Decoupling decorations on moduli spaces of manifolds
DDECOUPLING DECORATIONS ON MODULI SPACES OF MANIFOLDS
LUCIANA BASUALDO BONATTO
Abstract.
We consider moduli spaces of d -dimensional manifolds with embedded particles anddiscs. In this moduli space, the location of the particles and discs is constrained by the d -dimensional manifold. We will compare this moduli space with the moduli space of d -dimensionalmanifolds in which the location of such decorations is no longer constrained, i.e. the decorationsare decoupled. We generalise work by B¨odigheimer–Tillmann for oriented surfaces and obtain newresults for surfaces with different tangential structures as well as to higher dimensional manifolds.We also provide a generalisation of this result to moduli spaces with more general submanifolddecorations and specialise in the case of decorations being unparametrised unlinked circles. Introduction
The diffeomorphism group of a smooth manifold and its classifying space are fundamental objects intopology. In particular, for a closed smooth manifold W , the space B Diff( W ) classifies the smoothfibre bundles with fibre W . When W is a manifold with non-empty boundary, we consider Diff( W )to be the group of those diffeomorphisms which are the identity near ∂W . The classifying spacesof such groups are also extremely important, as they are crucial for instance to the constructionof topologically enriched categories of bordisms. To completely understand the classifying space ofa diffeomorphism group is extremely difficult and such a description is only available for very fewmanifolds. One key strategy when studying B Diff( W ) is to understand how its homology behaveswhen changing the manifold W by operations such as connected sum or gluing of cobordisms. Inthis paper, we use this strategy to study the stable homology of the decorated diffeomorphismgroup.A d -dimensional manifold W is said to be decorated if it is equipped with disjoint embeddings ofk points and m discs D d . The decorated diffeomorphism group of W , denoted Diff km ( W ) consistsof those φ : W → W which preserve the marked points and parametrized discs up to permutations.The classifying space B Diff km ( W ) has been studied from many different perspectives, for instance,considering the behaviour after increasing the number of marked points or discs (see [Til16]).For the case of W = S g,b the orientable surface of genus g and b boundary components,B¨odigheimer and Tillmann [BT01] studied the comparison between B Diff km ( S g,b ) and B Diff( S g,b ).They used a decoupling map(1.1) d : B Diff + ,km ( S g,b ) B Diff + ( S g,b ) × B Σ m × B (Σ k (cid:111) SO(2)) f × e m × e k where the map f is induced by the inclusion Diff km ( S g,b ) → Diff( S g,b ), the map e m is induced byDiff km ( S g,b ) → Σ m recording the permutation of the marked discs, and e k is induced by the mapDiff km ( S g,b ) → Σ k (cid:111) SO(2) recording the permutation of the marked points together with the inducedmap on their tangent space (see Figure 1 for a geometric representation of the decoupling map).B¨odigheimer and Tillmann showed that d induces a homology isomorphism in degrees ≤ g , therefore,in this range, we say that the decorations, which were bound to the manifold, get decoupled. Theproof of this result relies strongly on Harer’ stability theorem. Later, generalisations of Harer’sresult for non-orientable surfaces [Wah08] allowed Hanbury [Han09] to generalise the decouplingresult to such surfaces as well. In this paper, we further generalise this result to moduli spaces ofmanifolds in higher dimensions with tangential structures. Date : August 5, 2020. a r X i v : . [ m a t h . A T ] A ug LUCIANA BASUALDO BONATTO R ∞ R ∞ f e m e k B Diff km ( S g ) B Diff( S g ) C m ( R ∞ ) C m ( R ∞ ; B SO(2))
Figure 1.
Geometric representation of the decoupling map for the oriented modulispace of a surface S g as the product of three maps: forget the decorations, recordthe centre of the m marked discs, and record the k marked points and their orientedtangent spaces. For more details on the geometric interpretation see Section 1.5.1.1. Tangential Structures.
Orientation, framings, spin structures and maps to a backgroundspace are all examples of a more general type of structure which can be described just from dataon the tangent bundle of a manifold. A tangential structure is a topological space Θ equipped witha continuous GL d action. A Θ-structure on a d -dimensional manifold W is a GL d -equivariant map ρ W : Fr( T W ) → Θ, where Fr(
T W ) denotes the space of framings of the manifold W . A canonicalexample is Θ or = {± } with action given by multiplication with the sign of the determinant, andit is simple to see that a Θ or -structure on a manifold is a choice of orientation.The space of all Θ-structures has an action of Diff( W ) given by precomposition with the differ-ential. Given a closed compact connected smooth manifold W equipped with a Θ-structure ρ W ,the moduli space M Θ ( W, ρ W ) of W with Θ-structures concordant to ρ W is defined as the pathcomponent of ρ W in the Borel construction { GL d -equivariant maps ρ : Fr( T W ) → Θ } // Diff( W ) . Examples of this construction are the classifying spaces B Diff( W ) and, when W is orientable, B Diff + ( W ).Analogously, the decorated moduli space of ( W, ρ W ), denoted M Θ ,km ( W, ρ W ), is defined as thepath component of ρ W in the Borel construction { GL d -equivariant maps ρ : Fr( T W ) → Θ } // Diff km ( W ) . If W is a manifold with non-empty boundary, the moduli spaces M Θ ( W, ρ W ) and M Θ ,km ( W, ρ W )are defined analogously but only considering the GL d -equivariant maps ρ : Fr( T W ) → Θ whichagree with ρ W on Fr( T W | ∂W ).In this paper, we construct a decoupling map analogous to (1.1). For W a d -dimensional orientedmanifold with non-empty boundary, this map is given by D : M Θ ,km ( W, ρ W ) M Θ ( W, ρ W ) × Θ m // Σ m × (Θ // GL + d ) k // Σ kF × E m × E k The image of D is a path-component of the codomain, which we denote M Θ ( W, ρ W ) × Θ m // Σ m × (Θ // GL + d ) k // Σ k and we show that, when a stabilisation condition is satisfied, the decoupling induces a homologyisomorphism onto its image, in a range depending on the genus of W .1.2. Homology Stability.
A key ingredient in B¨odigheimer and Tillmann’s proof for orientedsurfaces is Harer’s stability theorem [Har85a]. It states that the map given by extending a diffeo-morphism by the identity Diff( S g,b +1 ) → Diff( S g,b ) ECOUPLING DECORATIONS ON MODULI SPACES 3 induces a map on classifying spaces which is homology isomorphism in degrees ≤ g (the originalbound by Harer was of g , and the most recent bound is due to Randal-Williams [RW16]). Likewise,the more general decoupling result will depend on an analogous result for moduli spaces of manifoldswith tangential structures: let W be a compact connected d -dimensional manifold, and let ρ W bea fixed Θ-structure on W . The manifold W \ int( D d ) is naturally endowed with a Θ-structure ρ (cid:48) W given by the restriction of ρ W , and we have a map(1.2) M Θ ( W \ int( D d ) , ρ (cid:48) W ) → M Θ ( W, ρ W )induced by extending a Θ-structure by ρ W | Dd and a diffeomorphism by the identity. As for orientedsurfaces, this map has been shown in many cases to induce a homology isomorphism in a rangedepending on the genus of W , for instance, this holds for surfaces with spin structures and framings.In dimensions 2 n ≥
6, this was shown to hold whenever ρ W is n -connected [GRW17, Corollary 1.7].1.3. Main results.
Throughout this paper, let W be a compact, connected manifold of dimension d ≥ Theorem A.
Let W be an orientable manifold with non-empty boundary and ρ W be a fixed Θ -structure on W . If the map (1.2) induces a homology isomorphism in degrees i ≤ α then thedecoupling map D : M Θ ,km ( W, ρ W ) −→ M Θ ( W, ρ W ) × Θ m // Σ m × (Θ // GL + d ) k // Σ k induces homology isomorphisms in degrees i ≤ α . In particular, the theorem above gives us new results on moduli spaces of surfaces with tangentialstructures, generalising the decoupling result of B¨odigheimer and Tillmann to surfaces with spinstructures, maps to a background space, framings, amongst others.The main corollary of Theorem A is obtained in the context of manifolds of high even dimension,where the assumption on the map (1.2) has been shown to hold whenever ρ W is n -connected.Although this connectivity assumption is quite restrictive, it is still possible to obtain furtherresults for more general tangential structures using the techniques of [GRW17, Section 9]. Inparticular, we prove the following: Theorem B.
Let W g, = g S n × S n \ D n , for n ≥ . Then for all i ≤ g − H i ( B Diff + ,km ( W g, )) ∼ = H i ( B Diff + ( W g, ) × SO[0 , n − m // Σ m × B SO(2 n ) (cid:104) n (cid:105) k // Σ k ) where SO[0 , n − is the n -truncation of SO and B SO(2 n ) (cid:104) n (cid:105) is the n -connected cover of B SO(2 n ) . We also provide a computation of the cohomology of B Diff + ,km ( W g, ) using Theorem B.1.4. More general decorations.
When studying surfaces, it is natural to look at decorations bymarked points and discs, however for high dimensional manifolds, one is allowed to explore moregeneral types of decorations. This has been studied for instance in the recent work [Pal12, Kup13,Pal18a, Pal18b]. We generalise the definition of the decorated moduli space of a manifold withmore general decorations, we define a decoupling map and show that under a homology stabilityhypothesis the decoupling map induces a homology isomorphism in a range. We analyse more closelythe case that the decorations are unlinked circles, because of its relation to the literature and alsoits relevance for string theory. The moduli space of a manifold W with k embedded circles and Θ-structure is denoted M Θ kS ( W, ρ W ) and is defined to be the moduli space of manifolds diffeomorphicto W , equipped with a Θ-structure and k marked unparametrised circles. As in Theorem A, wealso get a splitting result, now in terms of the space of configurations of circles in R ∞ with labelson a space L (see Definition 5.18), which we denote C kS ( R ∞ , L ). Theorem C.
Let W be a simply-connected spin manifold of dimension n ≥ with non-emptyboundary, equipped k marked unparametrised circles and with a Θ -structure ρ W : Fr( T W ) → Θ which is n -connected and such that Θ // GL d is simply-connected. Then for all i ≤ g − H i ( M Θ kS ( W, ρ W )) ∼ = H i ( M Θ ( W, ρ W ) × C kS ( R ∞ ; ( L Θ //L null GL + d − ) )) where L − is the free loop space, L null is the subspace of nullhomotopic loops, and ( − ) indicates apath-component that is specified in the proof. LUCIANA BASUALDO BONATTO
Geometric Interpretation.
The spaces and maps used in the decoupling result all have avery concrete geometrical interpretation, which we briefly introduce.Let Θ ∗ = ∗ be the point with the trivial GL d -action, then any manifold W admits a uniqueΘ ∗ -structure ρ W . By definition, the moduli space M Θ ∗ ( W, ρ W ) is equivalent to B Diff( W ). So wecan obtain a geometric interpretation for this moduli space from the specific model of B Diff( W )given by the quotient Emb( W, R ∞ ) / Diff( W ). Then M Θ ∗ ( W ) can be seen as the subspace of allsubmanifolds of R ∞ that are abstractly diffeomorphic to W . Analogously, fixing an arbitrary Θ-structure, ρ W , on a manifold W (for instance a choice of orientation), the moduli space M Θ ( W, ρ W )has a model as the space of all submanifolds of R ∞ that are diffeomorphic to W together with achoice of Θ-structure concordant to ρ W . A detailed description of this model can be found insections 6 and 7 of [GRW18a].Through this perspective, the decorated moduli space M Θ ,km ( W, ρ W ) is the space of all subman-ifolds of R ∞ that are diffeomorphic to W together with k marked points, m marked parametrizeddiscs, and a choice of Θ-structure concordant to ρ W .Moreover, note that the spaces Θ m // Σ m and (Θ // GL d ) k // Σ k also have geometric models in termsof unordered configuration spaces with labels, which we denote respectively by C m ( R ∞ , Θ ) and C k ( R ∞ , (Θ // GL d ) ).The decoupling map in Theorem A, is the product of three maps: M Θ ,km ( W, ρ W ) → M Θ ( W, ρ W )is the map that simply forgets the decorations; the map M Θ ,km ( W, ρ W ) → C m ( R ∞ , Θ )records the centre of the marked discs together with local tangential structure information; andfinally M Θ ,km ( W, ρ W ) → C k ( R ∞ , (Θ // GL d ) )is the map that records the positions of the marked points in R ∞ together with their tangent spacesand tangential structure information. See Figure 1 for an illustration of these maps.With this geometric interpretation, the decoupling result tells us that the homology of the spaceof decorated submanifolds of R ∞ of a fixed diffeomorphism type, in which the points and discs areconstrained to our manifolds, can be understood in terms of the homology of a space where thesepoints and discs are not constrained anymore, ie. they are decoupled.1.6. Outline of the paper.
Section 2 recalls the basic concepts and results needed throughout thepaper. We start by defining and giving examples of tangential structures and the topological modulispace of manifolds. Further, we prove auxiliary results on fibre sequences of Borel constructionsand a spectral sequence argument which will be needed throughout the paper.We define the decorated moduli space of manifolds and the decoupling map, and we proveTheorem A in Section 3 . Subsequently, in Section 4, we prove the corollaries of Theorem A whichprovide decoupling results for surfaces with many tangential structures, as well as for manifolds ofdimension 2 n ≥ W g, , we show Theorem B. Acknowledgements.
