Moduli spaces of manifolds: a user's guide
aa r X i v : . [ m a t h . A T ] M a r MODULI SPACES OF MANIFOLDS: A USER’S GUIDE
SØREN GALATIUS AND OSCAR RANDAL-WILLIAMS
Abstract.
We survey recent work on moduli spaces of manifolds with anemphasis on the role played by (stable and unstable) homotopy theory. Thetheory is illustrated with several worked examples. Introduction
The study of manifolds and invariants of manifolds was begun more than acentury ago. In this entry we shall discuss the parametrised setting: invariants of families of manifolds, parametrised by a base manifold X . The invariants we lookfor will be cohomology classes in X , characteristic classes.This article will be structured in the following way.(i) A discussion of the abstract classification theory. The main content here isthe precise definition of the kind of families we consider, and an outline of howa classifying space may be constructed. There is one such classifying space foreach pair ( W, ρ W ) consisting of a closed manifold W and a tangential structure ρ W , see §
2. The more general case where W is compact with boundary isbriefly discussed in § W, ρ W ).(iii) Statement of the main result of [GRW14, GRW18, GRW17] in their generalforms. These statements require a bit more homotopy theory to formulatebut apply quite generally, at least for even-dimensional manifolds, see thetheorems in § §§ § § H .Our theorems apply in even dimensions 2 n ≥
6, but were inspired by the break-through theorem of Madsen and Weiss [MW07] in dimension 2, building on earlierideas of Madsen and Tillmann [MT01] and Tillmann [Til97].2.
Smooth bundles and their classifying spaces
First, some conventions. By smooth manifold we shall always mean a Hausdorff,second countable topological manifold equipped with a maximal C ∞ atlas, and Mathematics Subject Classification.
Key words and phrases. characteristic classes, diffeomorphism groups, homological stability,moduli spaces. smooth map shall always mean C ∞ . We shall generally use the letter M withvarious decorations for variants of classifying spaces for families of manifolds, andthe letter F with various decorations for the functor it classifies.2.1. Smooth bundles.
Our notion of “family” of manifolds will be smooth fibrebundle , possibly equipped with extra tangent bundle structure, as follows. If V isa d -dimensional real vector bundle we shall write Fr( V ) for the associated framebundle, which is a principal GL d ( R )-bundle. Definition 2.1.
Let d be a non-negative integer.(i) A smooth fibre bundle of dimension d consists of smooth manifolds E and X (without boundary) and a smooth proper map π : E → X such that Dπ : T E → π ∗ T X is surjective and the vector bundle T π E = Ker( Dπ ) has d -dimensional fibres. The bundle T π E is called the vertical tangent bundle .(ii) If Θ is a space with a continuous action of GL d ( R ), a smooth fibre bundle with Θ -structure consists of a smooth fibre bundle π : E → X , together with acontinuous GL d ( R )-equivariant map ρ : Fr( T π E ) → Θ.Typical choices of Θ include the terminal one Θ = {∗} , and Θ = Z × = {± } on which GL d ( R ) acts by multiplication by the sign of the determinant. In theformer case a Θ-structure is no information, and in the latter it is the data of acontinuously varying family of orientations of the d -manifolds π − ( x ). (The spaceof equivariant maps Fr( T π E ) → Θ may be modelled in other ways, equivalent upto weak equivalence, see § f : X ′ → X will be transverse to any smooth fibre bundle π : E → X as above, and the pullback ( f ∗ π ) : f ∗ E → X ′ is again a smooth fibrebundle. Given a Θ-structure ρ on π , we shall write f ∗ ρ for the induced structureon ( f ∗ π ).2.2. Classifying spaces.
The natural equivalence relation between the bundlesconsidered above is concordance , which we recall.
Definition 2.2.
Let π : E → X and π : E → X be smooth bundles withΘ-structures ρ : Fr( T π E ) → Θ and ρ : Fr( T π E ) → Θ.(i) An isomorphism between ( π , ρ ) and ( π , ρ ) is a diffeomorphism φ : E → E over X , such that the induced map Fr( D π φ ) : Fr( T π E ) → Fr( T π E ) is overΘ.(ii) A concordance between ( π , ρ ) and ( π , ρ ) is a smooth fibre bundle π : E → R × X with Θ-structure ρ : Fr( T π E ) → Θ, together with isomorphismsfrom ( π , ρ ) and ( π , ρ ) to the pullbacks of ( π, ρ ) along the two embeddings X ∼ = { } × X ⊂ R × X and X ∼ = { } × X ⊂ R × X .Pulling back is functorial up to isomorphism and preserves being concordant. Definition 2.3.
