Formality and Rational Homotopy Theory of Relative Homotopy

Home , Connected sum, Homotopy category, Loop space, Rational homotopy theory

Bashar Saleh Formality and rational homotopy theory of relative homotopy

Formality and rational homotopy of relativetheory homotopy automorphisms automorphisms

Bashar Saleh

ISBN 978-91-7911-266-0

Doctoral Thesis in Mathematics at Stockholm University, Sweden 2020

Formality and rational homotopy theory of relative homotopy automorphisms Bashar Saleh Academic dissertation for the Degree of Doctor of Philosophy in Mathematics at Stockholm University to be publicly defended on Friday 23 October 2020 at 13.00 in online via Zoom, public link is available at the department web site.

Abstract This PhD thesis consists of four papers treating topics in rational homotopy theory. In Paper I, we establish two formality conditions in characteristic zero. We prove that a dg Lie algebra is formal if and only if its universal enveloping algebra is formal. We also prove that a commutative dg associative algebra is formal as a dg associative algebra if and only if it is formal as a commutative dg associative algebra. We present some consequences of these theorems in rational homotopy theory. In Paper II, which is coauthored with Alexander Berglund, we construct a dg Lie algebra model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a subspace, so called relative homotopy automorphisms. In Paper III, which is coautohored with Hadrien Espic, we prove that the group of homotopy classes of relative homotopy automorphisms of a simply connected finite CW-complex is finitely presented and that the rationalization map from this group to its rational analogue has a finite kernel. In Paper IV, we study rational homological stability for the classifying space of the monoid of homotopy automorphisms of iterated connected sums of complex projective 3-spaces.

Keywords: rational homotopy theory, formality, relative homotopy automorphisms.

Stockholm 2020 http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-184205

ISBN 978-91-7911-266-0 ISBN 978-91-7911-267-7

Department of Mathematics

Stockholm University, 106 91 Stockholm

FORMALITY AND RATIONAL HOMOTOPY THEORY OF RELATIVE HOMOTOPY AUTOMORPHISMS

Bashar Saleh

Formality and rational homotopy theory of relative homotopy automorphisms

Bashar Saleh ©Bashar Saleh, Stockholm University 2020

ISBN print 978-91-7911-266-0 ISBN PDF 978-91-7911-267-7

Printed in Sweden by Universitetsservice US-AB, Stockholm 2020 Abstract

This PhD thesis consists of four papers treating topics in rational homotopy theory. In Paper I, we establish two formality conditions in characteristic zero. We prove that a dg Lie algebra is formal if and only if its universal enveloping algebra is formal. We also prove that a commutative dg associative algebra is formal as a dg associative algebra if and only if it is formal as a commutative dg associative algebra. We present some consequences of these theorems in rational homotopy theory. In Paper II, which is coauthored with Alexander Berglund, we construct a dg Lie algebra model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a subspace, so called relative homotopy automorphisms. In Paper III, which is coautohored with Hadrien Espic, we prove that the group of homotopy classes of relative homotopy automorphisms of a simply connected finite CW-complex is finitely presented and that the rationalization map from this group to its rational analogue has a finite kernel. In Paper IV, we study rational homological stability for the classifying space of the monoid of homotopy automorphisms of iterated connected sums of complex projective 3-spaces. Sammanfattning

Denna doktorsavhandling består av fyra artiklar som behandlar ämnen inom rationell homotopiteori. I Artikel I, bevisar vi två formalitetsvillkor i karakteristik noll. Vi visar att en dg Liealgebra är formell om och endast om dess universella envelopperande algebra är formel. Vi visar även att en kommutativ differentialgraderad algebra är formell som en kommutativ algebra om och endast om den är formell som en associativ algebra. Vi presenterar konsekvenser av dessa satser inom rationell homotopiteori. I Artikel II, som är skriven tillsammans med Alexander Berglund, beräknas en dg Liealgebramodell för den universella täckningen av det klassificerande rummet av den grupplika monoiden av homotopiautomorfier av ett topologiskt rum som fixerar ett delrum, så kallade relativa homotopiautomorfier. I Artikel III, som är skriven tillsammans med Hadrien Espic, bevisar vi att gruppen av homotopiklasser av relativa homotopiautomorfier av ett enkelt sammanhängande ändligt CW-komplex är ändligt presenterad och att ratio- naliseringsavbildningen från denna grupp till dess rationella analog har en ändlig kärna. I Artikel IV, studererar vi rationell homologisk stabilitet för det klassificerande rummet för monoiden av homotopiautomorfier av itererade samman- hängande summor av komplexa projektiva 3-rum. List of Papers

The following papers, referred to in the text by their Roman numerals, are included in this PhD thesis.

