs 푓∗ Graphs ∗ Graphs 푅 퐵 퐴 which apply 푓 ∶ 푅 → 퐵 (resp. 휋 ∶ 퐵 → 퐴) to all the labels.

Proof. Filter the graph comodules by the number of edges. On each E0 page, only the differentials from 푅 (resp. 퐵, 퐴) remain. Since 푓 and 휋 are a quasi- isomorphism, it follows that the induced morphisms on E1 pages is an isomorphism. Standard spectral sequence arguments then imply that 푓∗ and 휋∗ are quasi-isomorphisms.

Graphs G̃ We can now define a morphism 퐴 → 퐴 which maps all graphs with internal vertices to zero, and which sends an edge 푒푣푣′ between external vertices to ̃휔푣푣′ . Then we can mimic the proof of Proposition 2.4.17 in a straightforward way (note that as soon as we filter by #edges − #vertices the perturbed relations of G퐴̃ become the usual relations of G퐴):

Proposition 3.5.35. This defines a quasi-isomorphism of right Hopf comodules:

Graphs Graphs G̃ e∨ ( 퐴, 푛) → ( 퐴, 푛 ).

We can also copy the proof of Proposition 2.4.31 and use Theorem 3.3.16 to obtain:

푆 푐휑,z휑 ∗ Proposition 3.5.36. The morphism 휔 ∶ SGraphs푅 (∅, −) → ΩPA(SFM푀(∅, −)) is a quasi-isomorphism.

Finally we can summarize this section as:

136 3.5 A model of right Hopf comodules

Theorem 3.5.37 (Precise version of Theorem D). Let 푀 be a simply connected, smooth, compact manifold with boundary of dimension at least 5, and assume either that it admits a surjective pretty model or that dim 푀 ≥ 7 so that it admits a PLD. Let 퐴 be the CDGA model built from the resulting PLD model. ̃ ∗ Then the symmetric collection of CDGAs G퐴 is quasi-isomorphic to ΩPA(SFM푀(∅, −)). ∨ If moreover 푀 is framed, then the Hopf right comodule (G퐴̃ , e푛 ) is quasi-isomorphic to ∗ ∗ (ΩPA(SFM푀(∅, −)), ΩPA(FM푛)). Corollary 3.5.38 (Corollary E). The real homotopy type of configuration spaces on a simply connected, smooth manifold 푀 with simply connected boundary only depends on the real homotopy type of the manifold as soon as:

• either dim 푀 ≥ 5 and 푀 admits a surjective pretty model;

• or dim 푀 ≥ 7.

Example 3.5.39. We can apply this to 푀 = 퐷푛, using the surjective pretty model from Example 3.1.16. Recall that in this case, 퐴 = ℝ, Δ퐴 = 0, 휎퐴 = 0, and G퐴̃ is ∨ isomorphic to e푛 as a Hopf right comodule over itself (see Example 3.3.26). We then “recover” the already known fact that SFM퐷푛 (∅, −) is (Hopf) formal as a right FM푛-module (though that “proof” is of course circular). Graphs Graphs Graphs The Hopf right 푛-comodule 퐴 is isomorphic to 푛 seen as a comodule over itself. The augmentation 휀퐵 ∶ 퐵 → ℝ[−푛 + 1] yields a Maurer– ∨ Cartan element z휀 in the abelian hoLie푛-algebra GC퐴 (see Remark 3.5.20), given in the dual basis by the graph with a single vertex labeled by vol푛. The twisted ∨,z휀 ∨ 7 hoLie푛-algebra GC퐴 is then isomorphic to GC푛 from Section 3.1.5.

3.5.4 Proof of Theorem F Proposition 3.5.40. Assume that 푛 = dim 푀 ≥ 5. Then for all finite sets 푈 and 푉, 푆 ∗ 푐휑,z휑 퐻 (SGraphs푅 (푈, 푉)) has the same dimension in each degree as ∗ ∗ ∗ 퐻 (SFM푀(푈, 푉)) ≅ 퐻 (Conf푈(휕푀)) ⊗ 퐻 (Conf푉(푀)). Proof. We use a spectral sequence argument similar to the proof in [Wil15, Sec- tion 5]. The assumption about the dimension of 푀 allows us to apply Theo- rem 3.5.37 to 푀 and Theorem C to 휕푀. 푆 푐휑,z휑 First filter SGraphs푅 (푈, 푉) by the number of edges minus the number of vertices as in the proof of the previous proposition. We get as an E0 page the SGraphs graded module 푅(푈, 푉) together with the differential which contracts 7The twisting ensures that dead ends are not contractible, see the proof of Proposition 2.4.13.

