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Infinity Structures and Higher Products in Rational Homotopy Theory

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Infinity Structures and Higher Products in Rational Homotopy Theory Infinity structures and higher products in rational homotopy theory Ph. D. Dissertation José Manuel Moreno Fernández Directed by: Dr. Urtzi Buijs Martín and Dr. Aniceto Murillo Mas Tesis Doctoral Departamento de Álgebra, Geometría y Topología Programa de Doctorado en Matemáticas Facultad de Ciencias Universidad de Málaga 2018 AUTOR: José Manuel Moreno Fernández http://orcid.org/0000-0002-0956-8407 EDITA: Publicaciones y Divulgación Científica. Universidad de Málaga Esta obra está bajo una licencia de Creative Commons Reconocimiento-NoComercial- SinObraDerivada 4.0 Internacional: http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode Cualquier parte de esta obra se puede reproducir sin autorización pero con el reconocimiento y atribución de los autores. No se puede hacer uso comercial de la obra y no se puede alterar, transformar o hacer obras derivadas. Esta Tesis Doctoral está depositada en el Repositorio Institucional de la Universidad de Málaga (RIUMA): riuma.uma.es Contents 1 Background 17 1.1 Notation and conventions . 17 1.2 Infinity structures . 17 1.3 The homotopy transfer theorem . 21 1.4 The infinity bar, cobar and Quillen constructions . 25 1.5 Rational homotopy theory . 28 1.6 The Quillen and Eilenberg-Moore spectral sequences . 30 2 Higher Whitehead products and L structures 33 1 2.1 Whitehead products . 33 2.2 Higher Whitehead products . 34 2.3 Higher Whitehead products and L structures . 36 1 2.4 Higher Whitehead products and Sullivan L algebras . 46 1 2.4.1 Higher Whitehead products in Sullivan models . 47 2.4.2 Sullivan L algebras . 48 1 2.4.3 Extending Andrews and Arkowitz’s theorem to L algebras . 50 1 3 Higher Whitehead products and formality 53 3.1 Formality of DGL’s . 53 3.2 Formality and higher Whitehead products . 55 3.3 Examples . 56 3.4 Intrinsic (co)formality . 60 4 Massey products and A structures 63 1 4.1 Massey products and A structures . 64 1 4.2 Discussion on a result of Lu et al. 69 4.3 Recovering Massey products . 71 5 6 CONTENTS INTRODUCTION Rational homotopy theory classically studies the torsion free phenomena in the homotopy cat- egory of topological spaces and continuous maps. Its success is mainly due to the existence of relatively simple algebraic models that faithfully capture this non-torsion homotopical infor- mation. Infinity structures are algebraic gadgets in which some axioms hold up to a hierarchy of co- herent homotopies. We will work particularly with two of the most important of these, namely, A and L algebras. The former can be thought of as a differential graded algebra (DGA, 1 1 henceforth, or CDGA if it is commutative) where the associativity law holds up to a homo- topy which is determined by a 3-product whose associativity up to homotopy is again given by a 4-product, and so on. The latter can be seen as a differential graded Lie algebra (DGL, henceforth) where similarly, the Jacobi identity holds up to a homotopy which is determined by a 3-product, and so on. In this work, we use infinity structures to shed light on classical matters of rational homo- topy theory (and beyond, as will be explained below). When attempting to classify (or, more modestly, just to distinguish) rational homotopy types, one is naturally led to consider sec- ondary operations in homotopy or cohomology. Among these, we focus on the higher White- head and Massey products, complementing the usual Whitehead product in homotopy and cup product in cohomology, respectively [26, 60]. These fundamental homotopical invariants are at the very heart of the theory, but their manipulation can be difficult at times. Infinity struc- tures also classify rational homotopy types [31] and are sometimes more amenable to compu- tations or better adapted to the problem at hand than are the secondary operations. The main achievement of this thesis is the description of the precise relationship between the higher arity operations of an infinity structure governing a given rational homotopy type and the higher products in homotopy and cohomology of it. We use the developed theory to recover and generalize a classical result in rational homotopy theory [3, Thm. 5.4], and to give some applications in the context of (co)formality. Furthermore, some of the main results on the entanglement between the higher arity operations and the higher order secondary opera- tions