Paper I: Noncommutative formality implies commutative and Lie formality First published in Algebraic & Geometric Topology, 17:4, 2523- 2542 (2017), published by Mathematical Sciences Publishers. Bashar Saleh

Paper II: A dg Lie model for relative homotopy automorphisms First published in Homology, Homotopy and Applications, 22:2, 105 -121 (2020), published by International Press of Boston. Alexander Berglund, Bashar Saleh

Paper III: On the group of homotopy classes of relative homotopy automorphisms Submitted Hadrien Espic, Bashar Saleh

Paper IV: Homological stability for homotopy automorphisms of connected sums of complex projective 3-spaces Bashar Saleh

Reprints were made with permission from the publishers. Paper I and Paper II appear in the author’s Licentiate thesis.

Acknowledgements

My advisor Alexander Berglund has been a huge source of inspiration, knowledge and support. I feel fortunate to have had the opportunity of writing this thesis under your supervision and no words are enough to express my gratitude to you.

I would also like to thank all the colleagues at the Department of Mathmetics. Thank you for making the department a pleasant workplace.

A special thanks go to the present and former members of the Stockholm Topology Center. Many of you have had a direct impact on this thesis via discussions, suggestions and proof readings.

I would also like to thank the mathematical society in general. I have during my PhD studies asked many mathematical questions, either to mathematicians I have interacted with, or at MathOverﬂow, and I have almost always received valuable feedback.

Lastly, I would like to thank my family and my friends for their support and love.

Contents

Abstract i

Sammanfattning ii

List of Papers iii

Acknowledgements v

1 Introduction 9 1.1 Rational homotopy theory ...... 9 1.2 Algebraic models and (co)formality ...... 10 1.3 Algebraic operads and P∞-algebras ...... 11 1.4 Classiﬁcation of ﬁbrations and homotopy automorphisms ...... 12 1.5 Homological stability for homotopy automorphisms ...... 13 1.6 The Sullivan-Wilkerson Theorem ...... 14

2 Summaries of Papers 17 2.1 Summary of Paper I ...... 17 2.2 Summary of Paper II ...... 18 2.3 Summary of Paper III ...... 19 2.4 Summary of Paper IV ...... 20

References 23

1. Introduction

The purpose of this introduction is to put the papers in this PhD thesis in a context of homotopy theory. Some parts of sections 1.1 - 1.4 appear in the author’s Licentiate thesis [Sal18].

1.1 Rational homotopy theory

We will give a brief introduction to rational homotopy theory of simply connected spaces. Much of what is presented here can be generalized to nilpotent spaces, but will not be treated in this introduction. Homotopy theory of topological spaces is sometimes very hard; for instance, until this moment there is no known closed formula for the higher homotopy groups of the n-dimensional sphere, for n ≥ 2. Continuing on this specific example; the hard part of determining the homotopy groups of spheres is to determine the torsion part of these groups. The free part is however com- pletely known and easily described. By tensoring a finitely generated abelian group by the field of rational num- bers Q, the torsion part is eliminated and we get a rational vector space with a rank that coincides with the rank of the abelian group. In this way, the in- formation about the free part of the finitely generated group is fully preserved while the torsion part is eliminated. Quillen proved in [Qui69] that there is a simplified approach to homotopy theory on the category of simply connected spaces where the notion of weak equivalence is weakened to be a map f : X → Y that induces isomorphisms

π∗( f ) ⊗ Q: π∗(X) ⊗ Q → π∗(Y) ⊗ Q on the rational homotopy groups. We call such a map for a rational homotopy equivalence. We say that two spaces X and Y are rationally equivalent if there exists a zig-zag of rational homotopy equivalences

X ←− Z1 −→··· ←− Zn −→ Y connecting these two spaces.