For a smooth manifold X without boundary, let F Θ [ X ] denotethe set of concordance classes of pairs ( π, ρ ) of a smooth fibre bundle π : E → X with Θ-structure ρ : Fr( T π E ) → Θ. (Note that X is fixed, but E is allowed tovary.) Theorem 2.4.
The functor X
7→ F Θ [ X ] is representable in the (weak) sense thatthere exists a topological space M Θ and a natural bijection F Θ [ X ] ∼ = [ X, M Θ ] , (2.1) where the codomain denotes homotopy classes of continuous maps. ODULI SPACES OF MANIFOLDS 3
There are various ways to prove this representability statement. In the series[GRW14, GRW18, GRW17] we did this by constructing explicit point-set modelsin terms of submanifolds of R ∞ . Instead of repeating what we said there, let usoutline an approach based on simplicial sets and the observation that the functor F Θ may be upgraded to take values in spaces , and the natural bijection (2.1) maybe upgraded to a natural weak equivalence to the mapping space. Definition 2.5.
Let ∆ • e be the cosimplicial smooth manifold given by the extendedsimplices ∆ pe = { t ∈ R p +1 | t + · · · + t p = 1 } .For a smooth manifold X let F Θ • ( X ) denote the simplicial set whose p -simplicesare all pairs ( π, ρ ) consisting of a smooth fibre bundle π : E → ∆ pe × X withΘ-structure ρ : Fr( T π E ) → Θ.As they stand, these definitions are not quite rigorous: F Θ [ X ] and F Θ p ( X ) arenot (small) sets for size reasons, and F Θ p ( X ) is not functorial in [ p ] ∈ ∆ becausepullback is not strictly associative on the level of underlying sets. This may befixed in standard ways, e.g. by requiring the underlying set of E to be a subset of X × Ω for a set Ω of sufficiently large cardinality. See e.g. [MW07, Section 2.1] formore detail.
Proposition 2.6.
Let
Man and sSets be the categories of smooth manifolds andsimplicial sets, and let F Θ • : Man op → sSets be the functor defined above. Then wehave a natural bijection F Θ [ X ] ∼ = π F Θ • ( X ) and a natural weak equivalence of simplicial sets F Θ • ( X ) ≃ −→ Maps(Sing( X ) , F • ( {∗} )) . (2.2) Consequently, we may take M Θ := |F Θ • ( {∗} ) | in Theorem 2.4. In this statement we take Sing( X ) to mean the smooth singular set, i.e. thesimplicial set [ p ] C ∞ (∆ pe , X ). It is equivalent to the usual simplicial set madeout of all continuous maps ∆ p → X by a smooth approximation argument, andin particular the evaluation map | Sing( X ) | → X is a homotopy equivalence byMilnor’s theorem. The codomain of (2.2) is the simplicial set of maps into the Kancomplex F • ( {∗} ), homotopy equivalent to the space of maps X → |F • ( {∗} ) | . Proof sketch.