still hold when the infinity structure is not necessarily modeling a rational space, making it possible to apply these results in other contexts. More precisely, some of the theorems that we prove are valid over fields of characteristic p 0 and/or for not necessarily finite type or È bounded (upper or below) complexes. We begin with a more accurate description of the results of the thesis (see the correspond- 7 8 Introduction ing section for complete details). The first chapter is a compendium of necessary background, and we properly start the original work in Chapter 2. We focus first on the homotopy side of things, and ignore the distinction between a map and its homotopy class hereafter. Higher or- der Whitehead products are homotopy invariant sets introduced in [55] and are constructed as follows. Let Sn1 ,...,Snk be simply connected spheres, denote by W Sn1 Snk and Æ _ ¢¢¢ _ T T (Sn1 ,...,Snk ) their wedge and fat wedge, respectively. There is a non trivial attaching map Æ N 1 ! : S ¡ T , with N n n , for which ! Æ 1 Å ¢¢¢ Å k Sn1 Snk T eN . £ ¢¢¢ £ Æ [w For k 2 given homotopy classes x ¼ (X ), consider the map these induce on the wedge ¸ j 2 n j f : W X and define the kth order Whitehead product set [x1,...,xk ] ¼N 1(X ) as the (possibly ! ⊆ ¡ empty) set © ª fe w fe: T X is an extension of f , ± j ! as depicted by the following diagram: f W X fe N 1 w S ¡ T Observe that for k 2, one recovers the classical two-fold Whitehead product. This construc- Æ tion is elegantly captured by Quillen’s DGL models for rational homotopy theory. As expertly explained in [63, Chap. V], given k 2 homology classes x j H (L) of a DGL L model of X , ¸ 2 ¤ define the kth order Whitehead bracket set [x1,...,xk ] H (L) as the (possibly empty) set of ho- ⊆ ¤ mology classes arising from the DGL maps closing the diagram below, arising from translating the previous topological diagram into the DGL setting: f L(u1,...,uk ) L fe L(w) w L(U) This translation identifies topological and algebraic higher Whitehead products. Recall that, given any DGL L there is a structure of minimal L algebra on H H (L), unique up to L 1 Æ ¤ 1 isomorphism, for which L and H are quasi-isomorphic L algebras. This structure can be 1 inherited from L by exhibiting H as a homotopy retract of it, but there is no canonical way of doing so. Hence, different choices give rise to different (although L quasi-isomorphic) L 1 1 structures on H. The rational homotopy type of a simply connected space X is governed by the Quillen model, which is in turn determined by many possibly different L structures on H. 1 One of the main results of Quillen’s rational homotopy theory lets us identify H H (L) ¼ (­X ) Q ¼ 1(X ) Q. Æ ¤ Æ» ¤ ­ Æ» ¤¡ ­ Roughly speaking, our aim in Chapter 2 (and in some sense, in the whole thesis, as will become clear by the end of this introduction) is to detect and recover (whenever defined) higher White- head products of order k of X via the operation of arity-k of an inherited L structure {`n} on 1 H. The first and most general result, which is essential for many others, reads as follows. Theorem 1 (Thm. 2.8) Let x ,...x H and assume that [x ,...,x ] is non empty. Then, for any 1 k 2 1 k homotopy retract of L, and for any x [x ,...,x ], 2 1 k k 1 X¡ "`k (x1,...,xk ) x ¡, ¡ Im`j , Æ Å Æ j 1 Æ 9 Pk 1 ¡ (k i) x where " ( 1) i 1 ¡ j i j. In particular, if ` 0 for j k 1, then up to a sign, ` (x ,...,x ) Æ ¡ Æ j Æ · ¡ k 1 k 2 [x1,...,xk ]. An immediate but interesting consequence of the result above is Corollary 2 (Cor. 2.9) If for some homotopy retract of L onto H, the induced higher brackets van- ish up to arity k 1 2, then for any x ,...,x H, the set [x ,...,x ] is non empty, and moreover, ¡ ¸ 1 k 2 1 k it consists of the single homology class [x ,...,x ] {x "` (x ,...,x )}. 1 k Æ Æ k 1 k We say that an L structure recovers higher Whitehead products if `k (x1,...,xk ) [x1,...,xk ]. 1 § 2 Theorem 1 suggests that it might not always be the case that `k recovers Whitehead products. In fact, it does not necessarily happen. Hence, we are led to define adapted homotopy retracts to a given x [x ,..,x ] (see Def. 2.11). We prove then 2 1 k Theorem 3 (Thm. 2.12)Let x [x ,...,x ]. Then, for any homotopy retract of L adapted to x, 2 1 k ` (x ,...,x ) x. k 1 k Æ We show (see thms. 2.13, 2.19 and 2.20, respectively) the existence of explicit spaces X and L structures on ¼ (­X ) Q governing the rational homotopy type of X for which 1 ¤ ­ (1) the evaluation `4(x1,x2,x3,x4) does not recover any Whitehead product.
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