9 The localization of the category of pointed simply connected spaces with respect to rational homotopy equivalences is called the rational homotopy category of simply connected spaces and has the following remarkable feature: The rational homotopy category of simply connected spaces is equivalent to the homotopy category of positively graded dg Lie algebras over the rationals (i.e. the localization of the category of positively graded dg Lie algebras at the quasi-isomorphisms). In particular, that means it is possible to model simply connected spaces and maps between those up to rational equivalence by dg Lie algebras and dg Lie algebra maps. Hence, any rational homotopy theoretic problem can be translated into an algebraic problem. Later, Sullivan showed in [Sul77] that it also possible to model the rational homotopy category of simply connected spaces of ﬁnite rational type (i.e. spaces where rank(πk(X)) < ∞ for every k) by the homotopy category of simply connected commutative dg algebras (i.e. the localization of the category of simply connected dg algebras at the quasi-isomorphisms). In particular, with the results of Quillen and Sullivan, we have for every simply conneced space X an associated algebraic model MX (where MX is either a dg Lie algebra (dgl) or a commutatuve dg associative algebra (cdga)) such that X is rationally homotopy equivalent to Y if and only if MX is quasi- isomorphic to MY (we say that two algebras are quasi-isomorphic if they are connected by a zig-zag of quasi-isomorphisms that preserve the algebra structure).

1.2 Algebraic models and (co)formality

A cdga model AX captures the rational homotopy type of X and is unique up to quasi-isomorphism in the category of cdga’s. Of course a cdga model ∗ could not be simpler than its cohomology H (AX ) (which in many cases is much smaller than AX and has also a trivial differential). It is not true that every cdga model is quasi-isomorphic to its cohomology, but sometimes that happens and in these cases we say that the cdga model AX and the space X ∗ are formal. In particular, if X is formal, then H (AX ), which coincides with ∗ the singular cohomology H (X;Q) of X with rational coefﬁctents, captures the ∗ ∼ ∗ rational homotopy type of X (since H (AX ) = H (X;Q) is in this case a cdga model for X). We have an analogous story for the dgl models. A dgl model LX for a space X captures the rational homotopy type of X and is unique up to quasi- isomorphism in the category of dgl’s. The homology H∗(LX ) of LX is isomorphic to the graded Lie algebra of rational homotopy groups π∗(ΩX)⊗Q of the based loop space on X, endowed by the Samelson Lie bracket (see [Whi78] for deﬁnition). When a dgl model LX is quasi-isomorphic to its homology

10 we say that LX is a formal dgl, and that X is a coformal space. In particular, we have that the rational homotopy type of a coformal space X is determined by the graded Lie algebra structure given on the rational homotopy groups π∗(ΩX) ⊗ Q. Formality and coformality simpliﬁes computations in many situations and are important concepts in rational homotopy theory ([DGMS75]), deformation quantization ([Kon03]), deformation theory ([GM88]), and other branches of mathematics where differential graded homological algebra is used.

1.3 Algebraic operads and P∞-algebras

In Paper I, we need results which apply to dga’s, cdga’s and dgl’s simultane- ously. In such situations the language of algebraic operads can be useful. An algebraic operad P is an object that encodes a type of algebra (e.g. associative, commutative, Lie, Poisson or Gerstenhaber algebras). If one proves a property for P-algebras without specifying P, then that property is valid for all types of algebras that are encoded by an operad (and thus we do not need to make separate proofs for every type of algebras). Over a ﬁeld of characteristic zero the category of dg P-algebras admits a model category structure, which basically means that there is a natural setting for homotopy theory in the category (see [Val14] for an introduction to the subject). The weak equivlances in the category of P-algebras are the quasi-isomorphisms, and two objects are called weakly equivalent if they are connected by a zig-zag of weak equivalences. It is always possible to embed the category of dg P-algebras in the category of so called ‘homotopy P-algebras’, also called the category of P∞- algebras with ∞-morphisms (denoted by ∞-P∞-alg). Proposition 1.3.1 ([LV12]). Let P be an algebraic operad. The category of dg P-algebras is embedded in a category ∞-P∞-alg such that:

(a) ∞-P∞-alg is the smallest category containing the category of dg P- algebras as a subcategory in which quasi-isomorphisms are invertible up to homotopy.

(b) Two P-algebras are weakly equivalent as P-algebras if and only if they are weakly equivalent in ∞-P∞-alg.

(c) Every P∞-algebra is weakly equivalent to a dg P-algebra; in particu- ∼ lar Ho(dgP-alg) = Ho(∞-P∞-alg). Corollary 1.3.2. A dg P-algebra A is a formal if and only if there exists an ∗ ∗ ∞-P∞-quasi-isomorphism A → H (A), where H (A) is given a dg P-algebra

11 structure with a trivial differential and a graded P-algebra structure induced by A.

