Representation Stability for Homotopy Automorphisms 3

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We define the notion of FI-space, FI-module and finite generation of such in Section 1.1, see Defini- tions 1.6 and 1.7 below.

For the analogous theorem for connected sums, we need the notion of a boundary relative homotopy automorphism of a manifold N (with boundary). A boundary relative homotopy automorphism of N is a homotopy automorphism of N that preserves the boundary ∂ := ∂N pointwise. The topological monoid of boundary relative homotopy automorphisms is denoted by aut∂ (N). We also need the following proposition in order to state the second main theorem: Proposition. Let M = M d be a closed, simply connected, d-dimensional manifold, such that ′ d n d M = M rD˚ is also simply connected and non-contractible. For any n ≥ 1, let Mn = (# M)rD˚ , be the space obtained by removing an open d-dimensional disk from the n-fold connected sum of Q Q Q M. Let ∂Q := ∂Mn ⊂ Mn be the rationalization of the inclusion ∂Mn ⊂ Mn. Then π0(aut∂Q (Mn )) Q contains a subgroup Σ(Mn ) that is isomorphic to Σn. ⋄

Q Q Since aut∂Q (Mn ) is a group-like monoid, it follows that π0(aut∂Q (Mn )) is a group that acts on Q Q Q π∗(aut∂Q (Mn )) (discussed § 5.3). In particular Σ(Mn ) acts on π∗(aut∂Q (Mn )). Theorem B. (a) Let M = M d be a closed, simply connected, d-dimensional manifold, such that M ′ = M r D˚d is also simply connected and non-contractible. For any finite set S, let S d MS = (# M) r D˚ , be the space obtained by removing an open d-dimensional disk from the S-fold connected sum of M. For each k ≥ 1, we can define an FI-module by Q S 7→ πk (aut∂(MS)) and this FI-module is finitely generated. Q (b) Let {πk (aut∂(Mn)), φn} be the associated consistent sequence. Then the Σn-action on Q Q ∼ Q ∼ Q πk (aut∂(Mn)) coincides with the action of Σ(Mn ) = Σn on πk(aut∂Q (Mn )) = πk (aut∂(Mn)) and φn is induced by the standard stabilization map. ⋄

We define the notion of consistent sequence in Section 1.1 below, see Definition 1.8. Remark 1.1. These results are somewhat analogous to those for unordered configuration spaces of manifolds. Rational homological stability for unordered configuration spaces of arbitrary connected manifolds was proven by Church [Chu12], following integral results for open1 manifolds by Arnol’d [Arn69], MacDuff [McD75] and Segal [Seg79]. It was later proven by Kupers and Miller [KM18] that the rational homotopy groups of unordered configuration spaces on connected, simply connected manifolds of dimension at least 3, satisfy representation stability, in the sense of Church and Farb.⋄

We will study in parallel both pointed homotopy automorphisms of wedge sums of spaces and boundary relative homotopy automorphisms of manifolds obtained by removing a disk from connected sums of manifolds. Their theory is similar due to the following: Let M and N be closed d-dimensional manifolds and let M#N r D˚d be the space obtained by removing a d-disk from their connected sum. We have that M#N r D˚d is homotopy equivalent to the wedge sum M ′ ∨ N ′ where

1Integral homological stability is known not to hold for closed manifolds. A simple counterexample is given already 2 2 ∼ by the 2-sphere S , where H1(Bn(S ), Z) = Z/(2n − 2)Z (see for example [Bir75, Theorem 1.11]). REPRESENTATION STABILITY FOR HOMOTOPY AUTOMORPHISMS 3

M ′ and N ′ are the manifolds obtained by removing an open d-disk from M and N respectively. This may indicate why the study of homotopy automorphisms of connected sums and wedges are appropriate to study in parallel. We will denote the homotopy automorphisms of (M#N) r D˚d d that preserve the boundary pointwise by aut∂((M#N) r D˚ ).

In the first case, let (X, ∗) be a pointed, simply connected space with the homotopy type of a finite n CW-complex. For each n ≥ 1, the symmetric group Σn acts on the wedge sum Xn := i=1 X, which induces a Σn-action on the homotopy automorphisms aut∗(Xn) and thus on the homotopy groups of QW aut∗(Xn). Theorem A describes the stabilization of the rational homotopy groups π∗ (aut∗(Xn)) := π∗(aut∗(Xn)) ⊗ Q as n →∞. Remark 1.2. Homotopy automorphisms of iterated wedge sums of spheres has been studied by Miller, Patzt and Petersen [MPP19]. Using representation stability, they prove that for d ≥ 2, the n d 1 sequence {B aut i=1 S }n≥1 satisfies homological stability with Z 2 -coefficients, which proves homological stability with the same coefficients for {GLn(S)}n≥1, where S is the sphere spectrum.⋄ W

In the second case, let M = M d be a compact, simply connected, d-dimensional manifold, such that ′ d n d M = M rD˚ is also simply connected. Let Mn = (# M)rD˚ , be the space obtained by removing an open d-dimensional disk from the n-fold connected sum of M. In this case, we show that Σn Q acts on π∗ (aut∂(Mn)). Theorem A describes their stabilization behaviour as Σn-representations as n →∞. ⋄

Remark 1.3. It is unknown to the authors whether πk(aut∂(Mn)) defines a Σn-representation. Q However, we can define a Σn-action on πk (aut∂(Mn) which is induced by some finite group action GΣn ⊂ π0(aut∂(Mn)) on πk(aut∂(Mn)) (Theorem 5.12). In particular we have that the order of GΣn is at least the order of Σn. We deduce the lower bounds on a family of groups of components of (homotopy) automorphisms. ⋄

Theorem. Let M = M d be a closed manifold such that M ′ = M r D˚d is a simply connected non-contractible space. Then the order π0(aut∂(Mn)) of is at least n!. ⋄ Remark 1.4. This theorem can be generalized and extended; see Theorem 5.15. ⋄ Remark 1.5. For a simply connected d-dimensional manifold M, with boundary ∂M =∼ Sd−1, the rational homotopy theory of aut∂(M) has been thoroughly studied by Berglund and Madsen [BM20], whose results we will use. ⋄

1.1. Representation stability. We will study the stabilization of the sequences of homotopy groups described above in the sense of representation stability, a notion which was introduced by Church and Farb [CF13]. Specifically, we will show that these sequences of homotopy groups are uniformly representation stable. We will work with representation stability in the framework of FI-modules, which was introduced by Church, Ellenberg and Farb [CEF15]. Definition 1.6. Let FI denote the category of finite sets and injective maps. If C is a category, a functor FI →C is called an FI-object in C. An FI-object in ModR, for a commutative ring R, is called an FI-module (over R). An FI-object in the category of topological spaces is called an FI-space. ⋄ 4 ERIK LINDELL AND BASHAR SALEH

There are natural notions of sub-FI-modules as well as quotients, direct sums and tensor products of FI-modules, which are all defined pointwise. Definition 1.7. Let n := {1, 2,...,n}. A FI-module V is said to be finitely generated if there FI exists a finite set S ⊂ n≥1 V(n) such that there is no proper sub- -module W of V such that S ⊂ W(n). ⋄ n≥1 F F The key result by Church, Ellenberg and Farb [CEF15] about FI-modules is that if R is a field of characteristic 0, finite generation corresponds to (uniform) representation stability, i.e. an FI- module V over R is finitely generated if and only if each term in the sequence {V(n)}n≥1 is finite dimensional, and the sequence is (uniformly) representation stable, in the sense of [CF13].