137 3 Configuration Spaces of Manifolds with Boundary edges connected to an aerial internal vertex. Filtering this new complex by the number of vertices that are not of the type “bivalent, internal, aerial, and labeled by 1푅” as in the proof of [Wil15, Section 5], and using the previous proposition, we obtain that E0 is quasi-isomorphic to its quotient 풬(푈, 푉) in which:

• graphs containing bivalent internal aerial vertices labeled by 1푅 other than those appearing in a zigzag of the type shown in Figure 3.5.5 are set to zero; • graphs with edges between aerial and terrestrial vertices other than those appearing in a zigzag of the type shown in Figure 3.5.5 are set to zero;

푛 • an edge 푒푣푣′ between aerial vertices is identified with (−1) 푒푣′푣.

1푅

푥 푦 1 2

Figure 3.5.5: A zigzag which represents an edge in Graphs (푈). 푅휕

We may identify the underlying graded module of 풬(푈, 푉) with the tensor product Graphs (푉) ⊗ Graphs (푈). The induced differential 푑0 contracts edges 푅 푅휕 in Graphs (푉) (but not Graphs (푉)). We obtain that the underlying graded 푅 푅휕 module of the E1 page is isomorphic to

퐻∗(Graphs (푉), 푑0) ⊗ Graphs (푉). 푅 푅휕 The differential 푑1 is induced by the part of the differential which decreases the filtration by exactly 1. By comparing with the proof of Theorem 3.5.37 (and by extension with the proof of Theorem C), we obtain that the complex ∗ Graphs 0 1 (퐻 ( 푅(푉), 푑 ), 푑 ) is isomorphic to G퐴(푉), which has the same Betti numbers as Conf푉(푀) by Theorem 3.3.16. Moreover, by inspection, 푑1 is given on Graphs (푈) by the 푅휕 contraction of edges (and the multiplication of labels). Applying Theorem C to 휕푀, we thus obtain that the E2 page of the spectral sequence is isomorphic to:

퐻∗(G (푉)) ⊗ G (푈) ≅ 퐻∗(Conf (푀)) ⊗ G (푈). 퐴 퐵휕 푉 퐵휕

푆 푐휑,z휑 Since any remaining cocycle is represented by a cocycle in SGraphs푅 (푈, 푉), we obtain that the spectral sequence abuts at this stage.

138 3.5 A model of right Hopf comodules

Proposition 3.5.41. Assume that 푛 = dim 푀 ≥ 5. Then the morphism

푆 푐휑,z휑 ∗ 휔 ∶ SGraphs푅,Ω∗(휕푀)(푈, 푉) → ΩPA(SFM푀(푈, 푉)) is a quasi-isomorphism of CDGAs for all 푈, 푉.

∗ Proof. We know that 퐻 (SFM푀(푈, 푉)) is isomorphic to the tensor product

∗ ∗ 퐻 (Conf푉(푀)) ⊗ 퐻 (Conf푈(휕푀)).

We can then use the previous proposition and an inductive argument similar to the proof of Proposition 2.4.31 to obtain the result. Indeed we can represent ∗ 휑 any cocycle in 퐻 (Conf푉(푀)) by a cocycle in Graphs푅(푉), and each cocycle in 휑 퐻∗(Conf (휕푀)) by a cocycle in Graphs 휕 (푈), and then take the disjoint union 푈 푅휕 of the resulting graphs. We can now bundle everything together into a theorem, with a summary of our hypotheses.

Theorem 3.5.42 (Precise version of Theorem F). Let 푀 be a smooth, simply connected manifold with a simply connected boundary, of dimension 푛 ≥ 5. Suppose also that 푀 admits a surjective pretty model, or that 푛 ≥ 7 so that it admits a PLD model as in Equation (3.2.17). 푐 ,z푆 Then for each 푈, 푉, then the CDGA SGraphs 푀 휑 (푈, 푉) is quasi-isomorphic to 푅,푅휕 ∗ ΩPA(SFM푀(푈, 푉)), and this is compatible with the symmetric group actions. If 푀 is framed, then together with Willwacher’s morphism [Wil15], this defines a quasi-isomorphism of right Hopf comodules

푆 푐푀,z휑 (SGraphs , SGraphs푐0 ) ≃ (Ω∗ (SFM ), Ω∗ (SFM )). 푅,푅휕 푛 PA 푀 PA 푛

푆 It would be an interesting question to know how much z휑 can be simplified.