I would like to thank my supervisor Ulrike Tillmann for her support andfor the many useful conversations. In addition, I would like to thank Jan Steinebrunner and TomZeman for all the helpful discussions and comments. This work was carried out with the supportof CNPq (201780/2017-8) and produced while the author was in residence at the MathematicalSciences Research Institute in Berkeley, California, during the Spring semester of 2020.
ECOUPLING DECORATIONS ON MODULI SPACES 5 Preliminaries
In this section we recall the definition of tangential structures and give the examples that will beused in Section 4. We also recall the moduli space of manifolds with tangential structures. We alsorecall basic results about descending fibre sequences to homotopy quotients as well as a spectralsequence argument that will be used throughout the paper.2.1.
Tangential structures.
Throughout, we consider W to be a smooth compact connected d -dimensional manifold, possibly with non-empty boundary. If W is a closed manifold, we denoteby Diff( W ) the group of diffeomorphisms of W with Whitney C ∞ topology. If W has non-emptyboundary, we assume it to be equipped with a collar neighbourhood of ∂W and we denote by Diff( W )the group of diffeomorphisms of W which restrict to the identity on this collar. If moreover W is anorientable manifold, we denote by Diff + ( W ) the subgroup of orientation preserving diffeomorphisms.Note that if W has a non-empty boundary, any element of Diff( W ) is automatically orientationpreserving.Given a vector bundle p : E → B , the frame bundle of E over B will be denote by Fr( E ).Recall that the fiber of Fr( E ) → B over a fixed b is the space of ordered bases of p − ( b ), and thisforms a GL d -principal bundle, with the action A · ( v , . . . , v n ) = ( A ( v ) , . . . , A ( v n )), for A ∈ GL d and ( v , . . . , v n ) an ordered basis of p − ( b ). Throughout this paper, we denote by T W the tangentbundle of the manifold W , and by ε n → B the trivial n -dimensional vector bundle over some basespace B . In this paper, we will consider only real vector bundles.We now define tangential structures following the terminology established by Galatius andRandal-Williams in [GRW18b]. Definition 2.1. A tangential structure for d -dimensional manifolds is a space Θ with a continuousaction of GL d := GL d ( R ). A Θ-structure on a d -manifold W is a GL d -equivariant map ρ : Fr( T W ) → Θ. Manifolds are usually equipped with data that can be described using tangential structures:
Examples 2.2.
Let W be a connected manifold.(a) An orientation consists of a coherent choice of which oriented bases of the tangent spaces areconsidered positive. Namely, this is the data of a GL d -equivariant map Fr( T W ) → {± } ,where the action on Θ or := {± } is given by multiplication by the sign of the determinant.Therefore, a Θ or -structure on a manifold is equivalent to a choice of orientation.(b) If we want to consider all manifolds with no extra data, we can use the trivial tangentialstructure Θ ∗ = {∗} with the trivial action. Any manifold admits a unique Θ ∗ -structure,and therefore it encodes no extra data.(c) Framings on a manifold also are a tangential structure described by Θ fr = GL d , since thedata of a framing is precisely a continuous choice of basis for the tangent space at eachpoint, which can be expressed as a GL d -equivariant map Fr( T W ) → GL d .(d) Given a space X , we define the tangential structure of maps to X , by taking Θ X = X withthe trivial action of GL d . Then a Θ X -structure on a manifold W is the data of a continuousa map W → X . Remark . Many authors approach tangential structures in a different way, namely by definingit as a fibration θ : B → B Ø( d ), and by setting a θ -structure on a manifold W to be a map W → B lifting the map W → B Ø( d ) which classified T W . There is a clear way of exchangingthe two approaches using the correspondence between spaces with a GL d action and spaces over B GL d (cid:39) B Ø( d ), made through the principal GL d -bundle E GL d → B GL d . Both the spaces Θ and B associated to a given tangential structure will come into the decoupling result, so it is worthmaking precise the relation between them: given a fibration θ , the pullback spaceΘ := E GL d × B GL d B is naturally equipped with a GL d action. On the other hand, given a GL d -space Θ, we can define B as the Borel construction Θ // GL d (ie. the quotient of E GL d × Θ by the diagonal action of GL d ).Then E GL d × Θ → B is a principal GL d -bundle, which means B comes equipped with a map θ : B → B GL d . Since E GL d is contractible, these processes are inverse up to equivariant fibre-wiseweak equivalence. LUCIANA BASUALDO BONATTO
Example 2.4.
Spin structures on an n -dimensional manifold are known to be classified by liftsalong the fibration θ Spin : B Spin → B Ø( d ) (cid:39) B GL d . So the corresponding Θ Spin is the pullbackspace E GL d × B GL d B Spinand it fits into the following diagram of fibre sequences {± } × B Z / Spin E GL d {± } × B Z / B Spin B GL d (cid:121) which implies that the space Θ Spin is homotopy equivalent to {± } × B Z / Definition 2.5.
Let W be a closed manifold and Θ a fixed tangential structure. We define the spaceof Θ -structures on W , denoted Bun Θ ( W ), to be the space of all GL d -equivariant maps Fr( T W ) → Θequipped with the compact-open topology.Let W be a manifold with non-empty boundary and a collar together with a GL d -equivariantmap ρ ∂ : Fr( T ∂W ⊕ ε ) → Θ. We define the space of Θ -structures on W restricting to ρ ∂ , denotedBun Θ ρ ∂ ( W ), to be the space of all GL d -equivariant maps Fr( T W ) → Θ that restrict to ρ ∂ on ∂W .Given a manifold W with non-empty boundary together with a GL d -equivariant map ρ ∂ :Fr( T ∂W ⊕ ε ) → Θ, it is possible that the space Bun Θ ρ ∂ ( W ) is empty, in the case where the chosenmap ρ ∂ cannot be extended to Fr( T W ). An example, if W is an orientable manifold with discon-nected boundary and ρ ∂ assigns non-compatible orientations for the different components of ∂W .These are not the cases we are interested in, and therefore throughout the paper, whenever W is amanifold non-empty boundary we assume it comes equipped with a map ρ ∂ which is the restrictionof a Θ-structure in W . Examples 2.6.
Let W be a manifold.(a) If Θ ∗ is the trivial tangential structure of Example 2.2(b), then there is only one Θ ∗ -structurefor any manifold, so Bun Θ ∗ ( W ) is a single point.(b) Consider the tangential structure Θ or for orientation described in Example 2.2(a). If W isa closed orientable manifold, it admits two Θ or -structures which implies that Bun Θ or ( W )consists of two points. On the other hand, if W has a non-empty boundary and ρ ∂ is afixed Θ structure on ∂W , then Bun Θ or ρ ∂ ( W ) consists only of those GL d -equivariant mapsFr( T W ) → Θ which restrict to ρ ∂ , and therefore consists of a single point.(c) Given a space X , consider the tangential structure Θ X defined in Example 2.2(d). Asdiscussed before, a Θ X structure on a manifold W is just a continuous map W → X , andtherefore, if W is closed, Bun Θ X ( W ) is the space of continuous maps from W to X .2.2. Moduli spaces of manifolds.
The action of the diffeomorphism group of W on the tangentbundle T W induces an action on the space Bun Θ ( W ) for any tangential structure Θ. Explicitly,given φ ∈ Diff( W ) and ρ ∈ Bun Θ ( W ), φ · ρ = ρ ◦ Dφ − where Dφ : Fr( T W ) → Fr(
T W ) is the map induced by the differential of φ . Definition 2.7.
Let W be a closed manifold and fix ρ W a Θ-structure on W , we define Bun Θ ( W, ρ W )to be the orbit of the path-component of ρ W in Bun Θ ( W ) under the action of the diffeomorphismgroup Diff( W ). If W has non-empty boundary Bun Θ ( W, ρ W ) is defined to be the orbit of thepath-component of ρ W in Bun Θ ρ ∂ ( W ), where ρ ∂ is the restriction of ρ W to the boundary.We define the moduli space of W with Θ -structures concordant to ρ W to be the Borel construction(ie. homotopy orbit space) M Θ ( W, ρ W ) := Bun Θ ( W, ρ W ) // Diff( W ) . Remark . In the above definition, when W is a manifold with boundary and ρ W a fixed Θ-structure, we have omitted the symbol ρ ∂ from the notation for the space Bun Θ ( W, ρ W ). However,it should always be understood that there is a fixed boundary condition which is determined by therestriction of the fixed ρ W to the boundary. ECOUPLING DECORATIONS ON MODULI SPACES 7
The most important examples of these moduli spaces come from the simplest tangential struc-tures: for the trivial tangential structure Θ ∗ , the space Bun Θ ∗ ( W ) consists of a single point forany W and therefore M Θ ∗ ( W, ρ W ) will be simply the classifying space B Diff( W ). On the otherhand, if W is an orientable manifold Bun Θ or ( W, ρ W ) consists of either one or two points dependingon whether Diff( W ) has an element that reverses the orientation of W . In either case, the modulispace M Θ or ( W, ρ W ) is homotopy equivalent to B Diff + ( W ).2.3. A lemma on fibre sequences and homotopy quotients.
In this section we prove a lemmathat will be used throughout the paper to construct fibre sequences of moduli spaces from equivariantfibre sequences of diffeomorphism groups and spaces of Θ-structures.
Lemma 2.9.
Given a commutative diagram
XY Z fh g such that f and h are Serre fibrations and h is surjective, then g is also a Serre fibration.Proof. We will show that g has the homotopy lifting property with respect to any inclusion D i ×{ } (cid:44) → D i × I using that both f and h have this property. XD i × { } YD i × I Z h fg
Given a lift of D i × { } → Y to X , we can construct a lift (cid:96) : D i × I → X using that f is aSerre fibration. Then h ◦ (cid:96) : D i × I → Y is a lift with respect to g . It only remains to see thatany map D i × { } → Y admits a lift to X , which we can prove by induction on i : for i = 0, this isprecisely the condition that h is surjective, for i >
0, this lift can be obtained using the identification D i (cid:39) D i − × I and the fact that h is a Serre fibration. (cid:3) Lemma 2.10.
Let G i be a topological group and p i : M i → M i /G i be a G i -principal bundle, for i = 1 , , .(a) If φ : G → G is a continuous homomorphism and f : M → M is a φ -equivariantfibration, then the induced map ψ : M /G → M /G is a fibration.(b) Given a short exact sequence → G → G → G → and a fibre sequence of equivariant maps M → M → M , the induced maps on quotientsform a fibre sequence M /G → M /G → M /G Proof. (a) By assumption, the map p is a surjective fibration and the composition ψ ◦ p is equalsthe composition of fibrations p ◦ f . Therefore, by Lemma 2.9, ψ is a fibration.(b) Diagrammatically, we want to show that given the diagram of fibre sequences below, thereexists a fibre sequence fitting into the bottom row: G G G M M M M /G M /G M /G ι φip fp p ψ LUCIANA BASUALDO BONATTO
By item (a), the map ψ is a fibration, so all that remains is to identify its fibres. The composition p ◦ f is a fibration with fibre G · i ( M ) ⊂ M . Then the fibre of ψ is p ( G · i ( M )) = p ( i ( M )).Since the action of G on M is free, we know that for any g ∈ G which is not in the kernel of φ ,the intersection ( g · i ( M )) ∩ i ( M ) is empty. So p ( i ( M )) is simply the quotient of i ( M ) by theaction of ker φ = ι ( G ). Then the map M /G → M /G is precisely the inclusion of the fibre of ψ . (cid:3) Corollary 2.11.
Let G i be a topological group and S i be a G i -space, for i = 1 , , .(a) If φ : G → G is a continuous homomorphism and f : S → S is a φ -equivariant fibration,then we can choose a model for the Borel constructions such that the induced map ψ : S //G → S //G is a fibration.(b) Given a short exact sequence → G → G φ −→ G → such that φ is a principal bundle, and a fibre sequence of equivariant maps S → S → S ,the induced maps on quotients form a homotopy fibre sequence S //G → S //G → S //G Proof.
Both statements follow from Lemma 2.10 and the following observations: fix EG , then theinclusion ι : G → G induces a G action on EG and since φ is a principal G -bundle, then sois EG → EG /G . Therefore the space EG is a model for EG as well. Also, any model forthe space EG carries an action of G via the map φ : G → G , and in particular, the quotientof EG × EG by G is still a G -principal bundle. Then the proof follows directly from applyingLemma 2.10 to the diagram G G G S × EG S × EG × EG S × EG S //G S //G S //G ι φ where the middle row is the product of the fibre sequence S → S → S with the trivial fibresequence EG EG × EG EG and the action of the groups is the diagonal action. The commutativity of the diagram follows fromthe fact that the action of G on EG induced by φ ◦ ι is trivial. (cid:3) In particular, applying the above corollary to the trivial fibration ∗ → ∗ , gives us the well-knownresult that a short exact sequence of groups G → G → G induces a fibre sequence on classifyingspaces BG BG BG The spectral sequence argument.
The decoupling result will be deduced from the compar-ison of the homology spectral sequence associated to fibre sequences of moduli spaces. The resultthen follows from a well-known spectral sequence result which we recall (for a proof see [Til16]):
Lemma 2.12 ( Spectral Sequence Argument ) . Let f : E • p,q → ˜ E • p,q be a map of homological firstquadrant spectral sequences. Assume that f : E p,q ∼ = −→ ˜ E p,q for ≤ p < ∞ and ≤ q ≤ l. Then f induces an isomorphism on the abutments in degrees ∗ ≤ l. ECOUPLING DECORATIONS ON MODULI SPACES 9 The Decoupling Theorem
In this section we introduce decorated manifolds, the decorated moduli space, and the maps thatare in the centre of the decoupling theorem: the forgetful map and evaluation map. We end bydefining the decoupling map and proving the decoupling theorem.For now, we focus on decorations being points and discs. These are the extreme cases: thesimplest embedded manifolds of lowest and highest possible dimension. In section 5 we show howthis can be extended to more general submanifold decorations, focusing on the case of manifoldsdecorated with embedded unlinked circles.3.1.