The first claim follows by identifying 1-simplices of F Θ • ( X ) with con-cordances between 0-simplices.For the second claim, we define the map (2.2) by pulling back. To check that itis a weak equivalence one first verifies that both sides send open covers X = U ∪ V to homotopy pullback squares and countable increasing unions to homotopy limits.Hence it suffices to check the case X = R n . Then one checks that both sides send X × R → X to a weak equivalence, so it suffices to check X = {∗} , which is obvious.The third claim follows by combining the first and the second claim and thehomotopy equivalence | Sing( X ) | → X given by evaluation. Alternatively it may bequoted directly from [MW07, Section 2.4]. (cid:3) The above proof of Theorem 2.4 gives an explicit bijection (2.1). Indeed, anelement ( π, ρ ) ∈ F ( X ) gives a morphism of simplicial sets Sing( X ) → F Θ • ( {∗} )and hence a canonical zig-zag X ev ←− | Sing( X ) | ( π,ρ ) −→ |F Θ • ( {∗} ) | = M Θ . (2.3)The map ev : | Sing( X ) | → X is a homotopy equivalence, and we may choose ahomotopy inverse. For example, a smooth triangulation of X gives such an inverse,which is even a section. The resulting homotopy class of map X → M Θ correspondsto [( π, ρ )] under the bijection (2.1). SØREN GALATIUS AND OSCAR RANDAL-WILLIAMS
For later purposes, let us explain how a universal fibration π Θ : E Θ → M Θ modelling the bundle π : E → X may be constructed by the same simplicial method.Let e F Θ • ( X ) be the simplicial set whose p -simplices are triples ( π, ρ, s ) where ( π, ρ )is as before and s : ∆ pe × X → E is a section of π , and set E Θ := | e F Θ • ( {∗} ) | . Thezig-zag (2.3) above associated to an element ( π, ρ ) ∈ F Θ0 ( X ) now extends to acanonical diagram E | Sing( E ) | E Θ X | Sing( X ) | M Θ π ≃ π Θ ≃ ( π,ρ ) in which both squares are homotopy cartesian, and the top right-hand map is thatassociated to the data (pr : E × X E → E, ρ ◦ D pr , diag : E → E × X E ).Finally, a p -simplex ( π : E → ∆ pe , ρ, s ) of E Θ gives a map ℓ = ( ρ/ GL d ( R )) : E ∼ =(Fr( T π E ) / GL d ( R )) → (Θ / GL d ( R )). Composing with the section s | ∆ p : ∆ p → E then gives rise to a map of simplicial sets e F Θ • ( {∗} ) −→ Sing(Θ / GL d ( R ))realising to a map E Θ → | Sing(Θ / GL d ( R )) | . The orbit space Θ / GL d ( R ) need notbe well behaved, and we would like to replace it by the homotopy orbit space B = Θ // GL d ( R ) := ( E GL d ( R ) × Θ) / GL d ( R ). Taking homotopy orbits of the mapΘ → {∗} yields a map θ : B → B GL d ( R ). Repeating the construction above with E GL d ( R ) × Θ instead of Θ allows us to construct a zig-zag E Θ ≃ ←− E E GL d ( R ) × Θ ℓ −→ | Sing( B ) | ev −→ B θ −→ B GL d ( R ) . Slightly less precisely, we shall summarise this situation as a diagram E E Θ B B GL d ( R ) X M Θ π π Θ ℓ θ (2.4)where the square is homotopy cartesian. The vector bundle classified by the com-position θ ◦ ℓ : E Θ → B GL d ( R ) is a universal instance of T π E , and shall be denoted T π E Θ . The factorisation through θ gives an equivariant map ρ Θ : Fr( T π E Θ ) → Θ,a universal instance of ρ : Fr( T π E ) → Θ.2.3.
Path connected classifying spaces.
The bundles classified so far are incon-veniently general. For example, we have not made any restrictions on the diffeomor-phism type of the fibres of π : E → X . Hence, if e.g. Θ = {∗} , the set π ( M Θ ) isin bijection with the set of diffeomorphism classes of compact smooth d -manifoldswithout boundary, a countably infinite set for any d ≥
0. The homotopy type of M Θ then encodes at once the classification of smooth manifolds up to diffeomor-phism and the classification of smooth bundles, because it is the “moduli space”(or “ ∞ -groupoid”) of all d -manifolds.We shall often study one path component of M Θ at a time, which correspondsto fixing the concordance class of the fibres of the classified bundles. We introducethe following notation. Definition 2.7.
Let Θ be a space with GL d ( R ) action, W be a compact d -manifoldwithout boundary, and ρ W : Fr( T W ) → Θ be an equivariant map. Consider-ing this as a family over a point yields a [(
W, ρ W )] ∈ π M Θ , and we shall write M Θ ( W, ρ W ) ⊂ M Θ for the path component containing ( W, ρ W ). This path com-ponent is a classifying space for smooth fibre bundles π : E → X with structure ρ : Fr( T π E ) → Θ, whose restriction to any point { x } ⊂ X is concordant to ( W, ρ W ). ODULI SPACES OF MANIFOLDS 5
Remark . In the special case Θ = {∗} the maps ρ : Fr( T π E ) → Θ are irrelevant,and in this case we shall write simply M ( W ). This space classifies smooth bundles π : E → X with fibres diffeomorphic to W with no further structure, and we havethe weak equivalence M ( W ) ≃ B Diff( W ) , where Diff( W ) is the diffeomorphism group of W in the C ∞ topology. Similarly,if Θ = Z × with the orientation action and W is given an orientation ρ W , we shallwrite M or ( W ) for M Θ ( W, ρ W ) and there is a weak equivalence M or ( W ) ≃ B Diff or ( W ) , where Diff or ( W ) ⊂ Diff( W ) is the subgroup of orientation preserving diffeomor-phisms. For general Θ, the homotopy type is described as the Borel construction M Θ ( W, ρ W ) ≃ { ρ : Fr( T W ) → Θ, equivariantly homotopic to ρ W } (cid:16) topological group of diffeomorphisms W → W preserving the equivariant homotopy class of ρ W (cid:17) . See [GRW14, Definition 1.5], [GRW14, Section 1.1], or [GRW17, Section 1.1] forfurther discussion of this point of view.3.