Deﬁnition 1.8. An FI-module V gives rise to a sequence {V(n)} of Σn-representations. The standard inclusion n ֒→ n + 1 induces a Σn-equivariant map φn : V(n) →V(n + 1). Such a sequence {V(n), φn} is called a consistent sequence. ⋄ Remark 1.9. Not every consistent sequence arises from an FI-module, see Lemma 2.5 for details.⋄

1.2. Structure of the paper. In Section 2, we review the necessary background on FI-modules that we will need. We also introduce the notion of FI-Lie models of FI-spaces, which is of key importance for proving the main theorems. In Section 3 we review necessary rational homotopy theory for homotopy automorphisms needed for proving the main theorems. In Section 4, we study homotopy automorphisms of wedge sums and prove Theorem A. In Section 5, we study homotopy automorphisms of connected sums and prove Theorem B.

1.3. Acknowledgements. The authors would like to thank Dan Petersen and Alexander Berglund for many useful comments and suggestions. The proof of Theorem 5.12 is based on an answer by Ryan Budney to a question asked by the second author on MathOverﬂow. The second author was supported by the Knut and Alice Wallenberg Foundation through grant no. 2019.0521.

2. FI-modules and Lie models of FI-spaces

2.1. Conventions. We will work over the field Q, so unless otherwise specifically stated, all vector spaces are over Q. We will use “dg” to abbreviate the term differential graded. Throughout the paper, FI denotes the category of finite sets with injective maps as morphisms.

If S is a ﬁnite set, we will use |S| to denote its cardinality, and we will write Σ(S) := AutFI(S) for the symmetric group on S. If S = {1, 2,...,n} we will simply write Σ(S)=Σn for brevity.

2.2. FI-modules. In this section, we review the theory of FI-modules that we will need to prove the main theorems. Deﬁnition 2.1. Let C be a category. A functor FI →C is called an FI-object in C. ⋄

Let us review the kinds of FI-objects that will be of interest to us: REPRESENTATION STABILITY FOR HOMOTOPY AUTOMORPHISMS 5

• An FI-object in (gr)ModR, the category of (Z-graded) R-modules where R is a commutative FI FI ring, will be called a (graded) -module. An -object in dgModR, the category of differential graded modules over R, will be called a dg FI-module. For a dg FI-module V, we will write H∗(V) for the composition with the homology functor and refer to it as the homology of V. FI FI • An -object in dgLieR, the category of dg Lie algebras, over R, will be called a dg -Lie algebra and abbreviated by FI-dgl. • An FI-object in Top will be called an FI-space. If P is a property of topological spaces, such as being simply connected, we will say that an FI-space X has property P if X (S) has property P, for every finite set S. If X is an FI-space with π1(X (S)) being abelian for every Q finite set S, composing with the (rational) homotopy groups functor π∗ (resp. π∗ ) naturally Q gives us a graded FI-module over Z (resp. Q). We will simply write π∗(X ) (resp. π∗ (X )) for this composite functor and refer to it as the (rational) homotopy groups of X .

We will generally consider the first two examples for R = Q and reserve the terminology for this case, unless otherwise specified. Let us recall some basics from the theory of FI-modules. There are natural notions of sub-FI-modules as well as quotients, direct sums and tensor products of FI-modules, which are all defined pointwise. Definition 2.2. Let n := {1, 2,...,n}. A (graded) FI-module V is said to be finitely generated if FI there exists a finite set S ⊂ n≥1 V(n) such that there is no proper (graded) sub- -module W of V such that S ⊂ W(n). ⋄ n≥1 F F What makes the category of FI-modules over Q particularly useful is that it is noetherian, i.e. a sub-FI-module of a finitely generated FI-module is itself finitely generated (see [CEF15, Theorem 1.3]). Finite generation is also preserved by tensor products and quotients. This means that to prove that a FI-module is finitely generated, we may show that it is a subquotient of a tensor product of some FI-modules that are more obviously finitely generated.

Sometimes it will be more convenient to work with FI-modules in “coordinates”, rather than working with arbitrary ﬁnite sets. Therefore we make the following deﬁnition:

Deﬁnition 2.3. An FI-module V gives rise to a sequence {V(n)} of Σn-representations. The standard inclusion n ֒→ n + 1 induces Σn-equivariant maps φn : V(n) →V(n + 1). Such a sequence {V(n), φn} is called a consistent sequence. ⋄ Remark 2.4. Not every consistent sequence arises from an FI-module.

Lemma 2.5 ([CEF15, Remark 3.3.1]). A consistent sequence {V(n), φn} is induced by some FI- module if and only if for every σ ∈ Σn+k with σ|n = id acts trivially on

im(φn+k−1 ◦··· φn : V(n) →V(n + k)). If two FI-modules give rise to isomorphic consistent sequences, then the two FI-modules are isomorphic. ⋄

Since we want to use rational homotopy theory to prove our result, we need to consider graded FI-modules in our proofs. Let us therefore review some facts about them that we will need. As 6 ERIK LINDELL AND BASHAR SALEH for ordinary FI-modules, we may define submodules, quotients, direct sums and tensor products of graded FI-modules pointwise, using the corresponding notions for graded modules. Definition 2.6. If V is a graded FI-module and m ∈ Z, let Vm be the degree m part of V. If Vm = 0 for m ≤ m′ (resp. m ≥ m′) we say that V is concentrated in degrees above (resp. below) m′. Such a graded FI-module is called bounded from below (above). ⋄ Definition 2.7. A graded FI-module V is said to be of finite type if Vm is finitely generated for each m ≥ 0. ⋄

By the noetherian property of FI-modules, a graded sub-FI-module of a graded FI-module of finite type is also of finite type. It is also clear that quotients of graded FI-modules of finite type are of finite type. Note that this implies that if a dg FI-module is of finite type, considered just as a graded FI-module, then so is its homology. For tensor products to preserve finite type, we however need to add extra assumptions on the tensor factors: Lemma 2.8. Let V and W be graded FI-modules of finite type where both are either bounded from below or from above. Then V⊗W is also a graded FI-module of finite type. ⋄

2.3. FI-Lie models. Now, let us introduce the notion of an FI-Lie model, which will be one of the main tools of the paper. For the basic theory of Lie models in rational homotopy theory, see for example [FHT01]. Deﬁnition 2.9. Let X be a simply connected based FI-space and let L be a FI-dg Lie algebra. We say that L is an FI-Lie model for X if

(1) for every S ∈ FI, L(S) is a dg Lie model for the space X (S) and for every morphism S ֒→ T in FI, the map (2) L(S) →L(T ) models the map X (S) →X (T ). ⋄ ∼ Q Remark 2.10. If L is a FI-Lie model for X , then H∗(L) = π∗ (X ) as graded FI-modules. ⋄ Remark 2.11. Definition 2.9 works for our purposes, but it is not the “philosophically” correct definition of FI-Lie model, seen from a modern homotopy theoretic perspective. There is an equivalence of ∞-categories ∼ (dgLieQ)≥1 = Top≥2, between the ∞-categories of connected dg Lie algebras, localized at the quasi-isomorphisms, and simply connected spaces, localized at the rational homotopy equivalences. The usual definition of dg Lie model in rational homotopy theory may be expressed by saying that a connected dg Lie algebra (L, d) is a Lie model for a simply connected space X if they are isomorphic under this equivalence. Equivalently, it suffices to require that they are isomorphic under the equivalence ∼ FI between the homotopy categories h(dgLieQ)≥1 = hTop≥2. The correct definition of -Lie model should therefore be that a dg FI-Lie algebra L is a FI-Lie model of a simply connected FI-space X if they are isomorphic under the equivalence of the homotopy categories FI ∼ FI hFun( , (dgLieQ)≥1) = hFun( , Top≥2). REPRESENTATION STABILITY FOR HOMOTOPY AUTOMORPHISMS 7

In contrast, our deﬁnition is requiring isomorphism under the equivalence of “ordinary” functor categories FI ∼ FI Fun( , h(dgLieQ)≥1) = Fun( , hTop≥2). ⋄

2.4. Schur functors. The underlying graded FI-modules of the FI-Lie models that we want to study will be constructed by composing Schur functors with simpler graded FI-modules, which is why they are of ﬁnite type. In this section we will review what we mean by Schur functors in this context and their properties when composed with graded FI-modules.