The decorated moduli space and the forgetful map.
Throughout this section, let W bea compact connected smooth manifold. We will study manifolds equipped with decorations: Definition 3.1. A d -dimensional manifold with decorations consists of a manifold W togetherwith a set of distinct marked points in its interior p , . . . , p k ∈ W \ ∂W and disjoint embeddings φ , . . . , φ m : D d (cid:44) → W \ ( ∂W ∪ { p , . . . , p k } ), with k, m ∈ N . If W is orientable, we requireall embeddings to be oriented in the same way. We refer to these choices as decorations on ourmanifold.Given a manifold W with decorations, we define the decorated diffeomorphism group Diff km ( W )to be the subgroup of Diff( W ) of the diffeomorphisms ψ such that ψ ◦ φ j = φ α ( j ) ψ ( p i ) = p β ( i ) for some α ∈ Σ m and β ∈ Σ k .In other words, we are looking at the diffeomorphisms that preserve the marked points andparametrized discs up to permutations. Note that the notation Diff km ( W ) does not record whichpoints and embedded discs comprise the decorations. The following lemma justifies this notation. Lemma 3.2. If d ≥ , the isomorphism type of Diff km ( W ) does not depend on the choice of the k points and m embedded discs that comprise the decorations.Proof. For any two collections of decorations in W denoted( p , . . . , p k , φ , . . . , φ m ) and ( p (cid:48) , . . . , p (cid:48) k , φ (cid:48) , . . . , φ (cid:48) m ) , there exists a diffeomorphism ψ of W such that ψ ( p i ) = p (cid:48) i ψ ◦ φ i = φ (cid:48) i which can be constructed recursively by extending isotopies of the points and discs to diffeotopiesof W as described in [Hir94, Chapter 8, Theorems 3.1, 3.2]. Then conjugation with ψ defines anisomorphism between the group of diffeomorphisms preserving ( p , . . . , p k , φ , . . . , φ m ) and the onepreserving ( p (cid:48) , . . . , p (cid:48) k , φ (cid:48) , . . . , φ (cid:48) m ). (cid:3) We are now ready to define the analogue of the moduli space, including the decorations:
Definition 3.3.
Given a manifold W with a Θ-structure ρ W , we define the decorated moduli spaceof W with k points and m discs to be M Θ ,km ( W, ρ W ) := Bun Θ ( W, ρ W ) // Diff km ( W ) . Recall that, if W has non-empty boundary, then Diff( W ) consists only of those diffeomorphismsfixing a collar of the boundary and the elements of Bun Θ ( W, ρ W ) agree with ρ W on ∂W .We define the forgetful map(3.1) F : M Θ ,km ( W, ρ W ) → M Θ ( W, ρ W )to be the one induced by the identity map on Bun Θ ( W ) and the subgroup inclusion Diff km ( W ) → Diff( W ). The evaluation map.
Let W be a decorated manifold with k marked points and m markeddiscs. For each marked point p i , we choose once and for all a frame of T p i W , and if W is oriented, weask that these frames have the same orientation. We also fix throughout this section N ⊂ W whichis the union of a tubular neighbourhood of the marked points and the interiors of the parametrizeddiscs. We denote by W m + k the manifold W \ N . The decoupling result follows from understandingthe difference between the decorated moduli space of W and the moduli space of W m + k .For instance, assume k = 0 and m = 1, then there is a group isomorphismDiff( W ) → Diff ( W )given by extending a diffeomorphism on W by the identity on the marked disc. More generally, if W is a manifold with m embedded discs, the map e m : Diff m ( W ) → Σ m taking a diffeomorphism φ to the α ∈ Σ m recording the permutation induced on the discs by φ , isa surjective homomorphism with kernel Diff( W m ), where, as above, W m is the manifold obtainedfrom W by removing the interior of the m embedded discs.Assume now W has k marked points { p , . . . , p k } and no marked discs. We still get a homomor-phism Diff( W k ) → Diff k ( W )by extending a diffeomorphism on W k by the identity on the removed neighbourhood of the points,but this is not an isomorphism, since the elements of Diff k ( W ) are not required to fix the entireneighbourhood of the marked points. A way to understand the diffeomorphisms around these is bylooking at the differential map on the chosen frames at the marked points. So we define a map tothe wreath product e k : Diff k ( W ) −→ Σ k (cid:111) GL d φ (cid:55)−→ ( D p φ, . . . , D p k φ, β )where β ∈ Σ k is the permutation induced on the marked points by φ . The image of e k depends themanifold W . Definition 3.4.
An orientable decorated manifold W with k marked points and m marked discsis called decorated-chiral if every φ ∈ Diff km ( W ) preserves the orientation. Remark . If the manifold W is decorated by m > ∂W (cid:54) = ∅ then it is immediatelydecorated-chiral. There are also many manifolds for which any diffeomorphisms (not necessarilydecorated) are orientation preserving, these are called chiral manifolds. A classical example is C P ,which can be deduced by analysing the automorphisms of its cohomology ring. Trivially, any chiralmanifold is always decorated-chiral.It follows from [Til16, Lemmas 2.3 and 2.4] that: Lemma 3.6 ([Til16]) . Let W be a compact connected decorated manifold, then:(a) the map e : Diff km ( W ) Σ m × (Σ k (cid:111) GL † d ) e m × e k is a surjective principal bundle, where the group GL † d is GL + d if W is decorated-chiral, and GL d otherwise.(b) Diff( W m + k ) is the homotopy fibre of e .Remark . Identifying the image of e is important because we want to use a Serre spectral sequenceto compare the homology of the total spaces of two fibre sequences. Therefore it is important toidentify precisely the images of the fibrations we define.A generalisation of the above lemma provides a fibre sequence on moduli spaces with tangentialstructures which is the key to the proof of the decoupling. Proposition 3.8.
Let W be a compact connected decorated manifold and ρ W a fixed Θ -structureon W , then: ECOUPLING DECORATIONS ON MODULI SPACES 11 (a) The homomorphism e induces an evaluation map E : M Θ ,km ( W, ρ W ) Θ m // Σ m × (Θ // GL † d ) k // Σ k which is a Serre fibration onto the path component which it hits, where the group GL † d is GL + d if W is decorated-chiral, and GL d otherwise.(b) Let W m + k be equipped with the Θ -structure ρ W m + k given by the restriction of ρ W . Then M Θ ( W m + k , ρ W m + k ) is the homotopy fibre of E over its image. To prove the Proposition, we will need the following lemma:
Lemma 3.9.
Let W be a connected manifold and S a smooth submanifold, then the restriction map r S : Bun Θ ( W ) → Map GL d (Fr( T W | S ) , Θ) is a Serre fibration.Proof. For any i ≥
0, a lift for the diagram D i × { } Bun Θ ( W ) D i × I Map GL d (Fr( T W | S ) , Θ) r S is equivalent to a GL d -equivariant extension of the following(3.2) ( D i × { } × Fr(
T W )) ∪ ( D i × I × Fr(
T W | S )) Θ D i × I × Fr(
T W ) ρ Since the inclusion
S (cid:44) → W is an embedding, there exists a strong deformation retract r : D i × I × W −→ ( D i × { } × W ) ∪ ( D i × I × S ) . If i denotes the inclusion of ( D i × { } × W ) ∪ ( D i × I × S ) into D i × I × W , we have an isomorphism f : D i × I × Fr(
T W ) ∼ = −→ r ∗ i ∗ ( D i × I × Fr(
T W ))which is the identity on ( D i × { } × Fr(
T W )) ∪ ( D i × I × Fr(
T W | S )).Therefore the composite D i × I × Fr(
T W ) r ∗ i ∗ ( D i × I × Fr(
T W )) i ∗ ( D i × I × Fr(
T W )) Θ f r ∗ ρ gives a lift to diagram 3.2. This implies that the map Bun Θ ( W ) → Map GL d (Fr( T W | S ) , Θ) is aSerre fibration. (cid:3)
Proof of Proposition 3.8. (a) Let P ⊂ W be the union of the k marked points and the centres ofthe m marked discs. By Lemma 3.9, the restriction map r P : Bun Θ ( W, ρ W ) → Map GL d (Fr( T W | P ) , Θ)is a Serre fibration. For each marked point, we chose a frame of its tangent space. Each point in thecentre of a marked disc, comes with a preferred frame induced by the parametrization of the disc. Soevery point in P is equipped with a frame of its tangent space, and this gives us a diffeomorphismFr( T W | P ) ∼ = GL d × P . Therefore the space of GL d -equivariant maps Fr( T W | P ) → Θ can beidentified with the space of continuous maps P → Θ, which is just Θ m × Θ k . The result thenfollows by applying Corollary 2.11 to combine the fibration r P with the homomorphism of Lemma3.6(3.3) Diff km ( W ) e −→ Σ m × (Σ k (cid:111) GL † d ) . We apply Corollary 2.11(a) by taking G = Diff km ( W ) and S = Bun Θ ( W, ρ W ), with the usualaction by precomposition with the differential. On the other hand, we take G = Σ m × (Σ k (cid:111) GL † d ),and S = Θ m × Θ k with the following action: the space Θ m × Θ k can be spit into the m factors corresponding to the marked discs and k factors corresponding to the marked points. Then we havean action of Σ m × (Σ k (cid:111) GL † d ) on Θ m × Θ k induced by the actionsΣ m Θ m Σ k (cid:111) GL † d Θ k . Then the fibration Bun Θ ( W, ρ W ) → Θ m × Θ k is e -equivariant and therefore, by Corollary 2.11(a),we have a fibration E : M Θ ,km ( W, ρ W ) Θ m // Σ m × Θ k // (Σ k (cid:111) GL † d ) . Since E Σ k × ( E GL d ) k is a model for E (Σ k (cid:111) GL † d ), then(Θ // GL † d ) k // Σ k is a model for Θ k // (Σ k (cid:111) GL † d ), and the result follows.(b) Recall that W m + k is defined as the submanifold W \ N , where N is the union of a tubularneighbourhood of the marked points and the interiors of the marked discs. The restriction ρ W m + k of ρ W is a Θ-structure on W m + k . In the remainder of the proof, we will show that M Θ ( W m + k , ρ W m + k )is the homotopy fibre of E .A description of the fibre of E can be obtained using Corollary 2.11(b) with the short exactsequence of groups being(3.4) ker e → Diff km ( W ) e −→ Σ m × (Σ k (cid:111) GL † d )and the fibre sequence of S → S → S being the one associated to the fibration r P of item(a). The fibre of r P over r P ( ρ W ) is the subspace of all elements of Bun Θ ( W, ρ W ) which restrictto r P ( ρ W ) over P , which we here denote Bun Θ P ( W, ρ W ). This space carries an action of ker e byprecomposition with the differential, and it is simple to check that this fibre sequence is equivariantwith respect to (3.4). Then by Corollary 2.11(b), the fibre of the evaluation map E is given byBun Θ P ( W, ρ W ) // ker e. Applying Lemma 3.9 to both submanifolds P and N , we obtain two fibrations fitting into thefollowing commutative diagramBun Θ ( W, ρ W ) Map GL d (Fr( T W | N ) , Θ)Bun Θ ( W, ρ W ) Map GL d (Fr( T W | P ) , Θ) ∼ = Θ m × Θ kr N = i ∗ r P where the right-hand vertical map is induced by the inclusion i : P (cid:44) → N . Since Fr( T D d ) isisomorphic to GL d × D d as GL d -bundles, and the spaces GL d × D d and GL d × {∗} are homotopyequivalent as GL d -spaces, the map i ∗ is a homotopy equivalence. In particular, this implies thatthe map from the fibre of r N to Bun Θ P ( W, ρ W ) is a homotopy equivalence.The fibre of r N over r N ( ρ W ) is by definition the space of all Θ structures on W which agree with ρ W on N . We claim that this space is homeomorphic to Bun Θ ( W m + k , ρ W m + k ), since the restrictionmap r Wm + k takes the fibre of r N bijectively to Bun Θ ( W m + k , ρ W m + k ) and it has an inverse given byextending an element by r N ( ρ W ).So we have a commutative diagram of principal fibre bundlesDiff( W m + k ) ker e Bun Θ ( W m + k , ρ W m + k ) × E Diff( W ) Bun Θ P ( W, ρ W ) × E Diff( W ) M Θ ( W m + k , ρ W m + k ) Bun Θ P ( W, ρ W ) // ker e (cid:39)(cid:39) where the top horizontal map is a homotopy equivalence by Lemma 3.6 and the middle map is ahomotopy equivalence by the discussion above. Therefore the map M Θ ( W m + k , ρ W m + k ) → Bun Θ P ( W, ρ W ) // ker e ECOUPLING DECORATIONS ON MODULI SPACES 13 is also a homotopy equivalence, as required. (cid:3)
Proof of the Decoupling.
In this section we prove the decoupling result by comparing thehomotopy fibration sequence from Proposition 3.8 to the product fibre sequence via the aforemen-tioned spectral sequence argument.
Definition 3.10.
The decoupling map D : M Θ ,km ( W, ρ W ) M Θ ( W, ρ W ) × Θ m // Σ m × (Θ // GL † d ) k // Σ kF × E is the product of the forgetful map (3.1) and the evaluation map E defined in Proposition 3.8.We now restate the decoupling theorem: Theorem 3.11.