In this setting the notion of formality is simpler since a zig-zag of quasi- isomorphisms can be replaced by a single quasi-isomorphism.

1.4 Classiﬁcation of ﬁbrations and homotopy automorphisms

The theory of fibrations is very fundamental for homotopy theory. A map π : E → B is called a Serre fibration if the map has the homotopy lifting property with respect to all CW-complexes, which means that for any CW-complex ˜ Y, any homotopy h: Y ×I → B and any map h0 : Y → E that lifts h|Y×{0}, there ˜ ˜ ˜ exists a homotopy h: Y × I → E which lifts h and where h|Y×{0} = h0. If B is connected, the fibers π−1(b), i.e. the inverse images of the points in B, are all weakly equivalent, so it makes sense to consider ‘the fiber’ X of a fibration E → B. We will refer to a fibration with fiber X as an X-fibration. Associated to an X-fibration X ,→ E −→π B there is a long exact sequence

δ ··· → πn(X,x) → πn(E,e) → πn(B,b) −→ πn−1(X,x) → ··· → π0(E,e), which is functorial with respect to maps of fibrations. This long exact sequence is one of the most useful tools homotopy theory. Moreover, any map Y → B can be ‘replaced’ by a fibration E → B where E is weakly equivalent to Y over B. A map of fibrations over B is a commutative triangle

f E / E0

π 0 π B where π : E → B and π0 : E → B are fibrations. We say that the map of fibrations above is a weak equivalence if f is a weak equivalence. Two fibrations are said to be weakly equivalent if there is a zig-zag of weak equivalences connecting these two fibrations. It follows that weakly equivalent fibrations have weakly equivalent fibers. The X-fibrations are classified by the classifying space Baut(X) of the monoid of homotopy automorphisms of the space X, in the following sense:

12 For every CW-complex B, there is a functorial bijection between the set of equivalence classes of X-fibrations over B and the set [B,Baut(X)] of homotopy classes of maps from B to Baut(X) (we refer to [May75] for details). Of course one could put restrictions on the X-fibrations, which result in other classifying spaces. In Paper II and III, we study the homotopy type of classifying spaces for so called relative fibrations. If A ⊂ X is a cofibration, e.g. a CW-pair, then an A-relative X-fibration is a fibration X → E → B under the trivial A-fibration A → A × B → B, such that over each b ∈ B, the canoni- cal map from A to im(A → Eb) is a weak equivalence. Fibrations of this type are classified by the classifying space of the topological monoid of so called A-relative homotopy automorphisms of X, i.e. homotopy automorphisms of X that preserve the subspace A pointwise. The monoid of relative homotopy autmorphisms is denoted by autA(X). When X is a manifold and A is its boundary, the monoid of boundary preserving homotopy autmorphisms is denoted by aut∂ (X), and will be of essential interest in the study of homological stability for homotopy automorphisms of iterated connected sums of manifolds.

1.5 Homological stability for homotopy automorphisms

∞ Given a family of groups {Gi}i=1 where Gk is a subgroup of Gk+1, the inclusions G1 ,→ G2 ,→ ··· yield for every m ∈ Z≥0 a sequence of homology groups Hm(G1) → Hm(G2) → ··· .

If the groups Gi are topological groups, then we take homology of their classifying spaces (the group homology of a discrete group coincides with the homology of its classifying space). We say that the homology groups stabilizes if Hm(Gk) → Hm(Gk+1) is an isomorphism for k large enough. If the homology stabilizes, then the stable homology is denoted by H∗(G∞). Given a family of groups {Gi} of the type described above, we may ask many questions regard- ing homological stability. Does the homology stabilize? If that is the case, what is the stability range, and what is the stable homology H∗(G∞)? Homological stability results has been established for a wide variety of families of groups. There are homological stability results for symmetric groups ([Nak61]), braid groups ([Arn69]), general linear groups ([vdK80]), mapping class groups of compact orientable surfaces ([Har85]), to name few. One of the remarkable things about the study of homological stability is that it sometimes connects objects from different areas. For instance, the Madsen-Weiss theorem [MW07], relates the stable homology of the mapping class groups of compact orientable surfaces, to the homology of a certain inﬁ- nite loop space. A generalization of this result to higher dimensional manifolds