Usually, the term Schur functor is used to describe a certain kind of endofunctor of VectQ coming from representations of symmetric groups. If W isaΣk-representation, we deﬁne the corresponding Schur functor by ⊗k V 7→ W ⊗Σk V , ⊗k considering V with the standard Σk-action. We will denote the Schur functor associated to W by SW .

The irreducible representations of Σk are indexed by partitions of weight k, i.e., non-increasing sequences of non-negative integers λ = (λ1 ≥ λ2 ≥···≥ λl ≥ 0) that eventually reaches zero such ∼ that |λ| = i λi = k. If W = Vλ is irreducible, the Schur functor associated to W is often denoted by Sλ and referred to as the Schur functor associated to λ. P Remark 2.12. Up to isomorphism, we can explicitly construct the irreducible representation Vλ as the image of a Young symmetrizer cλ ∈ Q[Σk] under the standard left action on Q[Σk] (see [FH91, §4.1]). We thus see that the Schur functor Sλ applied to V is given by image of cλ acting on ⊗k ∼ ⊗k Q[Σk] ⊗Σk V = V , which is commonly used as the deﬁnition of Sλ. ⋄

Schur functors are interesting to us since they preserve ﬁnite generation of FI-modules. More speciﬁcally:

Proposition 2.13. (Church, Ellenberg, Farb [CEF15, Theorem 3.4.3]) Let V : FI → VectQ be ﬁnitely generated. Then the FI-module Sλ ◦V, where ◦ denotes composition of functors, is also ﬁnitely generated. ⋄

Since any ﬁnite-dimensional representation of Σk decomposes into a ﬁnite direct sum of irreducible representations, we get the following corollary:

Corollary 2.14. Let W be a finite-dimensional representation of Σk and V : FI → VectQ be a finitely generated FI-module. Then the FI-module SW ◦V is also finitely generated. ⋄

Given a symmetric sequence W = (W (1),W (2),...) of (graded) vector spaces, i.e. a sequence in which W (k)isaΣk-representation, we can associate to it an endofunctor of (gr)VectQ given by ⊗k V 7→ W (k) ⊗Σk V , Mk≥1 8 ERIK LINDELL AND BASHAR SALEH which we will denote by SW and call the Schur functor associated to W . From the above results we get the following corollary: Corollary 2.15. Let W = (W (1),W (2),...) be a symmetric sequence of graded vector spaces, where each W (k) is finite dimensional and concentrated in non-negative degree, and let V : FI → FI grVectk be a graded -module, such that V(S) is concentrated in strictly positive degrees for every S ∈ FI. Suppose that V is finitely generated, considered as an FI-module. Then the graded FI-module SW ◦V is of finite type.

Proof. Note that by deﬁnition SW ◦V decomposes as the direct sum

SW (k) ◦V. Mk≥1 Since V is assumed to be finitely generated and each W (k) is assumed to be finite dimensional, each summand SW (k) ◦V is finitely generated by Corollary 2.14. Since each W (k) is concentrated in non-negative degree and each V(S) is concentrated in positive degree we get that SW ◦V is concentrated in positive degree and that m ⊗k (SW ◦V) ⊆ W (k) ⊗Σk V , 1≤Mk≤m for each m ≥ 1, which is finitely generated. Since the category of FI-modules is noetherian, it follows m that the (SW ◦ V ) is finitely generated, and thus SW ◦V is indeed of finite type.

Now let us turn to the examples that will be of interest to us. We make the following definition: Definition 2.16. Let H be a graded vector space of finite type, i.e. which is finite dimensional in each degree. We define a graded FI-module H by letting H(S) := H⊕S, for any S ∈ FI and for any ⋄ .i : S ֒→ T in FI letting H(i) be the natural inclusion H(S) ֒→H(T ) induced by i

In the following sections, H will be the desuspension of the reduced homology of a simply connected space. The FI-module H is clearly ﬁnitely generated (in fact, it is generated by any basis of H(1)), considered as an ordinary FI-module. Composing with the free graded Lie algebra functor L, we get a new graded FI-module, which we denote by LH. Corollary 2.17. Let H be a graded vector space concentrated in strictly positive degrees. Then the graded FI-module LH is of ﬁnite type.

Proof. If we let Lie = (Lie(1), Lie(2),...) denote the Lie operad over Q, the free Lie algebra on a graded vector space W is by definition ⊗k LW = Lie(k) ⊗Σk W . Mk≥1 We thus see that LH = SLie ◦H. Since H satisfies the hypothesis of Corollary 3.6 by assumption and since the Lie operad is finite dimensional and concentrated in degree zero in each arity, the corollary follows by Corollary 3.6. REPRESENTATION STABILITY FOR HOMOTOPY AUTOMORPHISMS 9

Next, we need to consider what happens when we dualize H. We may define a graded FI-module, which we will denote by H∗, as follows: (Definition 2.18. For S ∈ FI, let H∗(S) := H(S)∗. If i : S ֒→ T is a morphism in FI, φ ∈ H∗(S ⊕T ∗ and xα lies in the summand of H corresponding to α ∈ T , we define H (i)(φ) by

∗ 0 if α ∈ T \ i(S), (1) H (i)(φ) (xα)= −1 ( φ ◦H(i) (xα) if α ∈ i(S). ⋄

Just as for H, the following proposition is easily verified: Proposition 2.19. If H is a graded vector space of finite type, then the graded FI-module H∗, defined as above, is finitely generated. ⋄ Remark 2.20. The notion of FI-modules we have worked with could more properly be called left FI-modules, whereas we would define right FI-modules as a contravariant functors between the same categories. In general, dualizing a graded FI-module pointwise, as we have done above, does not produce graded left FI-module, but it does produce a right FI-module. ⋄ Definition 2.21. We define an FI-module Der(LH) as follows: for S ∈ FI, we let Der(LH)(S) := Der(LH(S)) be the graded Lie algebra of derivations on LH(S). For i : S ֒→ T in FI, we get a map ⊕S ⊕T Der(LH)(i) : Der(L(H )) ֒→ Der(L(H )), defined as follows: suppose xα ∈ H(T ) lies in the direct summand of H(T ) corresponding to α ∈ T and let D ∈ Der(L(H⊕S)). Then Der(LH)(i)D is determined by 0 if α ∈ T \ i(S), L (2) Der( H)(i)D (xα)= −1 ( LH(i) ◦ D ◦H(i) (xα) if α ∈ i(S). Note that a derivation on a L(H⊕T ) is uniquely determined by its restriction to H⊕T . Hence the equality of (2), determines Der(LH)(i)D completely.

′ Note that Der(L(H⊕S)) has a graded Lie algebra structure where [D, D′]= D◦D′−(−1)|D||D |D′◦D. Straightforward computation shows that Der(LH)(i)[D, D′] = [Der(LH)(i)(D), Der(LH)(i)(D′)] which means that the graded FI-module structure on Der(LH) is compatible with the graded Lie algebra structure. In particular Der(L(H⊕S)) defines a graded FI-Lie algebra. ⋄ Proposition 2.22. Let H be a finite dimensional graded vector space concentrated in strictly positive degrees. Then the graded FI-module Der(LH) is of finite type.