Let W be a smooth connected compact manifold equipped with a Θ -structure ρ W .If the map τ : H i ( M Θ ( W m + k , ρ W m + k )) → H i ( M Θ ( W, ρ W )) induces a homology isomorphism in degrees i ≤ α , then for all such i the decoupling map D inducesan isomorphism H i ( M Θ ,km ( W, ρ W )) ∼ = H i ( M Θ ( W, ρ W ) × Θ m // Σ m × (Θ // GL † d ) k // Σ k ) where ( − ) denotes a path component of the image of ρ W , and the group GL † d is GL + d if W isorientable and decorated-chiral, and GL d otherwise.Proof. By Proposition 3.8, E is a fibration. Since M Θ ,km ( W, ρ W ) is connected by definition, we knowthat the image of E is precisely the path componentΘ m // Σ m × (Θ // GL † d ) k // Σ k . So we have a homotopy fibre sequence M Θ ( W m + k , ρ W m + k ) M Θ ,km ( W, ρ W ) Θ m // Σ m × (Θ // GL † d ) k // Σ k . E The proof of the theorem follows from the comparison between this homotopy fibre sequence andthe trivial fibre sequence associated to the projection map M Θ ( W, ρ W ) × Θ m // Σ m × (Θ // GL † d ) k // Σ k Θ m // Σ m × (Θ // GL † d ) k // Σ k By definition of the maps in Proposition 3.8, the following is a commutative diagram of homotopyfibre sequences M Θ ( W m + k , ρ W m + k ) M Θ ,km ( W, ρ W ) Θ m // Σ m × (Θ // GL † d ) k // Σ k M Θ ( W, ρ W ) M Θ ( W, ρ W ) × Θ m // Σ m × (Θ // GL † d ) k // Σ k Θ m // Σ m × (Θ // GL † d ) k // Σ k τ ED where the middle vertical map is the decoupling map. This induces a map of the respective Serrespectral sequences f : E • p,q → ˜ E • p,q , and since τ is a homology isomorphism in degrees i ≤ α , themap between the E pages E p,q = H p (cid:16) Θ m // Σ m × (Θ // GL † d ) k // Σ k ; H q ( M Θ ( W m + k , ρ W m + k )) (cid:17) ˜ E p,q = H p (cid:16) Θ m // Σ m × (Θ // GL † d ) k // Σ k ; H q ( M Θ ( W, ρ W )) (cid:17) is an isomorphism for all q ≤ α . Then by the Spectral Sequence Argument (Lemma 2.12), D inducesan isomorphism H i ( M Θ ,km ( W, ρ W )) H i ( M Θ ( W, ρ W ) × Θ m // Σ m × (Θ // GL † d ) k // Σ k ) ∼ =4 LUCIANA BASUALDO BONATTO for all i ≤ α . (cid:3) We now discuss how the decoupling result can be re-stated with a geometric interpretation. Asdiscussed in Section 1.5, the space Emb( W, R ∞ ) is a model for E Diff( W ), and therefore it is also amodel for E Diff km ( W ). With this model, the elements of M Θ ,km ( W, ρ W ) are decorated submanifoldsof R ∞ diffeomorphic to W with k marked points and m disjoint embedded discs, with a choice ofΘ-structure concordant (ie. equivariantly homotopic) to ρ W . With this model, the forgetful map(3.5) F : M Θ ,km ( W, ρ W ) → M Θ ( W, ρ W )simply forgets the marked points and discs.To interpret the evaluation map E with this model, we need also a geometric model for E Σ s .Recall that the configuration space of s points in a manifold M is defined as C s ( M ) := Emb( { , . . . , s } , M ) / Σ s where the action of Σ s is given by permutation of the points in { , . . . , s } . In other words, C s ( M )is the space of unordered collections of s distinct points in M . More generally, given a space X , the configuration space of s points in M with labels in X is defined as C s ( M ; X ) := (Emb( { , . . . , s } , M ) × X s ) / Σ s where Σ s acts by permuting the factors of X s , and acts on the product diagonally. In other words, C s ( M ) is the space of unordered collections of s distinct points in M , where each point is labelledby a point in X .Since the space Emb( { , . . . , m } , R ∞ ) is weakly contractible, it is a model for the total space E Σ m . Therefore a model for Θ m // Σ m is precisely the space of unordered configurations of m pointsin R ∞ with labels in Θ. Analogously, a model for the (Θ // GL † d ) k // Σ k is given by the space ofunordered configurations of k points in R ∞ with labels in (Θ // GL † d ).Then the evaluation map E takes a decorated submanifold S in R ∞ together with a tangentialstructure ρ to the configurations given by the centres of the m marked points, and the k markeddiscs. The labels of such configurations are determined by the tangential structure ρ : let p bethe centre point of a marked disc and let V p be the canonical frame of the tangent space of W at p induced by the parametrization of the disc. Then the label of the point corresponding to p inthe configuration space C m ( R ∞ ; Θ) is given by ρ ( V p ) ∈ Θ. Analogously, if p is a marked pointand V p is our chosen frame of its tangent space, then the label of the point corresponding to p in C k ( R ∞ ; Θ // GL † d ) is simply the class of ρ ( V p ) in the Borel construction Θ // GL † d .With these models, the decoupling map can be interpreted geometrically (see Figure 1), and thedecoupling theorem can be re-stated as: Corollary 3.12.
Let W be a smooth connected compact manifold equipped with a Θ -structure ρ W .If the map τ : H i ( M Θ ( W m + k , ρ W m + k )) → H i ( M Θ ( W, ρ W )) induces a homology isomorphism in degrees i ≤ α , then for all such i the decoupling map D inducesan isomorphism H i ( M Θ ,km ( W, ρ W )) ∼ = H i ( M Θ ( W, ρ W ) × C m ( R ∞ ; Θ ) × C k ( R ∞ ; (Θ // GL † d ) )) where ( − ) denotes a path component of the image of ρ W , and the group GL † d is GL + d if W isorientable and decorated-chiral, and GL d otherwise. Applications of the Decoupling Theorem
The main hypothesis of the decoupling theorem is that the map τ : H i ( M Θ ( W m + k , r W m + k ρ W )) → H i ( M Θ ( W, ρ W ))induces an isomorphism in a certain range i ≤ α . We know this condition holds in many circum-stances for a variety of manifolds dimensions and tangential structures. In this section we recallsome of these cases and discuss the applications of the decoupling theorem, focusing on the resultsfor dimension 2 and for even dimensions greater than 4. ECOUPLING DECORATIONS ON MODULI SPACES 15
We remark that, for odd higher dimensions, many stability results on the homology of the modulispace have also been proven, but it is not yet known whether the map τ needed for the decouplinginduces isomorphisms in a stable range.4.1. Applications for surfaces.
We now consider the 2 dimensional case, where the stabilityholds in many circumstances.4.1.1.
Orientation.
For orientations, the classical stability result of Harer on the homology of map-ping class groups of surfaces shows that the hypothesis of the decoupling theorem is satisfied forevery oriented surface of genus g and b boundary components, S g,b . The range in which the isomor-phism holds has been improved throughout the years [Har85b, Iva87, Iva89, Iva93, Bol12, RW16].The most recent bound, by Randal-Williams in [RW16], implies that the map H i ( B Diff + ( S g,b +1 )) → H i ( B Diff + ( S g,b )) is an isomorphism for all 3 i ≤ g . Then applying the decoupling theorem, werecover the result of B¨odigheimer and Tillmann, now with an improved isomorphism range: Corollary 4.1 ([BT01]) . For all i ≤ gH i ( B Diff + ,km ( S g,b )) ∼ = H i ( B Diff + ( S g,b ) × B Σ m × B (Σ k (cid:111) SO(2))) . Proof.
This is a direct application of the decoupling theorem. In this case, Θ or = {± } andtherefore Θ or = ∗ . Moreover, Θ or // GL +2 is the disjoint union of two copies of B GL +2 (cid:39) B SO(2), so(Θ or // GL +2 ) (cid:39) B SO(2), and therefore(Θ or // GL +2 ) k // Σ k (cid:39) B (Σ k (cid:111) SO(2))as required. (cid:3)
Non-orientable surfaces.
Let N g,b be the decorated non-orientable surface g R P ∞ \ (cid:96) b D .Wahl showed in [Wah08] that the map H i ( B Diff( N g,b +1 )) → H i ( B Diff( N g,b )) is an isomorphismfor all 4 i ≤ g −
3. Applying the decoupling theorem, we recover the result of Hanbury in [Han09]:
Corollary 4.2 ([Han09]) . For all i ≤ g − H i ( B Diff km ( N g,b )) ∼ = H i ( B Diff( N g,b ) × B Σ m × B (Σ k (cid:111) Ø(2))) . Proof.
The result follows from applying the decoupling theorem for Θ ∗ = ∗ . Then Θ ∗ // GL ishomotopy equivalent to B GL (cid:39) B Ø(2), and the result follows. (cid:3)
Framings.
In [RW14], Randal-Williams showed that for the oriented surface S g,b with aframing ρ , the map H i ( M fr ( S g,b +1 , ρ S g,b +1 )) → H i ( M fr ( S g,b , ρ )) is an isomorphism for all 6 i ≤ g − Corollary 4.3.
Let ρ be a framing on S g,b , then for all i ≤ g − H i ( M fr ,km ( S g,b , ρ )) ∼ = H i ( M fr ( S g,b , ρ ) × SO(2) m // Σ m × B Σ k ) . Proof.
The result is a direct application of the decoupling theorem, together with the fact that anypath-component of Θ fr = GL is homeomorphic to GL +2 (cid:39) SO(2) and that Θ fr // GL +2 is equivalentto two points. (cid:3) Spin structures.
In [Har90, Bau04, RW14] it was shown that for the oriented surface S g,b with spin structure ρ , the map H i ( M Spin ( S g,b +1 , ρ S g,b +1 )) → H i ( M Spin ( S g,b , ρ )) is an isomorphismfor all 5 i ≤ g − Corollary 4.4.
Let ρ be a spin structure on S g,b . For all i ≤ g − , the group H i ( M Spin ,km ( S g,b , ρ )) is isomorphic to H i (cid:0) M Spin ( S g,b , ρ ) × B (Σ m (cid:111) Z / × B (Σ k (cid:111) Spin(2)) (cid:1) . Proof.
From Example 2.4, we know that Θ
Spin is weakly equivalent to {± } × B Z /
2, and therefore(Θ
Spin ) is weakly equivalent to B Z /
2. On the other hand, Θ
Spin // GL +2 is the disjoint union of twocopies of B Spin(2) and therefore(Θ
Spin // GL +2 ) k // Σ k (cid:39) B Spin(2) k // Σ k (cid:39) B (Σ k (cid:111) Spin(2)) . (cid:3) Maps to a background space.
Consider the tangential structure given by maps to a simply-connected background space X . It was shown in [CM06, CM11a, RW16] that if X is a simply-connected space, then the map H i ( M X ( S g,b +1 , ρ S g,b +1 )) → H i ( M X ( S g,b , ρ )) is an isomorphism forall 3 i ≤ g , and all Θ X -structure ρ . Applying the decoupling theorem in this case, we obtain ageneralisation of [CM11a, Theorem 8]: Corollary 4.5.
Let ρ : S g,b → X be a continuous map and let X be the path component containingthe image of ρ . Then for all i ≤ g , H i ( M X ,km ( S g,b , ρ )) ∼ = H i (cid:0) M X ( S g,b , ρ ) × ( X ) m // Σ m × ( X ) k // (Σ k (cid:111) SO(2)) (cid:1) . Proof.
The result follows from applying the decoupling theorem together with the fact that Θ X = X with the trivial action and Θ X // GL +2 (cid:39) X × B GL +2 (cid:39) X × B SO(2). (cid:3) r structures. The tangential structure called Spin r is a generalisation of Spin which wasthoroughly studied in [RW14, Section 2]. In this paper, Randal-Williams showed that for theoriented surface S g,b with a Spin r structure ρ , the map H i ( M Spin r ( S g,b +1 , ρ S g,b +1 )) → H i ( M Spin r ( S g,b , ρ ))is an isomorphism for all 6 i ≤ g − Corollary 4.6.
Let ρ be a Spin r structure on S g,b . For all i ≤ g − , the group H i ( M Spin r ,km ( S g,b , ρ )) is isomorphic to H i (cid:16) M Spin r ( S g,b , ρ ) × B (Σ m (cid:111) Z /r ) × B (Σ k (cid:111) Spin r (2)) (cid:17) . Proof.
Using the fibre sequence {± } × B Z /r B Spin r (2) B GL we can apply the procedure described in Example 2.4 to deduce that a path component of Θ Spin r is weakly equivalent to the Lens space B Z /r and Θ Spin r // GL +2 is the disjoint union of two copies of B Spin r (2). (cid:3) ± structures. The tangential structures called Pin + and Pin − are generalisations of Spinfor non-orientable manifolds, and they were thoroughly studied in [RW14, Section 4]. In this paper,Randal-Williams showed that for the non-orientable surface N g,b = g R P ∞ \ (cid:96) b D with a Pin + -structure, the map H i ( M Pin + ( N g,b +1 , ρ N g,b +1 )) → H i ( M Pin + ( N g,b , ρ ))is an isomorphism for all 4 i ≤ g −
6, and Pin + -structure ρ .It was also shown in [RW14] that the map H i ( M Pin − ( N g,b +1 , ρ N g,b +1 )) → H i ( M Pin − ( N g,b , ρ ))is an isomorphism for all 5 i ≤ g −
8, and Pin − -structure ρ . Corollary 4.7.