13 was obtained by Galatius and Randal-Williams [GRW14]. Another example in which homological stability results connects objects from different areas is obtained by Berglund and Madsen in [BM20]. They connect the stable rational homology of homotopy automorphisms and block diffeomorphisms of iterated connected sums of Sd × Sd to the homology of graph complexes in the sense of Kontsevich [Kon93], [Kon94]. We will now brieﬂy discuss rational homology stability for homotopy automorphisms of iterated connected sums of manifolds, since that is the topic of Paper IV. Given a closed d-dimensional topological manifold M, we can form a sequence of inclusions X1 ⊂ X2 ⊂ ··· , n d where Xn = # M r D˚ , i.e. the space obtained by removing a d-dimensional disk from n-fold connected sum of M. We note that the manifold Xn+1 is d d obtained from Xn by attaching to it a copy of K = M r (D˚ t D˚ ) along the boundary of Xn and a boundary component of K. A relative homotopy au- ∼ d−1 tomorphism of Xn that ﬁxes the boundary ∂Xn = S , extends to a relative homotopy automorphism of Xn+1 by identity on K. Thus, we have a stabilization map aut∂ (Xn) → aut∂ (Xn+1), which is a map of monoids, and induces therefore a map on the classifying spaces. The study of rational homological stability for homotopy automorphisms of highly connected manifolds was initiated in [BM20]. In loc. cit., rational homological stability results are proven for homotopy automorphisms of iterated connected sums of products of spheres of the same dimension. In [Gre19], the results were extended to certain products of spheres of possibly different dimensions. In Paper IV, we study rational homological stability for homotopy automorphisms of iterated connected sums of CP3 (more details in Section 2.4).

1.6 The Sullivan-Wilkerson Theorem

A linear algebraic group G over a ring R is a group such that there is some embedding i: G ,→ GLn(R), where i(G) is defined by polynomial equations on the entries of the matrices in GLn(R). In particular G can be viewed as an affine variety with a group structure where multiplication and inverses are maps of varieties. Examples of linear algebraic groups are the special linear groups SLn(R), the orthogonal groups On(R), the symplectic groups Spn(R), and all finite groups. There are many classical books treating the theory of linear algebraic groups, e.g. [Bor91].

14 An arithmetic subgroup GZ of an algebraic group G over Q, is a subgroup that is isomorphic to i(G) ∩ GLn(Z) for some embedding i: G ,→ GLn(Q). Different embeddings yield different arithmetic subgroups, but one can prove that all arithmetic subgroups of a linear algberaic group are commensurable. We say that two groups are commensurable if there is a finite zig-zag of group homomorphisms connecting these two groups, where each group homomorphism has a finite kernel and an image of finite index 1 . Some of the nice properties of arithmetic groups are that they are finitely presented and, more strongly, they are of finite type (which means that their classifying spaces are homotopy equivalent to a CW-complex with finitely many cells in each dimension). Classical references on arithmetic groups are [BHC61] and [Bor69]. The connection between the theory of homotopy automorphisms and the theory of algebraic and arithmetic groups was realized by Sullivan ([Sul77]) and Wilkerson ([Wil76]), independently, using different methods. One of their main results is the following theorem:

Theorem 1.6.1 (The Sullivan-Wilkerson Theorem). Let X be a nilpotent connected finite CW-complex, and let XQ be a rationalization of X. Then π0(aut(XQ)) is an algebraic group and π0(aut(X)) is commensurable with an arithmetic subgroup of π0(aut(XQ)). The commensurability relation preserves the property of being finitely presented, being of finite type, or being finite. From this we get two important consequences of this theorem. The first one is that π0(aut(X)) is finitely presented and of finite type. Moreover, if π0(aut(XQ)) is finite then π0(aut(X)) is also finite. Several generalizations of the Sullivan-Wilkerson Theorem have been established since the original proofs, e.g. for groups of homotopy classes of fiber homotopy equivalences [Sch80], groups of homotopy classes of homotopy automorphisms of virtually nilpotent spaces [DDK81], and groups of homotopy classes of homotopy automorphisms of spaces with finite fundamental group [Tri95]. In Paper III, we prove a result that could to some extent be viewed as relative version of the Sullivan-Wilkerson Theorem. The result is then applied in the proof of the homological stability result of Paper IV.

1Different sources use the notion of commensurability of groups differently; we discuss these notions brieﬂy in the introduction of Paper III.

15 16 2. Summaries of Papers

The summaries of Paper I and Paper II appear in the author’s Licentiate thesis [Sal18].