Proof. For every S ∈ FI, we have an isomorphism of graded vector spaces ∗ =∼ ΨS : H (S) ⊗ LH(S) → Der(LH)(S) given by sending φ ⊗ A ∈H∗(S) ⊗ LH(S) to the derivation in Der(LH)(S) deﬁned by x 7→ φ(x)A, 10 ERIK LINDELL AND BASHAR SALEH on x ∈ H(S). We want to prove that this deﬁnes a map of graded FI-modules, i.e. that for every morphism i : S ֒→ T , the diagram

ΨS H∗(S) ⊗ LH(S) Der(LH)(S)

(3) H∗(i)⊗LH(i) Der(LH)(i)

ΨT H∗(T ) ⊗ LH(T ) Der(LH)(T ) is commutative. This can be verified by applying the the definitions of ΨS and ΨT , together with the definition of H∗(i), given by (1), and the definition of Der(LH)(i), given by (2).

Thus Der(LH) =∼ H∗ ⊗LH, as graded FI-modules. Note that H∗ is concentrated in negative degrees, but which are bounded from below, by the assumption on H. Since LH is concentrated in strictly positive degrees, the result now follows by Corollary 2.17 and Proposition 2.19.

The FI-Lie models that we will construct in the next section will be graded FI-submodules of graded FI-modules of the type Der(LH), with H as above, so it will follow immediately from Corollary 2.22 that their homology is of ﬁnite type.

3. Rational homotopy theory for homotopy automorphisms

In this section, we will review some rational homotopy theory for homotopy automorphisms needed for this paper.

Let X be a simply connected topological space homotopy equivalent to a CW-complex. A homotopy automorphism of X is a self-map ϕ: X → X, that is a homotopy equivalence. We denote the monoid of unpointed and pointed homotopy automorphisms of X by aut(X) and aut∗(X), respectively. Given a subspace A ⊂ X, we denote the monoid of A-relative homotopy automorphisms of X, i.e the homotopy automorphisms that preserve A pointwise, by autA(X). The special case when X = N is a manifold with boundary A = ∂N, the monoid of boundary relative homotopy automorphisms of N is denoted by aut∂(N).

If X is well pointed and A ⊂ X is a cofibration of cofibrant spaces, then all of aut(X), aut∗(X) and autA(X) are group-like monoids, and thus equivalent to topological groups. For a group-like monoid G, all components are equivalent, and hence the homotopy groups are independent of the base point. In this paper πk(G) will denote πk(G, id) for k ≥ 1. We denote the classifying space of G by BG and its universal cover by BG. Let G◦ ⊂ G denote the connected component of the identity. We have that BG◦ ≃ BG. We observe that ∼ g ∼ ∼ πk(G) ⊗ Q = πk+1(BG) ⊗ Q = πk+1(BG◦) ⊗ Q = Hk(gBG◦ ) g for all k ≥ 1 and where gBG◦ is any dg Lie algebra model for BG◦. g The identity component of autA(X) is denoted by autA,◦(X). Remark 3.1. Recall that a functorial rationalization construction on nilpotent simplicial spaces that preserve cofibrations is constructed in [BK87]. Whenever we refer to a cofibration A ⊂ X, we REPRESENTATION STABILITY FOR HOMOTOPY AUTOMORPHISMS 11 are actually refering to a model for it in the model category of simplicial sets. The Bousfield-Kan rationalization yields thus a map r : autA(X) → autAQ (XQ) of topological monoids. ⋄

In this paper we will mainly be interested in the rational homotopy groups. For k ≥ 1 we have that ∼ πk(autA(X)) ⊗ Q = πk(autAQ (XQ)), since B autA,◦(X)Q ≃ B autAQ,◦(XQ), see [BS20, Proposition 2.4].

A model for B autA,◦(X) is given in terms of dg Lie algebras of derivations. Definition 3.2. Given a dg Lie algebra L, let Der(L) denote the dg Lie algebra of derivations of L where the graded Lie bracket is given by [θ, η]= θ ◦ η − (−1)|θ||η|η ◦ θ, and the differential is given by [dL, −] where dL is the differential of L. ⋄ + Definition 3.3. Given a chain complex C = C∗, the positive truncation of C, denoted by C , is given by Ci if i> 1 C+ = d i  ker(C1 −→ C0) if i = 1  0 if i< 1. ⋄  Recall that a dg Lie model for the space X is said to be quasi-free if it is free in the category of graded Lie algebras. Proposition 3.4 ([Tan83],[BS20]). (a) Let X be a simply connected space of the homotopy type of a finite CW-complex with a quasi-free dg Lie algebra model LX . A dg Lie model + for B aut∗,◦(X) is given by Der (LX ). (b) Let A ⊂ X be cofibration of simply connected spaces of the homotopy type of finite CW complexes, and let LA → LX be a cofibration (i.e a free map) that models the inclusion A ⊂ X. A dg Lie model for B autA,◦(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). ⋄

We recall the notion of geometric realizations of dg Lie algebras. For a detailed account on the subject refer the reader to [Hin97], [Get09], [Ber15] and [Ber20b].

Definition 3.5. Let Ω• denote the simplicial commutative dg algebra in which Ωn is the Sullivan- de Rham algebra of polynomial differential forms on the n-simplex. The geometric realization of a positively graded dg Lie algebra L is defined to be the simplicial set MC(L ⊗ Ω•) of Maurer-Cartan elements of the simplicial dg Lie algebra L ⊗ Ω•, denoted by MC•(L). A positively graded dg Lie algebra L is a Lie model for a simply connected space X if and only if there exists a zig-zag of rational homotopy equivalences between the geometric realization MC•(L) and X. ⋄

The functor MC• takes surjections to Kan fibrations and takes injections to cofibrations. In particular if LA → LX is a free map of dg Lie algebras that models a cofibration A ⊂ X, then the 12 ERIK LINDELL AND BASHAR SALEH cofibration MC•(LA) ֒→ MC•(LX ) is a simplicial model for the cofibration AQ ⊂ XQ. In particular we have that autAQ (XQ) and autMC•(LA)(MC•(LX )) are weakly equivalent as topological monoids. Definition 3.6. The exponential exp(h) of a nilpotent Lie algebra h concentrated in degree zero is the nilpotent group with the underlying set given by h and with multiplication given by the Baker- Campbell-Hausdorff formula. The exponential of a positively graded dg Lie algebra L, denoted exp•(L), is the simplicial group given by the exponential exp(Z0(L ⊗ Ω•)) of zero cycles in L ⊗ Ω• (see [Ber20b]). ⋄

Proposition 3.7. [Ber20b] For a positively graded dg Lie algebra L, we have that exp•(L) is weakly equivalent to the loop space Ω MC•(L). ⋄ Definition 3.8. Let L(V ) ⊂ L(V ⊕ W ) be a cofibration of free positively graded dg Lie algebras and let Der(L(V ⊕ W )kL(V )) denote the dg Lie algebra of derivations on L(V ⊕ W ) that vanish on + L(V ). Define a left action of exp•(Der (L(V ⊕ W )kL(V ))) on MC•(L(V ⊕ W )) given by Θi(x) (4) Θ.x = i! i X≥0 (see [BS20, § 3.2]). ⋄ Proposition 3.9 (cf. [Ber20a, Proposition 3.7]). Let A ⊂ X be a cofibration of simply connected spaces with homotopy types of finite CW-complexes, and let ι: L(V ) → L(V ⊕ W ) be a cofibration (i.e. a free map) of cofibrant dg Lie algebras that models the inclusion A ⊂ X. Then the topological monoid map + F : exp•(Der (L(V ⊕ W )kL(V ))) → autMC•(L(V )),◦(MC•(L(V ⊕ W ))) ≃ autAQ,◦(XQ), F (Θ)(x)=Θ.x is a weak equivalence.