Let ρ be a Pin + structure on N g,b , then for all i ≤ g − H i ( M Pin + ,km ( N g,b , ρ )) ∼ = H i ( M Pin + ( N g,b , ρ ) × B (Σ m (cid:111) Z / × B (Σ k (cid:111) Pin + (2))) . Corollary 4.8.
Let ρ be a Pin − structure on N g,b , then for all i ≤ g − H i ( M Pin − ,km ( N g,b , ρ )) ∼ = H i ( M Pin − ( N g,b , ρ ) × B (Σ m (cid:111) Z / × B (Σ k (cid:111) Pin − (2))) . Proof of Corollaries 4.7 and 4.8.
Using the fibre sequence B Z / B Pin ± ( d ) B GL d we can apply the procedure described in Example 2.4 to deduce that a path component of Θ Pin ± isweakly equivalent to B Z /
2, and Θ
Pin ± // GL is the space B Pin ± (2). (cid:3) ECOUPLING DECORATIONS ON MODULI SPACES 17
Applications for high dimensional manifolds.
One of the most interesting applicationsof the decoupling result appears when looking at higher dimensional manifolds. In [GRW17], it wasshown that the hypothesis of the decoupling theorem holds for many manifolds W of even dimensiongreater or equal to 6, and many tangential structures. The range of the homology isomorphism isgiven in terms of the stable genus of W , which we now recall, following the notation of [GRW18b].Analogously to the surface case, the genus will be measured by disjoint embeddings of the space( S n × S n ) \ {∗} , but now taking into account the tangential structure as well. Namely, Galatiusand Randal-Williams define what it means for a Θ-structure on ( S n × S n ) \ {∗} to be admissible (see [GRW18b, Section 3.2]) and define the genus of a manifold W with Θ-structure ρ W to be g ( W, ρ W ) = max (cid:26) g ∈ N (cid:12)(cid:12)(cid:12)(cid:12) there are g disjoint embeddings j : ( S n × S n ) \ {∗} (cid:44) → W such that j ∗ ρ W is admissible (cid:27) . The stable genus of (
W, ρ W ) is defined to be g ( W, ρ W ) = max (cid:110) g (cid:16) W W k, , ρ ( k ) W (cid:17) − k | k ∈ N (cid:111) where W W k, is obtained from W by removing k discs and attaching k copies of ( S n × S n ) \ int( D n )along the new boundary. The Θ-structure ρ ( k ) W is obtained by extending the restriction of ρ W byany admissible structure on ( S n × S n ) \ int( D n ). Lemma 4.9.
Let W be a smooth compact manifold of dimension n ≥ , L ⊂ int( W ) a closedsubmanifold of dimension ≤ n − , and N a tubular neighbourhood of L . Then the genus of W \ N is equal to the genus of W .Proof. Sard’s theorem implies that for any submanifold L (cid:48) ⊂ ( S n × S n ) \ {∗} with dim ( L (cid:48) ) ≤ n − S n × S n ) \ {∗} (cid:44) → ( S n × S n ) \ {∗} that avoids L (cid:48) and is isotopic to theidentity. In particular, this implies that for any φ : (cid:96) g ( S n × S n ) \ {∗} (cid:44) → W , there is an embedding φ (cid:48) : (cid:96) g ( S n × S n ) \ {∗} (cid:96) g ( S n × S n ) \ {∗} W φ that avoids L and is isotopic to φ . Since W \ L is diffeomorphic to int( W \ N ) via a diffeomorphismfixing everything but a collar of L , the result follows. (cid:3) Let W be a manifold with non-empty boundary P , and ρ W a Θ-structure on W . Given M acobordism from P to Q together with a Θ-structure ρ M on M which restricts to ρ W over P , thereis an induced map(4.1) − ∪ P ( M, ρ M ) : M Θ ( W, ρ W ) M Θ ( W ∪ P M, ρ W ∪ ρ M )which is induced by the Diff( W )-equivariant map Bun Θ ( W, ρ W ) → Bun Θ ( W ∪ P M, ρ W ∪ ρ M ) givenby extending a map by ρ M , and the homomorphism Diff( W ) → Diff( W ∪ P M ) given by extendinga map by the identity on M . Theorem 4.10 ([GRW17], Corollary 1.7) . Assume d = 2 n ≥ , and Θ is such that Θ // GL d issimply-connected. Let ρ W be an n -connected Θ -structure on W and let g = g ( W, ρ W ) . Given acobordism ( M, ρ M ) as above such that ( M, P ) is ( n − -connected, the map ( − ∪ P ( M, ρ M )) ∗ : H i ( M Θ ( W, ρ W )) H i ( M Θ ( W ∪ P M, ρ W ∪ P M )) is an isomorphism for all i ≤ g − . We recall that a map is called n -connected if the map induced on homotopy groups π i is anisomorphism for i < n and a surjection for i = n . Remark . In [GRW17, Corollary 1.7], the result above is given in much more generality, allowingarbitrary coefficient systems and providing a better stability range depending on the coefficientsystem and the tangential structure. We restrict ourselves to the case above, for simplicity, butremark that such generalisations can also be immediately carried out in the decoupling theorem.
Corollary 4.12.
Assume d = 2 n ≥ , and Θ is such that Θ // GL d is simply-connected. Let ρ W be an n -connected Θ -structure on W and let g = g ( W, ρ W ) . Then for all i ≤ g − we have anisomorphism H i ( M Θ ,km ( W, ρ W )) ∼ = H i ( M Θ ( W, ρ W ) × Θ m // Σ m × (Θ // GL † n ) k // Σ k ) where GL † n equals to GL +2 n if W is decorated-chiral, and is GL n otherwise.Proof. First notice that since the map ρ W is n -connected and the pair ( W, W m + k ) is (2 n − ρ W m + k is still an n -connected Θ-structure.The map τ : H i ( M Θ ( W m + k , ρ W m + k )) → H i ( M Θ ( W, ρ W )) in the hypothesis of the decouplingtheorem, is induced by attaching (cid:96) m + k D n along the m + k boundary sphere components of W m + k .Since ( M, P ) = ( (cid:96) m + k D n , ∂ (cid:96) m + k D n )is ( n − τ induces ahomology isomorphism in degrees 3 i ≤ g −
4. Applying Theorem 3.11, the result follows. (cid:3)
Example 4.13.
Let W g, = ( S n × S n ) D n . Since T W g, is trivialisable, we know W g, admits aframing ρ W g, : Fr( T W g, ) → GL n fitting into the following pullback diagram:Fr( T W g, ) GL n W g, E GL nρ Wg, (cid:121) The bottom arrow is necessarily n -connected because W g, is ( n − E GL n is weaklycontractible. Therefore, ρ is n -connected as well.Let g denote the stable genus g ( W g, , ρ W g, ). By Corollary 4.12, for all i ≤ g − , the group H i ( M fr ,km ( W g, , ρ W g, )) is isomorphic to H i (cid:0) M fr ( W g, , ρ W g, ) × SO(2 n ) m // Σ m × B Σ k (cid:1) . Example 4.14.
Let V d ⊂ C P be a smooth hypersurface determined by a homogeneous complexpolynomial of degree d . This is an orientable chiral 6-dimensional manifold whose diffeomorphismtype depends only on the degree d . In section 5.3 of [GRW18b], Galatius and Randal-Williamsshow that, if d is even, there exists a 3-connected Spin c -structure ρ V d on V d . They also compute anexpression for the stable genus g ( V d , ρ V d ) in terms of d .Applying the procedure of Example 2.4 to the fibre sequence {± } × B U(1) B Spin c ( d ) B GL d we get that Θ Spin c (cid:39) {± } × B U(1), and Θ
Spin c // GL +6 (cid:39) {± } × B Spin c (6). Therefore, byCorollary 4.12, for all i ≤ d − d +10 d − d +44 , the group H i ( M Spin c ,km ( V d , ρ V d )) is isomorphic to H i (cid:16) M Spin c ( V d , ρ V d ) × B (Σ m (cid:111) U(1)) × B (Σ k (cid:111) Spin c (6)) (cid:17) The conditions on the tangential structure in Theorem 4.10 are quite restrictive, for instancethe trivial tangential structure Θ ∗ does not satisfy the hypothesis because B GL d is not simplyconnected for any d . Moreover, the condition that we start with an n -connected Θ-structure ρ W excludes many of the cases we are interested in. For instance, it implies that the manifold W g, with an orientation does not satisfy the hypothesis of Theorem 4.10. However, in [GRW17, Section9], Galatius and Randal-Williams provided a generalisation of this result for general tangentialstructures. In Section 6, we use their techniques to prove a generalisation of Theorem A for highdimensional manifolds with any tangential structure.5. Decoupling Submanifolds
In Section 3, we proved a decoupling result for the decorated moduli space of a manifold with markedpoints and discs, following the works of [BT01, Han09, CM11b]. Recently, in [Pal12, Pal18a, Pal18b]Palmer has studied manifolds equipped with more general decorations, allowed to be any embeddedclosed manifold P . In this section, we show that there is a decoupling result for these generaliseddecorations. As a specific example, we focus on the case where the decorations are unlinked circles,which have also been closely studied in dimension 3 by Kupers in [Kup13]. ECOUPLING DECORATIONS ON MODULI SPACES 19
The L -decorated moduli space. In this section, we generalise the definition of a decoratedmanifold to allow more general submanifolds as decorations. Throughout, let W be a smoothconnected compact d -dimensional manifold. Definition 5.1. A d -dimensional L -decorated manifold is a pair ( W, L ) of a manifold W togetherwith a closed submanifold L ⊂ W .Given a L -decorated manifold ( W, L ), we define the decorated diffeomorphism group
Diff L ( W )to be the subgroup of Diff( W ) of the diffeomorphisms ψ such that ψ ( L ) = L. In other words, we are looking at the diffeomorphisms preserving the marked submanifold, butnot necessarily pointwise.
Definition 5.2.
Given a closed manifold W and a Θ-structure ρ W on W , we define the L -decoratedmoduli space of W to be M Θ L ( W, ρ W ) := Bun Θ ( W, ρ W ) // Diff L ( W ) . The inclusion of groups Diff L ( W ) → Diff( W ) induces a map(5.1) F L : M Θ L ( W, ρ W ) → M Θ ( W ρ W )which we call the forgetful map .5.2. The evaluation map E L . Let (
W, L ) be an L -decorated manifold and let ν L := ( T W | L ) /T L be the normal bundle of L in W . Let N be the tubular neighbourhood of the decoration identifiedas the image of an embedding Φ : ν L → W , and denote by W N the manifold W \ N . The theoremfor decoupling submanifolds relies on understanding the difference between the L -decorated modulispace of W and the moduli space of W N .We start by constructing an equivariant fibre sequence relating the decorated diffeomorphismgroups Diff L ( W ) and Diff( W N ). Recall that Diff( W N ) consists only of those diffeomorphisms fixinga collar neighbourhood of the boundary of W N , including the newly formed boundary obtained byremoving N . Extending a diffeomorphism by the identity on N , gives us a homomorphismDiff( W N ) → Diff L ( W ) . On the other hand, since any diffeomorphism φ ∈ Diff L ( W ) fixes L , the differential of φ inducesan isomorphism of the tangent bundle T W | L fixing T L (not necessarily pointwise). This gives amap: e L : Diff L ( W ) −→ Iso(
T W | L , T L )(5.2) φ (cid:55)−→ Dφ | L (5.3)where Dφ | L denotes the isomorphism of T W | L induced by the differential of φ , and Iso( T W | L , T L )denotes the group of bundle isomorphisms of T W | L fitting into the following diagram: T L T LT W | L T W | L L L.
Dfff
Definition 5.3.
For any subgroup G ⊂ Im e L , we define Diff G ( W ) to be the subgroup e − L ( G ).Given a closed manifold W and a Θ-structure ρ W on W , we define M Θ G ( W, ρ W ) := Bun Θ ( W, ρ W ) // Diff G ( W ) . Note that taking G = Im e L , one recovers precisely the definition of M Θ L ( W, ρ W ). The kernelof e L consists precisely of those elements of Diff L ( W ) which fix the submanifold L pointwise andwhose differential D p φ is the identity on every point of the submanifold L . We denote the kernel of e L by Diff( W, T W | L ). Lemma 5.4.
Let ( W, L ) be a compact connected L -decorated manifold, then (a) the homomorphism e L : Diff L ( W ) Iso( T W | L , T L ) is a principal bundle;(b) the map i : Diff( W N ) Diff( W, T W | L ) is a homotopy equivalence.Proof. Throughout this proof, we will use a generalisation of Palais’ theorem in [Pal60] proved byLima in [Lim64], which gives us a principal bundleDiff( W N ) Diff( W ) Emb( N, W ) . Let Emb L ( N, W ) be the subspace of embeddings f : N (cid:44) → W such that the core of N is taken toour marked submanifold L in W . Then taking the pullback along the inclusion Emb L ( N, W ) (cid:44) → Emb(
N, W ) gives as the principal bundle:(5.4) Diff( W N ) Diff L ( W ) Emb L ( N, W ) r Write N as the image of an embedding exp ◦ Φ : ν L (cid:44) → W , where Φ : ν L → T W | L . Consider theforgetful map d : Emb L ( N, W ) → Iso(
T W | L , T L )taking an embedding to the map induced on the normal bundle of the zero section L ⊂ N . Then e L = d ◦ r . We will show d is a fibre bundle, which implies e L is a principal bundle. It is enough toexhibit a local section of d at a neighbourhood of the identity (for details see [Ste99, Part I, Section7.4]).Given f : T W | L → T W | L in Iso( T W | L , T L ) we can define s f : ν L T W | L T W | L W. Φ f exp Since the assignment f (cid:55)→ s f is continuous and Emb( ν L , W ) is an open subset of C ∞ ( ν L , W ), thenthe space of maps f ∈ Iso(
T W L , T L ) such that s f is an embedding, is an open neighbourhood U ofthe identity. Therefore, the map U −→ Emb( ν L , W ) f (cid:55)−→ s f is a local section for d at the identity. Part (b):
We will show the map i is a homotopy equivalence, from the fact that it fits into thefollowing commutative diagram of fibre sequences:Diff( W N ) Diff L ( W ) Emb L ( N, W )Diff(
W, T W | L ) Diff L ( W ) Iso( T W | L , T L ) i = de L The fiber of the forgetful map d over the identity is simply the space of tubular neighbourhoods of L in W , which is contractible. This implies d is a homotopy equivalence and therefore so is i . (cid:3) We now use the map e L and Lemma 5.4 to construct the evaluation map: Proposition 5.5.