139

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Index

P ⊗ Q, see Voronov product Δ(푘), 112 P ⊗ Q , 36 Δ , 125 + 0 + 푅,푅휕 ∇(훾), 112 Δ퐴, diagonal class, 50, 109 ! fr 휓 , see shriek map Disk푀, 83 ⟨−⟩핃, derived realization, 35 fr Disk푛 , 83 (−)̂, completion, 26 D푛, little disks operad, 10 Drinfeld associator, 27 “almost” Hopf cooperad, 43 Drinfeld center, 19 Ass, associative operad, 35 Ass휇(ℚ), Drinfeld associators, 27 푒휕푀, 131 e푛 = 퐻∗(D푛), 35 bisymmetric collection, 95 ∨ ∗ e푛 = 퐻 (D푛), 45 e∨⟨휋⟩, 74 푐0, 119 푛 퐶0(푛, 푚), 13 factorization homology ∫ 퐵, 83 categorical equivalence, 18 푀 fe∨ ∗ fFM rational, 27 푛 = 퐻 ( 푛), 89 CE fFM C , Chevalley–Eilenberg complex, 푀, framed F–M module, 89 ∗ fFM 84 푛, framed F–M operad, 89 fG CD̂, chord diagrams, 27 퐴, 90 fGC , 103 푐Γ, 102 푛 fGC푅, full graph complex, 64, 134 푐푀, 131 ′ CoB, 10 fGC푅, 68 fGC0 , 68 Conf푘(푀), configuration space, 39 푅 fLoop , 68 Conf(푛, 푚), 12 푅 FM Conf푈,푉(푀), 99 푀, Fulton–MacPherson module, 44 휕 CoPB, col. permutations and braids, FM푀(푉), fiberwise boundary, 62 13 FM푛, Fulton–MacPherson operad, 43 CoPB+, 25 fSGC푛, 104 fSGC푅, 130 Δ퐴퐾, 110 Graphs 퐹푠 퐴, 73 Δ퐵, 124 Δ푖푗, 112 G퐴, Lambrechts–Stanley CDGA, 50,

151 Index

110 PLD model, 106 훾푗, 118 Poincaré duality algebra, 49 G퐴⟨휋⟩, 74 Poincaré–Lefschetz duality model, 106 GC푛, 103 pretty model, 97 Ger, Gerstenhaber operad, 35 propagator 휑, 58, 124 G퐴̃ (푉), 116 ℝ휀, 70 Gra푛, 46 Gra relative operad, 12, 95 푛 , 53 ′ Graphs 휌∗, 67 퐴, 72, 135 Graphs 휌∗, 71 푛, 49 Graphs휀 right (co)module, 42 푅, 70 z ℝ , 65, 134 Graphs 휀 , 135 휑 푅 푆 휑 ℝ휑, 130 Graphs푅, 65 z휑 Graphs푅 , 132 SC, Swiss-Cheese operad, 11 Gra푅, 57 sc = 퐻∗(SC), 36 vor Gra푅, 56 SC , Voronov’s SC operad, 12 SFM , 99 hoLie 푀 푘, 47 SFM , 99 Hopf cooperad, 35 푛 SGC푛, 104 SGra I, unit operad, 85 푛, 101 SGraphs 휄 , 50, 110 푛, 100 푣 푆 푐휑,z휑 SGraphs ∗ , 130 ℓ , 131 푅,Ω (휕푀) 1 Sh , shuffles, 13 Λ, operadic suspension, 85 푛,푚 Sh , 29 Lie , shifted Lie operad, 47 + 푘 shriek map, 97 L , 85 푛 shuffle-type morphism, 23 mapping cone, 97 휎퐴, 116 morphism of Hopf right comodules, Stokes formula, 62 42 TotCof, total cofiber, 115 Tw C, twisted cooperad, 47 PaCD̂ +, 28 PaPB, par. perm. and braids, 18 Tw Gra푛, 48 Tw Gra , 60 PaPB+, 25 푅 ′ Tw Gra , 59 PaPB+, 30 푅 휑 Tw SGra , 102 PaPCD̂+, 33 푛 partition function, 65, 134 Tw SGra푅,Ω∗(휕푀), 127 PaSh +, 30 푈 (픤), 84 픭, Drinfeld–Kohno operad, 27 푛 Π, permutation operad, 28 vol푛−1, 43 푝푖푗, 43 Voronov product, 35

152 Index wedge, 119 Ω, magma operad, 16 ΩΩ, 16 ΩΩ+, 25 휔(Γ), 49, 61, 103, 129, 130 Ω∗(Δ1), 69 휔′(Γ), 47, 58, 101, 126 ∨ 휔푢푣 ∈ e푛 (푈), 45 ∗ ΩPA(−), PA forms, 43 풵(C), see Drinfeld center Z휀, 70, 134 z휀, 137 Z휑, see partition function 푆 Z휑, 130 푆 z휑, 130

153