2.1 Summary of Paper I

In Paper I we prove two formality conditions in characteristic zero and present some consequences of these theorems in rational homotopy theory. It is known that the universal enveloping algebra functor U : DGLk → DGAk preserves formal objects ([FHT01, Theorem 21.7]). That means that the formality of a dg Lie algebra L implies the formality of UL (as a dg associative algebra). But what about the converse? Does the formality of UL imply the formality of L? In Paper I we show that this holds for dg Lie algebras over a field of characteristic zero. Theorem 2.1.1. A dg Lie algebra L over a field of characteristic zero is formal if and only if its universal enveloping algebra UL is formal as a dga. The rational homotopy type of a simply connected space X is algebraically modeled by Quillen’s dg Lie algebra λ(X) over the rationals ([Qui69]). The space X is called coformal if λ(X) is a formal dgl. It is known that there exists a zig-zag of quasi-isomorphisms connecting Uλ(X) to the algebra C∗(ΩX,Q) of singular chains on the Moore loop space of X ([FHT01, Chapter 26]). From Theorem 2.1.1 the following corollary follows immediately: Corollary 2.1.2. Let X be a simply connected space. Then X is coformal if and only if C∗(ΩX;Q) is formal as a dga. Our second formality result is concerning the forgetful functor from the category of commutative dg associative algebras to the category of dg associative algebras. This functor preserves formality; a cdga which is formal as a cdga is obviously formal as a dga. Again, we ask whether this relation is re- versible or not. We will prove that over a field of characteristic zero the answer is positive. Theorem 2.1.3. Let A be a cdga over a field of characteristic zero. Then A is formal as dga if and only if it is formal as a cdga.

17 Recall that a space X is called rationally formal if the Sullivan-de Rham algebra APL(X;Q) is formal as a cdga ([FHT01, Chapter 12]). In that case the rational homotopy type of X is a formal consequence of its cohomology ∗ ∗ H (X;Q), meaning that H (X;Q) determines the rational homotopy type of X. Moreover, it is known that there exists a zig-zag of quasi-isomorphisms con- ∗ necting APL(X;Q) with the singular cochain algebra C (X;Q) of X ([FHT01, Theorem 10.9]). An immediate topological consequence is the following corollary:

Corollary 2.1.4. A space X is rationally formal if and only if the singular ∗ cochain algebra C (X;Q) of X is formal as a dga.

2.2 Summary of Paper II

Paper II is coauthored with Alexander Berglund. In this paper, we compute a dg Lie algebra model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that ﬁx a subspace.

Theorem 2.2.1 ([BM20, Theorem 3.4.]). Let A ⊂ X be a cofibration of simply connected spaces with homotopy types of finite CW-complexes, and let i: LA → LX be a cofibration that models the inclusion A ⊂ X and where LA and LX are cofibrant Lie models for A and X respectively. A Lie model for the universal covering of BautA(X) is given by the positive truncation of the dg Lie algebra of derivations on LX that vanish on LA, denoted by Der(LX kLA)h1i. This theorem is stated in [BM20] together with the suggestion that a proof can be given by generalizing [Tan83, Chapitre VII], but no detailed proof exists in the literature. One purpose of this paper is to fill this gap. However, instead of following the suggested route (which seems to yield a rather tedious proof), we give a proof that is perhaps more interesting. Namely, we show that the model for relative homotopy automorphisms can be derived from the known model for based homotopy automorphisms together with general result on rational models for geometric bar constructions. We give two computational applications of this theorem.

Example 2.2.2. For any k ∈ Z>0 one can show that the rational homotopy k+1 groups of BautCPk (CP ) are given by

Q k+1 Q k+1 π∗+2k+1(BautCPk (CP )) = π∗ (CP ). Example 2.2.3. If X is a simply connected manifold with boundary ∂X =∼ n−1 S , then there exists a dg Lie model for Baut∂ (X) with the underlying graded −1 Lie algbebra given by Der(L(s H˜∗(X;Q),u,v)ku)h1i.

18 2.3 Summary of Paper III

Paper III is coauthored with Hadrien Espic. In this paper, we prove that the group of homotopy classes of relative homotopy automorphisms of a simply connected finite CW-complex is finitely presented and that the rationalization map from this group to its rational analogue has a finite kernel.