Let W be a compact connected manifold and ρ W a fixed Θ -structure on W , then:(a) the homomorphism e L , induces an evaluation map E L : M Θ G ( W, ρ W ) Map GL d (Fr( T W | L ) , Θ) //G which is a Serre fibration onto the path component which it hits.(b) Let W N be equipped with the Θ -structure ρ W N given by the restriction of ρ W . Then M Θ ( W N , ρ W N ) is the homotopy fibre of E L over its image. ECOUPLING DECORATIONS ON MODULI SPACES 21
Proof. (a) By Lemma 3.9, the restriction map r L : Bun Θ ( W, ρ W ) → Map GL d (Fr( T W | L ) , Θ)is a Serre fibration. Then the result follows by applying Corollary 2.11 to combine the fibration r L with the homomorphism(5.5) e L : Diff G ( W ) → G. We apply Corollary 2.11(a) by taking G = Diff G ( W ) and S = Bun Θ ( W, ρ W ), with the usualaction by precomposition with the differential. On the other hand, we take G = G , and S =Map GL d (Fr( T W | L ) , Θ) with the action induced byIso(
T W | L , T L ) Fr( T W | L ) . Then the fibration Bun Θ ( W, ρ W ) → Map GL d (Fr( T W | L ) , Θ) is e L -equivariant and therefore, byCorollary 2.11(a), we have a fibration M Θ G ( W, ρ W ) Map GL d (Fr( T W | L ) , Θ) //G E L onto the path components which it hits.(b) Recall that W N is defined as the submanifold W \ N , where N is a tubular neighbourhoodof the submanifold L . Then the restriction ρ W N is a Θ-structure on W N . In the remainder of theproof, we will show that M Θ ( W N , ρ W N ) is the homotopy fibre of E L .A description of the fibre of E L can be obtained using Corollary 2.11(b) with the short exactsequence of groups being(5.6) ker e L → Diff G ( W ) e L −→ G and the fibre sequence of S → S → S being the one associated to the fibration r L . The fibreof r L over r L ( ρ W ) is the subspace of all elements of Bun Θ ( W, ρ W ) which restrict to r L ( ρ W ) over L , which we here denote Bun Θ L ( W, ρ W ). This space carries an action of ker e L by precompositionwith the differential, and it is simple to check that this fibre sequence is equivariant with respect to(5.6). Then by Corollary 2.11(b), the fibre of the evaluation map E L is given byBun Θ L ( W, ρ W ) // ker e L . Applying Lemma 3.9 to both submanifolds L and N , we obtain two fibrations fitting into thefollowing commutative diagramBun Θ ( W, ρ W ) Map GL d (Fr( T W | N ) , Θ)Bun Θ ( W, ρ W ) Map GL d (Fr( T W | L ) , Θ) r N = i ∗ r L where the right-hand vertical map is induced by the inclusion i : L (cid:44) → N . Since i is a strongdeformation retract, the map i ∗ is a homotopy equivalence. In particular, this implies that the mapfrom the fibre of r N to Bun Θ L ( W, ρ W ) is a homotopy equivalence.The fibre of r N over r N ( ρ W ) is by definition the space of all Θ structures on W which agree with ρ W on N . We claim that this space is homeomorphic to Bun Θ ( W N , ρ W N ), since the restriction map r WN takes the fibre of r N bijectively to Bun Θ ( W N , ρ W N ) and it has an inverse given by extendingan element by r N ρ W .So we have a commutative diagram of principal fibre bundlesDiff( W N ) ker e L Bun Θ ( W N , ρ W N ) × E Diff( W ) Bun Θ L ( W, ρ W ) × E Diff( W ) M Θ ( W N , ρ W N ) Bun Θ L ( W, ρ W ) // ker e L (cid:39)(cid:39) where the top horizontal map is a homotopy equivalence by Lemma 5.4 and the middle map is ahomotopy equivalence by the discussion above. Therefore the map M Θ ( W N , ρ W N ) → Bun Θ L ( W, ρ W ) // ker e L is also a homotopy equivalence, as required. (cid:3) Proposition 3.8 can be recovered as a special case of Proposition 5.5: let L be comprised of m + k points (which are the k marked points and the centres of the m marked discs), then T L is a zero-dimensional bundle and
T W | L is a trivial bundle of dimension d . Defining G ⊂ Iso(
T W | L , T L ) ∼ =Iso( (cid:96) m + k R d ) to be the subgroup (Σ k (cid:111) GL d ) × Σ m , we recover precisely the case analysed in Proposition3.8. Note that an element of Diff G ( W ) can permute the k marked points with no restrictions on themap induced on their tangent bundle, on the other hand, the m points are allowed to be permuted,but the map induced on their tangent spaces has to be the identity.5.3. Decoupling L -decorations. In this section we prove the decoupling result by comparinga homotopy fibre sequence constructed in Proposition 5.5 to the product fibre sequence via theaforementioned spectral sequence argument.
Definition 5.6.
The decoupling map D L : M Θ G ( W, ρ W ) M Θ ( W, ρ W ) × Map GL d (Fr( T W | L , Θ)) //G F L × E L is the product of the forgetful map 5.1 and the evaluation map E L defined in Proposition 5.5.We now state the decoupling theorem: Theorem 5.7.
Let ( W, L ) be an L -decorated manifold, with W a connected compact manifoldequipped with a Θ -structure ρ W , and G ⊂ Im e L . If τ : H i ( M Θ ( W N , ρ W N )) → H i ( M Θ ( W, ρ W )) isan isomorphism in degrees i ≤ α , then for all such i the decoupling map D L induces an isomorphism H i ( M Θ G ( W, ρ W )) ∼ = H i ( M Θ ( W, ρ W ) × (Map GL d (Fr( T W | L ) , Θ) //G ) ) where ( − ) denotes a path component of E L ( ρ W ) .Proof. By Proposition 3.8, E L is a fibration onto the path-components which it hits, thereforethe restriction of E L to the subspace M Θ G ( W, ρ W ) is a fibration onto the path-component ofMap GL d (Fr( T W | L ) , Θ) //G which it hits. We denote it(Map GL d (Fr( T W | L ) , Θ) //G ) . Therefore, we have a homotopy fibre sequence M Θ ( W N , ρ W N ) M Θ G ( W, ρ W ) (Map GL d (Fr( T W | L ) , Θ) //G ) E L The proof of the theorem follows from the comparison between this homotopy fibre sequence andthe one associated to the trivial fibration M Θ ( W, ρ W ) × (Map GL d (Fr( T W | L ) , Θ) //G ) (Map GL d (Fr( T W | L ) , Θ) //G ) By definition of the maps in Proposition 5.5, the following is a commutative diagram of homotopyfibre sequences M Θ ( W N , ρ W N ) M Θ G ( W, ρ W ) (Map GL d (Fr( T W | L ) , Θ) //G ) M Θ ( W, ρ W ) M Θ ( W, ρ W ) × (Map GL d (Fr( T W | L ) , Θ) //G ) (Map GL d (Fr( T W | L ) , Θ) //G ) τ E L D L where the middle vertical map is the decoupling map. This induces a map of the respective Serrespectral sequences f : E • p,q → ˜ E • p,q , and since τ is a homology isomorphism in degrees i ≤ α , themap between the E pages ECOUPLING DECORATIONS ON MODULI SPACES 23 E ∗∗ = H ∗ (cid:0) (Map GL d (Fr( T W | L ) , Θ) //G ) ; H ∗ ( M Θ ( W N , ρ W N )) (cid:1) ˜ E ∗∗ = H ∗ (cid:0) (Map GL d (Fr( T W | L ) , Θ) //G ) ; H ∗ ( M Θ ( W, ρ W )) (cid:1) is an isomorphism for all q ≤ α . Then by the Spectral Sequence Argument recalled in Lemma 2.12, D induces an isomorphism H i ( M Θ G ( W, ρ W )) H i ( M Θ ( W, ρ W ) × (Map GL d (Fr( T W | L ) , Θ) //G ) ∼ = for all i ≤ α . (cid:3) We give an application of this theorem for high even-dimensional manifolds using [GRW17,Corollary 1.7], which we recalled in Theorem 4.10.
Corollary 5.8.
Let ( W, L ) be an L -decorated manifold, with W a compact simply-connected man-ifold of dimension n ≥ , and L of dimension less than n . Let ρ W be an n -connected Θ -structureon W , and denote by g the stable genus of W . Then for all i ≤ g − and G ⊂ Im e L , the decouplingmap D L induces an isomorphism H i ( M Θ G ( W, ρ W )) ∼ = H i (cid:0) M Θ ( W, ρ W ) × (Map GL d (Fr( T W | L ) , Θ) //G ) (cid:1) where ( − ) is the path component of the image of ρ W .Proof. We know that N is homotopy equivalent to L , and that the boundary of N is a spherebundle over L with fibre S c − , where c is the codimension of L and W . The dimension assumptionon L implies that c ≤ n + 1, and therefore the pair ( N, ∂N ) is ( n − W N is equal to g . Hence we are under the hypothesis of Theorem4.10 and τ : H i ( M Θ ( W N , ρ W N )) → H i ( M Θ ( W, ρ W ))is an isomorphism for all i ≤ g − . By Theorem 5.7, the result follows. (cid:3) Decoupling unlinked circles.
In this section, we apply Theorem 5.7 to the specific casewhere L is a collection of k unlinked circles. Therefore, throughout this section, we assume W to bea compact simply-connected manifold of dimension 2 n ≥
6, to satisfy the hypothesis of Corollary5.8. Note that, since W is simply-connected, it is always orientable. Definition 5.9.
An embedding f : (cid:96) k S → W \ ∂W is said to be unlinked if it extends to anembedding f : (cid:96) k D → W \ ∂W . If W is oriented and 2-dimensional we also assume that theembedding f is orientation preserving. Notation 5.10.
Throughout this section, we let kS denote the space (cid:96) k S , and kD denote thespace (cid:96) k D .In this section, we will repeatedly use the following result, which follows from [Hir94, Chapter 8,Theorems 3.1, 3.2]. Lemma 5.11.
Let W be a connected d -manifold and and f, g : kD (cid:44) → W embeddings of k disjointdiscs into W . If d = 2 and W is oriented, assume also that f and g both preserve, or both reverse,orientation. Then there is a diffeomorphism φ of W which is diffeotopic to the identity, such that φ ◦ f = g . An immediate consequence of the result above is the following
Lemma 5.12.
The isomorphism type of
Diff f ( kS ) ( W ) does not depend on the choice of the unlinkedembedding f : kS (cid:44) → W .Proof. For any two unlinked embeddings f, g : kS (cid:44) → W , there are embeddings f , g : kD (cid:44) → W extending f, g . By Lemma 5.11, there is a diffeomorphism φ of W with φ ◦ f = g . Then conjugationwith φ defines an isomorphism between the group of diffeomorphism preserving f ( kS ) and the onepreserving g ( kS ). (cid:3) From here on, we denote by Diff kS the isomorphism type of Diff f ( kS ) ( W ) for any embedding f : kS (cid:44) → W , which is well-defined by Lemma 5.12.We want to use Theorem 5.7 for the case where the submanifold L is an collection of unlinkedcircles, but instead of choosing a subgroup of G , we will take G = Im e kS , so we start by analysingwhat this image is. Fix an unlinked embedding of kS in W (we will refer to it as kS ⊂ W ), since W is orientable and by fixing a Riemmannian metric we get an explicit isomorphism T W | kS ∼ = T kS ⊕ ν kS Lemma 5.13.
There is a quotient map q : Iso( T W | kS , T kS ) → Iso( ν kS ) which is a homomor-phism, a Serre fibration and a homotopy equivalence.Proof. Any isomorphism f ∈ Iso(
T W | kS , T kS ) satisfies f ( T kS ) = T kS . Therefore, it induces amap on the quotient bundle [ f ] : T W | kS /T kS = ν kS → ν kS . Using the inclusion ν kS → T W | kS induced by the choice of a Riemannian metric, it is simple to check that this map satisfies thehomotopy lifting property of Serre fibrations. Moreover, using the identification T W | kS ∼ = T kS ⊕ ν kS , we can verify easily that the fibre of q over the identity is the space of sections of the vectorbundle Hom( ν kS , T kS ) → kS which is contractible. (cid:3) Since the normal bundle of the marked circles is also orientable and any orientable vector bundleover a circle is trivial, we know there is a bundle isomorphism ν kS ∼ = kS × R d − giving a shortexact sequence C ∞ ( S , GL d − ) k Iso( νkS ) Diff( kS ) f where f takes an isomorphism of ν kS to the underlying diffeomorphism of the base kS . Themap Diff( kS ) → Iso( νkS ) defined by taking φ to the isomorphism φ × Id is a section for f , andtherefore(5.7) Iso( ν kS ) ∼ = C ∞ ( S , GL d − ) k (cid:111) Diff( kS ) . Fixing such isomorphism, the evaluation map (5.2) together with the quotient map of Lemma 5.13induce a homomorphism e kS : Diff kS ( W ) C ∞ ( S , GL d − ) k (cid:111) Diff( kS ) . We want to determine the image of the map e kS , which is equivalent to identifying the isomorphismsof the normal bundle of the circles that can actually be realised by a diffeomorphism of W . Figure 2.