+ Proof. Note that the action of exp•(Der (L(V ⊕W )kL(V ))) on MC•(L(V ⊕W )) ﬁxes MC•(L(V )) ⊂ MC•(L(V ⊕ W )) pointwise. In particular, the group action yields a map + exp•(Der (L(V ⊕ W )kL(V ))) → autMC•(L(V ))(MC•(L(V ⊕ W ))). + Moreover, since exp•(Der (L(V ⊕ W )kL(V ))) is connected we may replace the codomain by

autMC•(L(V )),◦(MC•(L(V ⊕ W ))) (i.e. the component of the identity). We proceed by following the proof of [Ber20a, Proposition 3.7], and adapting it to our situation. Given a positively graded dg Lie algebra h, there is an isomorphism of abelian groups G: Hk(h) → πk(exp•(h)) where a homology class of a cycle z ∈ Zk(h) is sent to the homotopy class of the k-simplex z ⊗ νk ∈ ∗ Z0(h ⊗ Ωk) where νk is the fundamental class k!dt1 · · · dtk. That G deﬁnes an isomorphism is motivated in the proof of [Ber20a, Proposition 3.7].

2 We have that νk = 0, and consequently

F (θ ⊗ νk)=id+θ ⊗ νk. REPRESENTATION STABILITY FOR HOMOTOPY AUTOMORPHISMS 13

Let us now analyze πk(autMC•(L(V )),◦(MC•(L(V ⊕ W )))), k ≥ 1, as in the proof of [BM20, Theorem 3.6]. In order to simplify notation, MC•(L(V ) is denoted by AQ and MC•(V ⊕ W ) by XQ. We have that an element f ∈ πk(autAQ,◦(XQ)) is represented by a map k f : (S ⊔ ∗) ∧ XQ → XQ k where f(∗,x)= x for every x ∈ XQ and and f(s, a)= a for every a ∈ AQ and s ∈ S .

k k A dg Lie algebra model for (S ⊔ ∗) ∧ XQ is given by (L(U ⊕ s U), ∂) where U = V ⊕ W and with a diﬀerential determined by the following: Let d be the diﬀerential on L(U), then ∂(u) = d(u) for every u ∈ U and ∂(sku) = (−1)kskd(u) for every sku ∈ skU.

k k Now, f : (S ⊔ ∗) ∧ XQ → XQ is modelled by some map ϕf : L(U ⊕ s U) → L(U) that satisfy k ϕf (u) = u for every u ∈ U and ϕf (s v) = 0 for every v ∈ V ⊂ U. Now, let θf be the unique k derivation on L(U) that satisfy θf (u)= ϕf (s u) for every u ∈ U. Note that θf is a cycle and that + it vanishes on L(V ), i.e. θf ∈ Zk(Der (L(U)kL(V ))). Also note that if f = πk(F )[θ ⊗ νk] then + θf = θ. Let K : πk(autAQ,◦(XQ)) → Hk(Der (L(U)kL(V ))) be given by K(f)= θf . It follows from [BM20] and [LS07] that this map is well-deﬁned and is an isomorphism.

Set h = Der+(L(U)kL(V )). The composition

G πk(F ) K Hk(h) −→ πk(exp•(h)) −−−−→ πk(autAQ,◦(XQ)) −→ Hk(h) is the identity map, which forces πk(F ) to be an isomorphism. This proves that F is a weak equivalence.

4. Homotopy automorphisms of wedge sums and Theorem A

Let (X, ∗) be a simply connected pointed space of the homotopy type of a finte CW-complex. For S any finite set S, let XS := X. For any morphism S ֒→ T in FI, there is an obvious induced base point preserving map XS ֒→ XT given by inclusion of wedge summands in the order specified by W . the injection S ֒→ T . Thus the functor S 7→ XS is an FI-space, which we will denote by X

We deﬁne an FI-space aut∗(X ) as follows: for S ∈ FI, we let aut∗(X )(S) := aut∗(XS ). For i : S ֒→ T in FI, we get a map aut∗(XS ) ֒→ aut∗(XT ), deﬁned as follows: suppose xα ∈ XT lies in the wedge summand of XT corresponding to α ∈ T and let f ∈ aut∗(XS). Then

xα if α ∈ T \ i(S), aut∗(X )(i)f (xα)= −1 ( X (i) ◦ f ◦X (i) (xα) if α ∈ i(S). Note that aut∗(X )(i)f is in some sense an extension by identity of f. For instance, if is : n → n + 1 is the standard inclusion, then aut∗(X )(i)f is the homotopy automorphism of Xn+1 that coincides with f on the ﬁrst n wedge summands, and is the identity on the last summand.

Restricting to the identity components gives a sub-FI-space aut∗,◦(X ). We are interested in the rational homotopy groups of this FI-space. 14 ERIK LINDELL AND BASHAR SALEH

Remark 4.1. Now it is tempting to say that we will construct an FI-Lie model for aut∗(X ). However, this FI-space is generally not simply connected. Instead, we take a functorial classifying space construction B : Monoids → Top, and consider the FI-space B aut∗,◦(X ), where aut∗,◦(XS) is the identity component of aut∗(XS), for every S ∈ FI. For every S, we have B aut∗,◦(XS) ≃ ^ B aut∗(XS) and so this is a simply connected FI-space, which enables us to apply our tools from rational homotopy theory. Furthermore, for every k ≥ 1, we have Q ∼ Q πk (aut∗(XS)) = πk+1(B aut∗,◦(XS)), so Q ∼ Q πk (aut∗(X )) = πk+1(B aut∗,◦(X )), as FI-modules. ⋄

A quasi-free dg Lie algebra (L(H), d) is said to be minimal if d(H) ⊂ [L(H), L(H)]. One can show −1 −1 that if (L(H), d) is a minimal model for a space X, then H =∼ s H˜∗(X; Q) (where s denotes desuspension). The differential will often be omitted from the notation, unless necessary. If L(H) is a quasi-free dg Lie model for X, then the free product L(H)∗S := L(H) ∗···∗ L(H) =∼ L(H⊕S) ⊕S is a dg Lie model for XS (see [FHT01, § 24 (f)]). We have that the association S 7→ L(H ) defines a , dg FI-Lie algebra LH. Given a morphism i : S ֒→ T in FI, we get an induced inclusion H⊕S ֒→ H⊕T which induces an inclusion L(H⊕S) ֒→ L(H⊕T ) that models the map X (i) : X (S) →X (T ). By the definition of the free product of dg Lie algebras, the map L(H⊕S) ֒→ L(H⊕T ) is a map of dg Lie algebras, which makes this into a dg FI-Lie algebra which is an FI-Lie model of X .

+ ⊕S We have by Proposition 3.4 (a) that a model for B aut∗,◦(XS ) is given by Der (L(H )), with ⊕S ⊕T diﬀerential given by [dL(H⊕S ), −]. Since the inclusion LH(i): L(H ) ֒→ L(H ) induced by an inclusion i: S ֒→ T is a dg Lie algebra map, it follows that Der+(L(H⊕S)) ֒→ Der+(L(H⊕T )) is also a dg Lie algebra map. This together with Deﬁnition 2.21, yields that we have a dg FI-Lie algebra + (Der (LH), [dLH, −]).