Theorem 2.3.1. Let A ⊂ X be a cofibration of simply connected spaces of the homotopy type of finite CW-complexes. Then π0(autA(X)) is finitely presented and the map (aut (X)) → (aut (X )) has a finite kernel. π0 A π0 AQ Q The results of this paper are, to some extent, an extension of the classical Sullivan-Wilkerson Theorem [Sul77; Wil76] (see §1.6). The proof of the finite presentation property depends on the classical Sullivan-Wilkerson Theorem, which involves some theory of algebraic and arithmetic groups. In order to be able to use the classical Sullivan-Wilkerson Theorem, we prove the following:

Theorem 2.3.2. Let A ⊂ X be a cofibration of simply connected spaces of the homotopy type of finite CW-complexes. Then (aut (X )) is a linear alge- π0 AQ Q braic group and the map (aut (X )) → (aut(X )) induced by the inclu- π0 AQ Q π0 Q sion aut (X ) ,→ aut(X ) is a homomorphism of linear algebraic groups. AQ Q Q The algebraicity of the non-relative homotopy automorphisms of a rational space XQ are usually proved by showing that the homotopy classes of automorphisms of a minimal Sullivan model for XQ forms an algebraic group. However, this approach is not suitable for modelling relative homotopy automorphisms. Rather one should consider homotopy classes of relative automorphisms of so called minimal relative dg Lie algebra models. The authors could not find an explicit treatment of the theory of minimal relative dg Lie algebra models in the literature, so this is explicitly treated in this paper. In particular we prove the following:

Theorem 2.3.3. Given a map f : L(V) → g of positively graded connected dg ∼ Lie algebras, there exists a minimal relative model q: L(V ⊕W) −→ g for f in the following sense:

(a) L(V) is a dg subalgebra of L(V ⊕W) and f = q ◦ ι, where ι : L(V) → L(V ⊕W) is the inclusion.

(b) Given a quasi-isomorphism g: L(V ⊕W) → L(V ⊕W), where g restricts to an automorphism of L(V), then g is an automorphism.

Here, L(A) denotes a quasi-free dg Lie algebra generated by A, i.e. a dg Lie algebra whose underlying graded Lie algebra structure is free.

19 2.4 Summary of Paper IV

In Paper IV, we prove a rational homological stability result for the iterated connected sums of complex projective 3-spaces. This constitutes, to the knowledge of the author, the ﬁrst rational homological stability result for homotopy automorphisms of iterated connected sums of a manifold M where the reduced cohomology ring on the punctured manifold ∗ H˜ (M r {x}) has a non-trivial cup product. The purpose of this project is to investigate to what extent the techniques in [BM20] could be used for simply connected closed manifolds with more complicated cohomology ring struc- tures. The easiest such example is CP3 and thus a very natural manifold to start with (note that CP2 is highly connected, and almost all of the techniques in [BM20] applies directly). The study of rational homological stability for homotopy automorphisms of iterated connected sums of highly connected manifolds was initiated in [BM20]. In loc. cit., stability results are proven for connected sums of products of spheres of the same dimension. In [Gre19], the results were extended to iterated connected sums of certain products of spheres of possibly different dimensions (but still, the reduced cohomology of these spaces are trivial in the sense discussed above). It turns out that the rational homology of the classifying space of the topological monoid of the relative homotopy automorphisms of the n-fold connected sum of CP3, is easily described given Theorem 2.3.1 in the summary of Paper III. n 3 6 Let Xn = # CP rD˚ denote the space obtained by removing a 6-dimensional open disk from the n-fold connected sum of CP3. A relative homotopy automorphisms of Xn is a homotopy automorphisms of Xn that preserve the bound- ∼ 5 ary ∂Xn = S pointwise. We will denote the topological monoid of relative homotopy automorphisms of Xn by aut∂ (Xn). 3 6 The space Xn+1 may be obtained from Xn by attaching K = CP r (D˚ t 6 D˚ ) along the boundary of Xn and one boundary component of K. A relative homotopy automorphism of Xn extends to a relative homotopy automorphism of Xn+1 by identity on K. In particular we get a stabilization map

s: aut∂ (Xn) → aut∂ (Xn+1).

The stabilization map induces a map in rational homology of the classifying spaces

s∗ : Hk(Baut∂ (Xn);Q) → Hk(Baut∂ (Xn+1);Q), which we will show is an isomorphism for n large enough.

20 Theorem 2.4.1. The stabilization map induces an isomorphism

s∗ : H`(Baut∂ (Xn);Q) → H`(Baut∂ (Xn+1);Q) for (` + 1)/2 − 1 < n.

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