A non trivial isomorphism of the normal bundle of S in a 3-dimensional manifold.For simplicity, we first look at the surface case: Lemma 5.14.
Let S g,b be the oriented surface of genus g and b ≥ boundary components. Thenthe image of e kS is C ∞ ( S , GL +1 ) (cid:111) Diff + ( kS ) . Proof.
Fix once and for all an embedding f : kD (cid:44) → S g,b , and consider f ( ∂kD ) to be the kS decoration in S g,b . We start by showing that any diffeomorphism of Diff + ( kS ) can be realised by anelement in Diff kS ( S g,b ). Any diffeomorphism φ ∈ Diff + ( kS ) can be extended to φ ∈ Diff + ( kD )[Hir94, Chapter 8, Theorem 3.3], so it is sufficient to find a diffeomorphism ψ of W such that ψ ◦ f = f ◦ φ . But this can always be done, by Lemma 5.11.This implies that the coset e kS ( Id ) · Diff + ( kS ) is contained in the image of e kS , and sincethis map is surjective on path components, we know that C ∞ ( S , GL +1 ) k (cid:111) Diff + ( kS ) is con-tained in the image of e kS . Moreover, since we assume b ≥
1, then for any k , all elements ofDiff kS ( S g,b ) are orientation preserving and fix f ( kD ) as a set, which means that Im e kS is con-tained in C ∞ ( S , GL + d − ) k (cid:111) Diff + ( kS ), as required. (cid:3) ECOUPLING DECORATIONS ON MODULI SPACES 25
We now analyse Im e kS for a manifold W of higher dimension. In particular, we need to under-stand when there exists a diffeomorphism of W that induces a loop with non-trivial homotopy classin π (GL d − ) as depicted in Figure 2. It is clear that determining this image does not only dependon the orientability of W , as it was for the case of points and discs, but it will also depend on thespinnability of W . Lemma 5.15.
Let W be a simply-connected manifold of dimension d ≥ . Then the image of e kS is C ∞(cid:5) ( S , GL d − ) k (cid:111) Diff( kS ) where C ∞(cid:5) ( S , − ) is equal to the subspace C ∞ null ( S , − ) of nullhomotopic loops if W is spinnable,and is equal to C ∞ ( S , − ) otherwise. Here we are not thinking of the spaces as pointed, so we consider a nullhomotopic loop to be onethat is homotopic to a constant loop, not necessarily at a base point.
Proof.
Fix once and for all an embedding f : kD (cid:44) → W , and consider f ( ∂kD ) to be the kS decoration in W . By the same argument as in the proof of Lemma 5.14, we know that any diffeo-morphism of Diff( kS ) can be realised by an element in Diff kS ( W ). Without loss of generality, weassume from now on that k = 1.Since e S is surjective on path components, we know that C ∞ null ( S , GL + d − ) (cid:111) Diff( S ) is containedin the image of e S because it is the path component of e S ( Id ).By Lemma 5.11 there exists an orientation preserving diffeomorphism φ c ∈ Diff S ( W ) thatrestricts to complex conjugation along the marked circles. This implies that φ c induces an orien-tation reversing diffeomorphism on the normal bundle of S . Since the marked circle bounds anembedded 2-disc, we can define such a φ c by taking an embedded disc D d in W containing themarked circle in its equator, and applying a rotation that flips the circle. By the Isotopy Exten-sion Theorem, such a rotation can be extended to an isotopy in W . Then e S ( φ c ) is contained in C ∞ null ( S , GL d − ) (cid:111) Diff( S ). Since e S is surjective on path-components, we conclude that C ∞ null ( S , GL d − ) (cid:111) Diff( S ) ⊂ Im e S . We now show that a smooth curve γ / ∈ C ∞ null ( S , GL d − ) is in the image of e S if, and only if, W is not spin.First, assume W is spin, and choose φ ∈ Diff S ( W ). We know that any diffeomorphism of thecircle is isotopic either to the identity or to complex conjugation, so without loss of generality, wecan assume that φ restricts to one of these two maps on the marked circle. Start by assuming that φ restricts to the identity on the marked circle. Then using the embedding f : D (cid:44) → W bounding themarked circle, we can define a continuous function g : S → W by sending the bottom hemisphere D − to f ( D ), and the top hemisphere D to φ ◦ f ( D ). Since we assume W to be spin, we knowthat w ( g ∗ ( T W )) = g ∗ ( w ( T W )) = 0. Since d ≥
5, the class w detects the only obstruction tolifting to to E SO( d ) the map S → B SO( d ) classifying the bundle g ∗ ( T W ). Since w ( g ∗ ( T W )) = 0,this implies that g ∗ ( T W ) is a trivial bundle, and in particular, its clutching function S → GL + d isnullhomotopic. But note that Dφ along the marked circle is a clutching function of g ∗ ( T W ), andtherefore e S ( φ ) is contained in C ∞ null ( S , GL + d − ) (cid:111) Diff( S ).On the other hand, if φ restricts to complex conjugation on the marked circle, then composingwith the map φ c constructed above, we get a map restricting to the identity. By the same argumentsas above, we can conclude that e S ( φ ) ∈ C ∞ null ( S , GL d − ) (cid:111) Diff( S ).Now assume W is not spin. Since W is simply-connected, by Hurewicz theorem, all its secondhomology classes are represented by maps S → W and by [Tho54, Theorem II.27] we can alwayspick a representative given by an embedding. Since W is not spin, there exists an embedding h : S → W such that w ( h ∗ ( T W )) (cid:54) = 0, and since d ≥
5, we can pick one such h not intersecting f ( D ) by the Transversality Theorem (see [Kup19, Corollary 12.2.7]). By Lemma 5.11, we knowthere exists a diffeomorphism φ h of W taking φ h ◦ h | D − to h | D , and which restricts to the identityon the image of f ( D ). By definition, Dφ h | h ( S ) is a clutching function for h ∗ ( T W ) and thereforeis not nullhomotopic as h ∗ ( T W ) is non-trivial.Let ψ be a diffeomorphism taking f ( D ) to h | D − . Then ψ − ◦ φ h ◦ ψ is a diffeomorphism of W whose image through e S is not in C ∞ null ( S , GL + d − ) (cid:111) Diff( S ). Since e S is surjective on pathcomponents, we conclude that the image of e S contains C ∞ ( S , GL + d − ) (cid:111) Diff( S ). Analogously, looking at the composition ψ − ◦ φ f ◦ ψ ◦ φ c , we conclude that the image of e S is C ∞ ( S , GL d − ) (cid:111) Diff( S ). (cid:3) Remark . Lemma 5.15 can be generalised for dimension 4 assuming the manifold is spin, usingexactly the same argument as above.Now that we have analysed the image of e kS we apply Theorem 5.7. As before, we start bylooking at the surface case. Corollary 5.17.
Let S g,b be the oriented surface of genus g and b ≥ boundary components. Thenfor all i ≤ g H i ( B Diff kS ( S g,b )) ∼ = H i ( B Diff( S g,b ) × B (Σ k (cid:111) SO(2))) . Proof.
Start by fixing an unlinked embedding f : kS (cid:44) → S g,b and f : kD (cid:44) → S g,b extending f . Theproof follows from applying Theorem 5.7 taking L to be the embedded circles. To do this, we startby verifying that the hypothesis of the Theorem are satisfied, ie. that the map defined by extendingdiffeomorphisms by the identity ε N : Diff( S g,b \ N ) → Diff( S g,b )induces isomorphisms on the homology groups of the classifying spaces in the range 3 i ≤ g .Since the embedded circles are unlinked, we know they are nullhomotopic and therefore S g,b \ N is diffeomorphic to S g,b + k ∪ (cid:96) k D . ThenDiff( S g,b \ N ) ∼ = Diff( S g,b + k ) × Diff( D ) k . The inclusion S g,b \ N → S g,b induces maps ε S : Diff( S g,b + k ) → Diff( S g,b ) ε D : Diff( D ) k → Diff( S g,b )given by extending the the diffeomorphisms by the identity, and it is simple to check that thefollowing diagram commutes Diff( S g,b ) × Diff( D ) k Diff( S g,b + k ) × Diff( D ) k Diff( S g,b ) cε S × id ε N where c ( φ, ψ ) := φ ◦ ε D ( ψ ).By [RW16, Theorem 7.1], we know that the map ε S induces a homology isomorphism in the range3 i ≤ g . Moreover, the map c is a homotopy equivalence since the space Diff( D ) is contractible.Therefore, ε N induces a homology isomorphism in the range 3 i ≤ g .Hence we are under the hypothesis of Theorem 5.7. Moreover, we know that T W | kS is trivialso Map GL d (Fr( T W | kS ) , Θ or ) (cid:39) Map( S , {± } ) k is simply a disjoint union of points. And, by Lemma 5.14,Im e kS (cid:39) Im e kS ∼ = C ∞ ( S , GL +1 ) k (cid:111) Diff + ( kS ) (cid:39) Diff + ( kS ) (cid:39) Σ k (cid:111) SO(2) . Applying Theorem 5.7, the result follows. (cid:3)
Corollary 5.17 can be re-stated with a geometric interpretation. As discussed in Section 1.5, thespace Emb( S g,b , R ∞ ) is a model for E Diff( S g,b ), and therefore it is also a model for E Diff kS ( S g,b ).With this model, the elements of B Diff kS ( S g,b ) are oriented submanifolds of R ∞ diffeomorphic to S g,b with k marked unlinked circles. With this model, the forgetful map(5.8) F kS : B Diff kS ( S g,b ) → B Diff( S g,b )simply forgets the marked circles.To interpret the evaluation map E kS in this model, we recall a definition that will also be usefulfor the interpretation of the evaluation map for the moduli space in higher dimensions with generaltangential structures. ECOUPLING DECORATIONS ON MODULI SPACES 27
Definition 5.18.
Let W be a manifold and X be a space with an action of Diff( S ), the space of k -unlinked circles in W with labels in X is defined to be C kS ( W ; X ) := Emb unl ( kS , W ) × X k / Diff( kS )where Emb unl denotes the space of unlinked embeddings.Note that if W is a simply connected manifold of dimension d ≥
5, all embeddings of kS into W are unlinked.Then a model for B Diff + ( kS ) is precisely the configuration space C kS ( R ∞ ; {± } ) of k circlesin R ∞ with labels in {± } , and the evaluation map E kS simply takes an oriented decoratedsubmanifold S in R ∞ , to the configurations given by the marked k oriented circles. Corollary 5.19.
Let S g,b be the oriented surface of genus g and b ≥ boundary components. Thenfor all i ≤ g H i ( B Diff kS ( S g,b )) ∼ = H i ( B Diff( S g,b ) × C kS ( R ∞ ; {± } )) . Analogous results for other tangential structures can be obtained by the same arguments. Wenow look at how the result above generalises for higher dimensions. From now on, we let L ( − ) :=Map( S , − ) denote the free loop space. Corollary 5.20.
Let W be a compact simply-connected manifold of dimension n ≥ , ρ W an n -connected Θ -structure on W , and denote by g the stable genus of W . Then for all i ≤ g − , thedecoupling map D kS induces an isomorphism H i ( M Θ kS ( W, ρ W )) ∼ = H i (cid:0) M Θ ( W, ρ W ) × C kS ( R ∞ ; ( L (Θ) //L (cid:5) (GL d − )) ) (cid:1) where ( − ) is the path component of the image of ρ W , L (cid:5) ( − ) is equal to the subspace L null ( − ) ofnullhomotopic loops if W is spinnable, and is equal to L ( − ) otherwise.Proof. The result follows from applying Corollary 5.8, taking G = Im e kS (cid:39) Im e kS , which wasidentified in Lemma 5.15. Moreover, since T kS is orientable, it is a trivial bundle and thereforethe space Map GL d (Fr( T kS ) , Θ) is equivalent to the space of continuous maps kS → Θ, which isprecisely L (Θ) k .Then, by Corollary 5.8, for all i ≤ g − , the decoupling map D kS induces an isomorphism H i ( M Θ kS ( W, ρ W )) ∼ = H i (cid:0) M Θ ( W, ρ W ) × ( L (Θ) k //C ∞(cid:5) ( S , GL d − ) k (cid:111) Diff( kS )) (cid:1) Moreover, the space ( L (Θ) k //C ∞(cid:5) ( S , GL d − ) k (cid:111) Diff( kS )) is homotopy equivalent to(5.9) ( L (Θ) //C ∞(cid:5) ( S , GL d − )) k // Diff( kS )) Taking Emb( kS , R ∞ ) as the model for E Diff( kS ), we get a model for the space in 5.9, whichis precisely the configuration space of k circles in R ∞ with labels in ( L (Θ) //C ∞(cid:5) ( S , GL d − )) , asrequired. Since the space of smooth loops is homotopy equivalent to the free loop space, the resultfollows. (cid:3) Decoupling for general tangential structures in higher dimensions
In this section we show how Corollaries 4.12 and 5.8 can be generalised for other tangential struc-tures, based on the techniques used by Galatius and Randal-Williams in [GRW17, Section 9]. Recallthat the decoupling theorems (3.11 and 5.7) relied on the hypothesis that the map M Θ ( W N , ρ W N ) → M Θ ( W, ρ W )induces a homology isomorphism in a range. In even dimensions at least 6, this assumption wasshown to hold in several cases in [GRW18a, Corollary 1.7] as recalled in 4.10, but only when theΘ-structure ρ W : Fr( T W ) → Θ is n -connected (ie. the induced map π i (Fr( T W )) → π i (Θ) is anisomorphism for i < n and an epimorphism for i = n ). In [GRW18a, Section 9] Galatius andRandal-Williams provide a generalisation of the result to general tangential structures. In thissection we introduce the tools used to construct this generalisation and show how they also providean extension of the decoupling result in higher dimensions for general tangential structures.One could hope that for any manifold W and any Θ-structure ρ W , the decoupling map would stillinduce a homology isomorphism, but this is not the case, as it is shown by the following example. Example 6.1.