+ We will show that (Der (LH), [dLH, −]) deﬁnes an FI-Lie model for the FI-space B aut∗,◦(X ). + FI Theorem 4.2. Let (L(H), dL(H)) be a Lie model for X. Then (Der (LH), [dLH, −]) is an -Lie model for the FI-space B aut∗,◦(X ).

Proof. It follows from Lemma 2.5 that it is enough to prove the following two properties:

(a) Let is : n → n + 1 denote the standard inclusion. Then a dg Lie algebra model for

B aut∗,◦(X )(is): B aut∗,◦(Xn) → B aut∗,◦(Xn+1) is given by + + ⊕n + ⊕n+1 ϕn := Der (LH)(is): Der (L(H )) → Der (L(H )) + ⊕n (b) The Σn-action on B aut∗,◦(Xn) is modelled by the Σn-action on Der (L(H )). REPRESENTATION STABILITY FOR HOMOTOPY AUTOMORPHISMS 15

⊕k Proof of (a): In order to simplify notation, let L(H ) be denoted by Lk, and let Hℓ =∼ H denote ⊕n+1 the last summand of H , and let Lℓ denote L(Hℓ). In particular we have that Ln+1 = Ln ∗ Lℓ. Let cn : MC•(Ln) → MC•(Ln+1) and cl : MC•(Lℓ) → MC•(Ln+1) denote the coﬁbrations induced by the standard inclusion Ln → Ln ∗ Lℓ and Lℓ → Ln ∗ Lℓ respectively.

From Proposition 3.9 we get topological monoid equivalences + Fn : exp•(Der (Ln)) → aut∗(MC•(Ln)). Those maps have adjoints ˜ + Fn : exp•(Der (Ln)) × MC•(Ln) → MC•(Ln).

We have by the explicit formulas for {Fn} that ˜ ˜ Fn+1 ◦ (exp•(ϕn) × cn)= cn ◦ Fn. + In particular, Fn+1 ◦ exp•(ϕn)(Θ) is an extension of Fn(Θ), for Θ ∈ exp•(Der (Ln)). We also have that ˜ Fn+1 ◦ (exp•(ϕn) × cℓ)= cℓ.

In particular Fn+1 ◦ exp•(ϕn)(Θ) restricts to the identity on MC•(Lℓ) ⊂ MC•(Ln+1). That means that exp•(ϕn) is a simplicial model for aut∗,◦(X )(is): aut∗,◦(Xn) → aut∗,◦(Xn+1). This gives (a).

+ + Proof of (b): The Σn-action on Der (Ln) induces a Σn-action on exp (Der (Ln)). We also have that n • the Σn-action on Ln that models the Σn-action on X induces a Σn-action on aut∗,◦(MC•(Ln)). + We need to show that the Σn-action on exp•(Der (Ln)) models the Σn-action on aut∗,◦(MC•(Ln)). This is satisﬁed if following diagram W + / + Σn × exp(Der (Ln)) / Der (Ln)

id ×F ∼ ∼ F / Σn × aut∗,◦(MC•(Ln)) / aut∗,◦(MC•(Ln)) commutes. We have that this diagram commutes since

(σ ◦ θ ◦ σ−1)i(x) θi F (σ.θ)(x)= ∗ ∗ = σ ◦ ◦ σ−1(x) = (σ.F (θ))(x) i! ∗  i!  ∗ i i X≥0 X≥0   for any σ ∈ Σn.

We have now all ingredients needed for proving Theorem A. Theorem A. Let (X, ∗) be a pointed, simply connected space with the homotopy type of a finite S CW-complex and let XS := X, for any finite set S. This defines an FI-space in a natural way. For each k ≥ 1, the FI-module defined by W Q S 7→ πk (aut∗(XS)) is finitely generated. ⋄ 16 ERIK LINDELL AND BASHAR SALEH

Proof. We will use the established terminology in this section. We have already seen in Theorem + 4.2 that (Der (LH), [dLH, −]) is an FI-Lie model for B aut∗,◦(X ). Hence, it is enough to prove that + + Hk(Der (LH)) is ﬁnitely generated (since Hk(Der (LH)) =∼ πk(aut∗(X )), see Remark 4.1).

−1 Since H = s H˜∗(X), it follows that H is finite dimensional and concentrated in positive degrees. This means that H satisfies the hypothesis of Corollary 2.22, and so Der(LH) is a graded FI-module of finite type. By the noetherian property of FI-modules it follows that Der+(LH) is also of finite + + type. Since (Der (LH), [dLH, −]), defines a dg FI-Lie algebra, it follows that H∗(Der (LH)) is a + + graded FI-module. Since H∗(Der (LH)) is a subquotient of Der (LH), it is also of finite type. + Hence the FI-module Hk(Der (LH)) is finitely generated for each k ≥ 1.

5. Homotopy automorphisms of connected sums and Theorem B

This section is divided into four subsections. In the ﬁrst subsection we discuss brieﬂy stabilization maps of homotopy automorphisms of iterated connected sums of manifolds. We conclude that it is unknown to us whether the homotopy automorphisms of iterated connected sums of manifolds form an FI-space in a natural way.

In the second subsection we observe that for any ﬁnite set S, we may construct dg Lie models gS S d for B aut∂,◦(# M r D˚ ), in a way so that S 7→ gS becomes a dg FI-Lie algebra. In particular we ∼ Q S ˚d FI have that S 7→ Hk(gS) = πk (aut∂,◦(# M r D )) deﬁnes an -module. In the third subsection we do topological interpretations of the results of the second subsection and in the fourth subsection we prove Theorem B.

5.1. General theory for homotopy automorphisms of iterated connected sums. Let M = M d be a closed, simply connected, d-dimensional manifold, such that M ′ = M r D˚d is also simply n d connected. For every n ≥ 1, let Mn = (# M) r D˚ , be the space obtained by removing an open d-dimensional disk from the n-fold connected sum of M.

d d We observe that Mn+1 is obtained from Mn by attaching K = M r D˚ ⊔ D˚ along the boundary of MS and one boundary component of K.

A homotopy automorphism of Mn does not extend to a homotopy automorphism of Mn+1 in any canonical way in general. However, boundary relative homotopy automorphisms of Mn extends by identity on K to a boundary relative homotopy automorphism of Mn+1. In particular, there is a stabilization map

s: aut∂(Mn) → aut∂(Mn+1).

n ′ We moreover observe that Mn ≃ i=1 M , so there exists some “up to homotopy” Σn-action on Mn. However, it is not clear to us whether this action can be made to preserve the boundary pointwise W (i.e. whether Σn ⊂ π0(aut∂(Mn))). In particular, it is for us unknown whether it is possible to S d deﬁne an FI-space given by S 7→ aut∂(# M r D˚ ). REPRESENTATION STABILITY FOR HOMOTOPY AUTOMORPHISMS 17

5.2. Rational homotopy theory for homotopy automorphisms of iterated connected sums and FI-modules. In this section we discuss a speciﬁc dg Lie model for B aut∂,◦(Mn) that is constructed in [BM20].