Consider the manifold W g = g S n × S n with one embedded disc as a decoration.Let W g, = g ( S n × S n ) \ int( D n ) and recall there is an isomorphismDiff + ( W g, ) ∼ = −→ Diff +1 ( W g )given by extending the diffeomorphism of W g, by the identity on the marked disc (see Lemma 3.6).Therefore, the decorated moduli space M or ( W g , ρ W ) (cid:39) B Diff +1 ( W g ) is weakly equivalent to M or ( W g, , ρ W ) (cid:39) B Diff + ( W g, ). In this case, the decoupling map(6.1) M or ( W g , ρ W g ) M or ( W g , ρ W g ) × Θ or M or ( W g, , ρ W g, ) M or ( W g , ρ W g ) D (cid:39) (cid:39) does not induce a homology isomorphism on integral coefficients in a stable range as was shown in[GRW18b, Sections 5.1 and 5.2]. This implies that the decoupling as stated in Corollary 4.12 is nottrue for general tangential structures.Let W be a 2 n -dimensional manifold, 2 n ≥
6, with possibly non-empty boundary, and λ W aΛ-structure on W . If the map λ W : Fr( T W ) → Λ is not n -connected we will use an “intermediate”tangential structure Θ which is better behaved. Precisely, let the following be the Moore-Postnikov n -stage of λ W ,(6.2) ΘFr( T W ) Λ . uλ W ρ W This means that Θ is a GL d -space, u is an n -co-connected (ie. the induced map π i (Fr( T W )) → π i (Θ)is an isomorphism for i > n and a monomorphism for i = n ) equivariant fibration and ρ W an n -connected equivariant cofibration. Such a factorization always exist and it is unique up to homotopyequivalence.Denote by ρ ∂ and λ ∂ the restriction of ρ W and λ W respectively to Fr( T W ) | ∂W . Any Θ-structureon W induces a Λ-structure by postcomposition with u , giving us a mapBun Θ ( W, ρ W ) → Bun Λ ( W, λ W ) . Lemma 6.2 ([GRW17], Lemma 9.4) . If W is a manifold equipped with a Λ -structure λ W and Fr(
T W ) ρ W −−→ Θ u −→ Λ is a Moore-Postnikov n -stage of λ W , then the stable genus g ( W, ρ W ) is equalto g ( W, λ W ) . We now define a topological monoid that is crucial to the comparison between the moduli spaces M Λ ( W, λ W ) and M Θ ( W, ρ W ). Definition 6.3. If W is a closed manifold, denote by hAut( u ) the group-like topological monoidconsisting of equivariant weak equivalences Θ → Θ over u , ie. GL d -equivariant maps Θ → Θ fittinginto the following commutative diagramΘ Θ . Λ (cid:39) u u If W has non-empty boundary, let ρ ∂ be the restriction of ρ W to ∂W . Denote by hAut( u, ρ ∂ ) thegroup-like topological monoid consisting of equivariant weak equivalences Θ → Θ over u and under ρ ∂ Fr(
T W ) | ∂W Θ Θ . Λ ρ ∂ ρ ∂ (cid:39) u u ECOUPLING DECORATIONS ON MODULI SPACES 29
The monoid hAut( u, ρ ∂ ) acts on the space of Θ-structures on W by post-composition, and thefollowing result shows that this action encodes precisely the relation between Θ and Λ-structureson W . Lemma 6.4 ([GRW17], Lemma 9.2) . In the context defined above, the map induced by postcompo-sition with u Bun Θ ρ ∂ ( W ) // hAut( u, ρ ∂ ) → Bun Λ λ ∂ ( W ) is a homotopy equivalence onto the path components which it hits. Let hAut( u, ρ ∂ ) [ W,ρ W ] denote the components of hAut( u, ρ ∂ ) that map Bun Θ ( W, ρ W ) to itself.By the orbit-stabiliser theoremBun Θ ( W, ρ W ) // hAut( u, ρ ∂ ) [ W,ρ W ] → Bun Λ ( W, λ W )is also a homotopy equivalence onto the path components which it hits. Taking a further Borelconstruction with the groups Diff( W ), Diff km ( W ), Diff L ( W ), we get that the induced maps M Θ ( W, ρ W ) // hAut( u, ρ ∂ ) [ W,ρ W ] → M Λ ( W, λ W )(6.3) M Θ ,km ( W, ρ W ) // hAut( u, ρ ∂ ) [ W,ρ W ] → M Λ ,km ( W, λ W )(6.4) M Θ L ( W, ρ W ) // hAut( u, ρ ∂ ) [ W,ρ W ] → M Λ L ( W, λ W )(6.5)are weak homotopy equivalences. Therefore, analysing hAut( u, ρ ∂ ) [ W,ρ W ] and applying Corollaries4.12 and 5.8 we get decoupling results for general Λ. Lemma 6.5 ([GRW18b]) . If ( W, ∂W ) is c -connected for some c ≤ n − , then the monoid hAut( u, ρ ∂ ) is a non-empty ( n − c − -type. In particular, it is contractible if ( W, ∂W ) is ( n − -connected. We now focus on applying these techniques to the manifold W g, = g S n × S n \ D n , for n ≥ Proposition 6.6.
Let W g, = g S n × S n \ D n , for n ≥ , and λ W a Λ -structure on W . Let g denote the stable genus g ( W, λ W ) . For all i ≤ g − , the group H i ( M Λ ,km ( W g, , λ W )) is isomorphicto H i ( M Λ ( W g, , λ W ) × Θ m // Σ m × (Θ // GL +2 n ) k // Σ k ) where ( − ) denotes the path-component of E ( ρ W ) , for ρ W as in (6.2) . Note that in the above proposition, the decorations on M Λ ,km ( W ) get decoupled into componentsdepending on Θ, the tangential structure that appeared in the Moore-Postnikov n -stage factorisationof λ W . This is quite different than what was obtained in Corollary 4.12 as well as in the otherdecoupling theorems of sections 3 and 4, where the decoupled components corresponding to themarked points and discs depended on the original chosen tangential structure Λ. Proof.
By Lemma 6.5, we know that hAut( u, ρ ∂ ) is contractible, and therefore M Θ ( W g, , ρ W ) (cid:39) M Λ ( W g, , λ W ) M Θ ,km ( W g, , ρ W ) (cid:39) M Λ ,km ( W g, , λ W )are weak homotopy equivalences. Since ρ W is n -connected, we can apply Corollary 4.12 to under-stand the homology of M Θ ,km ( W g, , ρ W ). Putting this together with the above identifications we getthat the group H i ( M Λ ,km ( W g, , λ W )) is isomorphic to i th homology group of M Λ ( W g, , λ W ) × Θ m // Σ m × (Θ // GL +2 n ) k // Σ k . (cid:3) We now look at the case where Λ is the tangential structure for orientations: a GL d -equivariantmap λ W : Fr( T W g, ) → {± } determines, up to a contractible choice, a map (cid:96) (cid:48) W : W g, → B SO(2 n )fitting into the following homotopy pullback squareFr( T W g, ) {± } W g, B SO(2 n ) . λ W (cid:121) (cid:96) (cid:48) W Then an equivariant Moore-Postnikov factorization of λ W can be obtained from a Moore-Postnikovfactorization of (cid:96) (cid:48) W . Since W g, is ( n − n -stageof this factorization is given by maps W g, B Ø(2 n ) (cid:104) n (cid:105) B SO(2 n ) (cid:96) W u where B Ø(2 n ) (cid:104) n (cid:105) is the n -connected cover of B Ø(2 n ). Taking the pullback of {± } → B SO(2 n )along these maps, we get Fr( T W g, ) Ø[0 , n − {± } W g, B Ø(2 n ) (cid:104) n (cid:105) B SO(2 n ) (cid:121) (cid:121) (cid:96) W u where Ø[0 , n −
1] is the ( n − , n −
1] ishomotopy equivalent to SO[0 , n − n − Corollary 6.7.
Let W g, = g S n × S n \ D n , for n ≥ . Then for all i ≤ g − , the group H i ( B Diff + ,km ( W g, )) is isomorphic to H i ( B Diff + ( W g, ) × SO[0 , n − m // Σ m × B Ø(2 n ) (cid:104) n (cid:105) k // Σ k ) . The proof is a direct application of Proposition 6.6 using the factorization described above, andthe fact that for an orientation ρ W g, : Fr( T W g, ) → {± } , the stable genus g ( W g, , ρ W g, ) is equalto g (see [GRW18b, Section 3.2]).We end by using the result above to explicitly compute the cohomology of B Diff + ,km ( W g, ) withrational coefficients, in the stable range. As an immediate consequence of Corollary 6.7 and KunnethTheorem, the elements of H ∗ ( B Diff + ,km ( W g, ); Q ) of degree i ≤ g − , are given by the elements ofsuch degrees in the tensor product of the cohomology rings of B Diff + ( W g, ), SO[0 , n − m // Σ m and B Ø(2 n ) (cid:104) n (cid:105) k // Σ k .By [GRW18a, Corollary 1.8], in degrees i ≤ g − , the ring H ∗ ( B Diff + ( W g, ); Q ) is isomorphic to Q [ κ c | c ∈ B , | c | > n ]where B denotes the set of monomials in the classes e , p n − , p n − , . . . , p (cid:100) n +14 (cid:101) of H ∗ ( B SO(2 n )) and | κ c | = | c | − n .By the Cartan-Leray spectral sequence, we also know that H ∗ ( B Ø(2 n ) (cid:104) n (cid:105) k // Σ k ; Q ) ∼ = H ∗ ( B Ø(2 n ) (cid:104) n (cid:105) k ; Q ) Σ k the fixed points by the action of Σ k which permutes the factors of ( B Ø(2 n ) (cid:104) n (cid:105) ) k . We know that H ∗ ( B Ø(2 n ) (cid:104) n (cid:105) ; Q ) is simply the subalgebra of H ∗ ( B Ø(2 n ); Q ) = Q [ p , . . . , p n − , e ] with no gener-ators of degrees ≤ n . So H ∗ ( B Ø(2 n ) (cid:104) n (cid:105) ; Q ) ∼ = Q [ e, p (cid:100) n +14 (cid:101) , . . . , p n − ] . Then H ∗ ( B Ø(2 n ) (cid:104) n (cid:105) k ; Q ) Σ k is isomorphic to (cid:32)(cid:79) k Q [ e, p (cid:100) n +14 (cid:101) , . . . , p n − ] (cid:33) Σ k the fixed points by the action of Σ k which permutes the factors of the k -fold tensor product.Analogously, H ∗ (SO[0 , n − m // Σ m ; Q ) ∼ = H ∗ (SO[0 , n − m ; Q ) Σ m . Using the fibre sequence SO(2 n ) (cid:104) n − (cid:105) SO(2 n ) SO[0 , n − Q -module isomorphism H ∗ (SO(2 n ); Q ) ∼ = H ∗ (SO[0 , n − Q ) ⊗ H ∗ (SO(2 n ) (cid:104) n − (cid:105) ; Q ) . Together with the fact that we have a canonical ring monomorphism H ∗ (SO(2 n ) (cid:104) n − (cid:105) ; Q ) H ∗ (SO(2 n ); Q ) ECOUPLING DECORATIONS ON MODULI SPACES 31 we conclude that H ∗ (SO[0 , n − Q ) ∼ = (cid:94) [ y , . . . , y (cid:98) n − (cid:99) ]with | y i | = 4 i − H ∗ (SO(2 n )[0 , n − m ; Q ) Σ m is isomorphic to (cid:32)(cid:79) m (cid:94) [ y , . . . , y (cid:98) n − (cid:99) ] (cid:33) Σ m the fixed points by the action of Σ m which permutes the factors of the m -fold tensor product.Therefore in degrees i ≤ g − , the ring H ∗ ( B Diff + ,km ( W g, )) is isomorphic to the graded commu-tative algebra Q [ κ c | c ∈ B , | c | > n ] ⊗ (cid:32)(cid:79) m (cid:94) [ y , . . . , y (cid:98) n − (cid:99) ] (cid:33) Σ m ⊗ (cid:32)(cid:79) k Q [ e, p (cid:100) n +14 (cid:101) , . . . , p n − ] (cid:33) Σ k . References [Bau04] Tilman Bauer,
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E-mail address : [email protected]@maths.ox.ac.uk