Let M = M d be a simply connected closed manifold and let M ′ = M r D˚d. The intersection form ′ on H∗(M) induces a graded symmetric inner product of degree d on the reduced homology H˜∗(M ). This in turn induces a graded anti-symmetric inner product of degree d − 2 on H = s−1H˜ ∗(M ′). Deﬁnition 5.1. Let H be a graded symmetric anti-symmetric inner product space of degree d − 2 −1 ˜ ′ # # (e.g. s H∗(M )) with a basis {α1,...,αm}. The dual basis {α1 ,...,αm} is characterized by the following property # hαi, αj i = δij. 2 Let ωH ∈ L (H) be given by m 1 ω = [α#, α ]. H 2 i i i X=1 It turns out that ωH is independent of choice of basis {α1,...,αm}; see [BM20] for details. ⋄

The next proposition is due to Stasheﬀ [Sta83], also discussed in [BM20]. Proposition 5.2 ([Sta83]). Let M = M d be a closed oriented d-dimensional manifold, let M ′ = d −1 ′ d−1 ′ ′ M r D˚ and let H = s H˜∗(M ). Then inclusion S =∼ ∂M ֒→ M is modelled by a dg Lie algebra map (ι: L(x) ֒→ L(H d ι(x) = (−1) ωH . ⋄

−1 ′ ⊕n Given a fixed basis {α1,...,αm} for H = s H˜∗(M ) we get a basis for H˜∗(Mn) =∼ H , which is of the form j {αi | 1 ≤ i ≤ m, 1 ≤ j ≤ n}. j # j ⊕n We denote ωH⊕n = i,j[(αi ) , αi ] ∈ L(H ) by ωn. We have that ωn is invariant under the Σ -action on L(H⊕n) that permutes the summands of H⊕n. n P ⊕n Note that ι: L(x) → L(H ) is not a cofibration. In order to model the inclusion ∂Mn ⊂ Mn by a cofibration in the model category of dg Lie algebras we need a new model for Mn. Lemma 5.3. Let L(H⊕n,x,y) be the dg Lie algebra that contains L(H⊕n) as a dg Lie subalgebra and where |x| = d − 2 and |y| = d − 1 and where

dx = 0, dy = x − ωn. Then ˆι: L(x) → L(H⊕n,x,y) ˆι(x)= x d−1 is a coﬁbration that models the inclusion of ∂Mn =∼ S into Mn. ⋄ 18 ERIK LINDELL AND BASHAR SALEH

We have by Proposition 3.4 (b), that a dg Lie algebra model for B aut∂,◦(Mn) is given by Der+(L(H⊕n,x,y)kL(x)). Proposition 5.4 ([BM20, Theorem 3.12]). Let L(H⊕n) and L(H⊕n,x,y) be the equivalent models ⊕n ⊕n of Mn (see Lemma 5.3). Let Der(L(H )kωn) denote the dg Lie algebra of derivations on L(H ) that vanish on ωn and where the diﬀerential is given by [dL(H⊕n), −]. Then there is an equivalence of dg Lie algebras + ⊕n + ⊕n Der (L(H )kωn) → Der (L(H ,x,y)kL(x)), θ 7→ θˆ ˆ + ⊕n where θ|L(H) = θ and θ(x) = θ(y) = 0. In particular Der (L(H )kωn) is a dg Lie algebra model for B aut∂,◦(Mn). ⋄

Definition 5.5. With the terminology established in § 2, we define a dg FI-Lie algebra Der(LHkωH) as follows: for S ∈ FI, we let Der(LHkωH)(S) := Der(LH(S)kωS) be the dg Lie algebra of derivations ⊕S on LH(S)= L(H ) that vanish on ωS. For i: S ֒→ T in FI, we get a map ⊕S ⊕T ,( Der(LHkωH)(i): Der(L(H )kωS) ֒→ Der(L(H )kωT defined as follows: suppose xα ∈H(T ) lies in the direct summand of H(T ) corresponding to α ∈ T ⊕S and let D ∈ Der(L(H )kωS). Then Then Der(LHkωH)(i)D is determined by 0 if α ∈ T \ i(S), L Der( HkωH)(i)D (xα)= −1 ( LH(i) ◦ D ◦H(i) (xα) if α ∈ i(S). ⋄

We conclude from having such a dg FI-Lie algebra the following: Corollary 5.6. For k ≥ 1, ⊕S ∼ Q S ˚d S 7→ Hk(Der(L(H )kωS)) = πk (aut∂(# M r D )) deﬁnes an FI-module. ⋄ + Corollary 5.7. Applying the geometric realization functor to Der (LHkωH) gives an FI-space + MC•(Der (LHkωH)) ⋄

In the next subsection, we give topological interpretations of the results of this subsection.

5.3. Topological interpretations. Associated to the FI-space + ⊕S S d S 7→ MC•(Der (L(H )kωS)) ≃ B aut∂,◦(# M r D˚ )Q, we get a consistent sequence (see Deﬁnition 2.3) + ⊕n {MC•(Der (L(H )kωn)), φn}n≥1. + ⊕n In this section we will give topological interpretations of the Σn-action on MC•(Der (L(H )kωn)) and the topological interpretation of φn. REPRESENTATION STABILITY FOR HOMOTOPY AUTOMORPHISMS 19

′ ′ ′ −1 A topological group G acts on itself by conjugation G → Aut(G ), g 7→ κg where κg(h) = ghg . ′ ′ This gives that π0(G ) acts on π∗(G ). This group action restricts to a group action on the identity ′ ′ ′ ′ component G◦ of G , which in turn induces an action of G on BG◦. Since a group-like monoid G ′ is equivalent to a topological group G , we have that π0(G) acts on π∗(G◦).

Q Q Let Mn denote the rationalization of Mn. We will show that π0(aut∂Q (Mn )) contains a subgroup Q Σ(Mn ) that is isomorphic to Σn. Thus we have that Σn acts on Q ∼ Q π∗(aut∂Q (Mn )) = π∗ (aut∂(Mn)).

In order to prove that, we need to recall some theory from [ES20, § 3] Deﬁnition 5.8. Let ˆι: L(x) → L(H⊕n,x,y), ˆι(x)= x be the dg Lie algebra model for the inclusion ⊕S ∂Mn ⊂ Mn, deﬁned in Lemma 5.3. We say that a dg Lie algebra endomorphism ϕ of L(H ,x,y) is ˆι-relative if the diagram L(x) rr ▲▲▲ ˆι rr ▲▲ ˆι rrr ▲▲▲ rx rr ▲▲& ⊕n / ⊕n L(H ,x,y) ϕ L(H ,x,y) commutes strictly. We say that two ˆι-relative endomorphisms ϕ and ψ are ˆι-equivalent if there exists a homotopy h: L(H⊕n,x,y) → L(H⊕n,x,y) ⊗ Λ(t,dt) from ϕ to ψ that preserves L(x) ⊂ L(H⊕n,x,y) pointwise; i.e. h(x)= x ⊗ 1. ⋄ Q Proposition 5.9. The group π0(aut∂Q (Mn )) is isomorphic to the group given by {ϕ: L(H⊕n,x,y) → L(H⊕n,x,y) | ϕ is an ˆι-relative automorphism}/ˆι-equivalence.

Proof. We have that ˆι: L(x) → L(H⊕n,x,y) is a minimal relative dg Lie algebra model for the inclusion ∂Mn ⊂ Mn in the sense of [ES20, § 3]. Now the result follows from [ES20, Corollary 4.5]. Proposition 5.10. Let M = M d be a closed manifold such that M ′ = M rD˚d is a simply connected Q Q non-contractible space. Then π0(aut∂Q (Mn )) contains a subgroup Σ(Mn ) that is isomorphic to Σn.

⊕n ⊕n Proof. Let Autˆι L(H ,x,y) denote the group of ˆι-relative automorphisms of L(H ,x,y), and let ⊕n Q ξ : Autˆι L(H ,x,y) → π0(aut∂Q (Mn )) be the projection map that sends an ˆι-relative automorphism to its ˆι-equivalence class.

⊕n We ﬁx a basis basis {α1,...,αm} for H and give H a basis of the form j {αi | 1 ≤ i ≤ m, 1 ≤ j ≤ n}.

⊕n ⊕n Given a permutation σ ∈ Σn, let ψσ : L(H ,x,y) → L(H ,x,y) be the ˆι-relative automorphism j σ(j) ⊕n given by ψσ(αi )= αi and ψσ(y)= y. Consider now the following subgroup of Autˆι L(H ,x,y) Q ∼ Σ(Mn )= {ψσ | σ ∈ Σn} = Σ(S).

e 20 ERIK LINDELL AND BASHAR SALEH

We have that ψσ and ψτ are equivalent if and only if σ = τ, which means that they represent Q diﬀerent classes in π0(aut∂Q (Mn )). Hence Q Q Σ(Mn ) := ξ(Σ(Mn )) Q is a subgroup of π0(aut∂ (M )) isomorphic to Σn. Q n e ⊕n Q Corollary 5.11. There exists a Σn-action on autMC•(L(x))(MC•(L(H ,x,y))) (≃ aut∂Q (Mn )) given by (σ, α) 7→ MC•(ψσ) ◦ α ◦ MC•(ψσ−1 ). ⋄

The authors do not know whether π0(aut∂(Mn)) contains a subgroup isomorphic to Σn. We will Q however show that Σ(Mn ) “comes from” π0(aut∂(Mn)) in the following sense: Q Theorem 5.12. Let r : π0(aut∂ (Mn)) → π0(aut∂Q (Mn )) be induced by the Bousﬁeld-Kan ratio- Q Q nalization functor. For every element ψσ ∈ Σ(Mn ) ⊂ π0(aut∂Q (Mn )) there exists some ησ ∈ π0(aut∂ (Mn)) such that r(ησ) = ψσ. In particular there exists a short exact sequence of ﬁnite groups −1 Q Q 1 → G → r (Σ(Mn )) → Σ(Mn ) → 1.

n+1 d d Proof. Let K = i=1 Di be a disjoint union of d-dimensional disks, d ≥ 2, and let E0 ∈ Emb K,S d d be some fixed topological embedding of K into S , which enables us to view Di as a subspace of d F d−1 d d d S . In particular E0 is a cofibration. Set Si := ∂Di ⊂ S . We have that Emb K,S is endowed by a Σn-action induced by the Σn-action on K that permutes the n first disks and lefts the last d one fixed. Since Emb K,S is connected, we have that E0 and σ.E0 =: Eσ are isotopic, i.e. there d exists a homotopy of embeddings hσ : K × I → S from E0 to Eσ. Since E0 is a cofibration, the d d homotopy hσ extends to a homotopy hˆσ : S × I → S from the identity map to some homotopy automorphism Eˆσ = hˆσ(−, 1), that satisfies Eˆσ|K (x)= σ.x. For each σ ∈ Σn we fix some homotopy automorphism Eˆσ that way.

Given a closed and oriented d-dimensional manifold M, we observe that #nM may be obtained d as follows: First take n copies of M, which we index M1,...,Mn. Then we attach Mi r D˚ to d n+1 ˚d d−1 ˚d S r i=1 Di along Si and the boundary of Mi r D . Now we see that for every σ ∈ Σn there n exists some homotopy automorphism ησ of # M, given by F σ.x if x ∈ M r D˚d η (x)= i σ Eˆ (x) if x ∈ Sd r D˚d. σ i Q Q We have that r(ησ)= ψσ ∈ Σ(Mn ), proving the exactness at Σ(FMn ).

−1 Q Q That G = ker(r (Σ(Mn )) → Σ(Mn )) is ﬁnite follows from the fact that r is ﬁnite to one [ES20, Theorem 1.1].

Remark 5.13. An interesting question is whether the exact sequence above splits, i.e. whether Σn is isomorphic to some subgroup of π0(aut∂ (Mn)). ⋄

Remark 5.14. We may replace the homotopy automorphisms Eˆσ and ησ by boundary relative diﬀeomorphisms by making use of the isotopy extension theorem (see for instance [Hir76, Theorem REPRESENTATION STABILITY FOR HOMOTOPY AUTOMORPHISMS 21

1.3, Chapter 8])). We may now ask whether it is possible to choose diffeomorphisms ησ ∈ Diff(Mn) in a way so that the map Σn → Diff(Mn) given by σ 7→ ησ becomes a group homomorphism (i.e. that Σn acts on Mn). ⋄

As a consequence of Theorem 5.12 and Remark 5.14 we get following bounds on groups of components (not relevant for proving the main theorems): Theorem 5.15. Let M = M d be a closed manifold such that M ′ = M r D˚d is a simply connected non-contractible space and let A ⊂ ∂Mn be any subspace, possibly empty. Then the orders of π0(autA(Mn)), π0(DiﬀA(Mn)), π0(HomeoA(Mn)) are at least n!.

−1 Q Proof. We start proving the assertion for π0(aut∂(Mn)). Since r (Σ(Mn )) is a subgroup of π0(aut∂ (Mn)), it follows that −1 Q ord(π0(aut∂ (Mn))) ≥ ord(r (Σ(Mn ))) (where ord denotes the order of a group). By Therorem 5.12 it follows that −1 Q Q ord(r (Σ(Mn )) = ord(G) · ord(Σ(Mn )) = ord(G) · n!. Q This gives a lower bound on the order of π0(aut∂ (Mn)). Since diﬀerent maps in Σ(Mn ) are non- equivalent as non-relative maps as well, the same lower bound on the order of π0(autA(Mn)) applies for any A ⊂ ∂Mn.

By Remark 5.14 and the arguments above we get the same lower bounds on the orders of π0(DiﬀA(Mn)) and π0(HomeoA(Mn)).

We conclude this subsection with the following theorem:

⊕n n d Theorem 5.16. Let L(H ) be a model for Mn =# M r D˚ .

(a) A dg Lie algebra model for the stabilization map B aut∂,◦(Mn) → B aut∂,◦(Mn+1) is given by + ⊕n + ⊕n+1 ϕn : Der (L(H )kωn) → Der (L(H )kωn+1) ⊕n+1 where ϕs(θ) coincides with θ on the ﬁrst n summands of H and vanishes on the last summand. ⊕n (b) The Σn-action on B autMC•(L(x)),◦(MC•(L(H ,x,y))) deﬁned in Corollary 5.11 is modeled + ⊕n −1 by the Σn-action on Der (L(H )kωn) given by σ.θ = σ∗ ◦ θ ◦ σ∗ .

The proof is omitted since it is very similar to the proof of Theorem 4.2.

5.4. Proof of Theorem B. Theorem B. (a) Let M = M d be a closed, simply connected, d-dimensional manifold, such that M ′ = M r D˚d is also simply connected and non-contractible. For any ﬁnite set S, let 22 ERIK LINDELL AND BASHAR SALEH

S d MS = (# M) r D˚ , be the space obtained by removing an open d-dimensional disk from the S-fold connected sum of M. For each k ≥ 1, we can deﬁne an FI-module by Q S 7→ πk (aut∂(MS)) and this FI-module is ﬁnitely generated. Q (b) Let {πk (aut∂(Mn)), φn} be the associated consistent sequence. Then the Σn-action on Q Q ∼ Q ∼ Q πk (aut∂(Mn)) coincides with the action of Σ(Mn ) = Σn on πk(aut∂Q (Mn )) = πk (aut∂(Mn)) and φn is induced by the standard stabilization map. ⋄

+ ⊕ ⊕n Proof. (a) We have that Der (LH kωn) ⊆ Der(LH )is an inclusion of graded FI-modules, and + ⊕ so we may apply the noetherian property of FI-modules in order to deduce that Der (LH kωn) is ﬁnitely generated. Now the assertion follows as in the proof of Theorem A.

(b) This is a straightforward consequence of Theorem 5.16.

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