Characteristic classes
The main topic of this article is the study of characteristic classes of the sort ofbundles described above, i.e. the calculation of the cohomology ring of the classifyingspaces M Θ ( W, ρ W ). We first recall the conclusions in rational cohomology, whichare easier to state and often gives an explicit formula for the ring of characteristicclasses.3.1. Characteristic classes.
As before, let B = Θ // GL d ( R ) denote the Borelconstruction. For a smooth bundle π : E → X with Θ-structure ρ : Fr( T π E ) → Θ,we may form the Borel construction with GL d ( R ) and obtain a correspondence, i.e.a diagram X π ←− E ≃ ←− Fr( T π E ) // GL d ( R ) ρ// GL d ( R ) −−−−−−→ B. (3.1)We write ℓ : E → B for the homotopy class of maps associated to ρ// GL d ( R ). Weshall let Z w denote the coefficient system on B arising from the non-trivial actionof π (GL d ( R )) = Z × on Z , and let A w = A ⊗ Z w for an abelian group A , and weshall use the same notation for these coefficient systems pulled back to E along (3.1).Then we have a homomorphism ℓ ∗ : H k + d ( B ; A w ) −→ H k + d ( E ; A w ) . (3.2)We also have a fibre integration homomorphism Z π : H k + d ( E ; A w ) −→ H k ( X ; A ) , (3.3)where d is the dimension of the fibres of π . It may be defined e.g. as the composition H k + d ( E ; A w ) ։ E k,d ∞ ⊂ E k,d in the Serre spectral sequence for the fibration π , orby a Pontryagin–Thom construction as in § c ∈ H d + k ( B ; A w ), we may combine (3.2) and (3.3) and define the Miller–Morita–Mumford classes (or MMM-classes) by the push-pull formula κ c ( π ) = Z π ℓ ∗ c ∈ H k ( X ; A ) . This class is easily seen to be natural with respect to pullback along smooth maps X ′ → X , and in fact comes from a universal class κ c ∈ H k ( M Θ ( W, ρ W ); A ) SØREN GALATIUS AND OSCAR RANDAL-WILLIAMS for any (
W, ρ W ), any A , and any c ∈ H k + d ( B ; A w ), defined by the analogouspush-pull formula in the universal instance (2.4).In the special case k = 0 and X = {∗} , the definition of κ c ∈ H ( X ; A ) = A simply reproduces the usual characteristic numbers , c.f. [MS74, § d = 2 n and e comes from the Euler class in H d ( B GL d ( R ); Z w ), then κ e ∈ H ( X ; Z ) = Z is the Euler characteristic of W . Henceforth we shall mostly beinterested in the case k > A = k is a field, we may combine with cup product to get a map of gradedrings k [ κ c | c ∈ basis of H >d ( B ; k w )] −→ H ∗ ( M Θ ( W, ρ W ); k ) , (3.4)whose domain is the free graded-commutative k -algebra on symbols κ c , one foreach element c in a chosen basis (or more invariantly, on a degree-shifted copy ofthe graded k -vector space H >d ( B ; k w )).3.2. Genus.
All of our results will hold in a range of degrees depending on genus ,a numerical invariant that we first introduce.Assume that d = 2 n and let a GL n ( R )-space Θ be given. We shall assumethat n > // GL n ( R ) is connected, i.e. that π (GL n ( R )) = Z × acts transitively on π (Θ).The genus will be defined in terms of the manifold obtained from S n × S n byremoving a point, which plays a special role in this theory. Up to diffeomorphismthis manifold may be obtained as a pushout S n × R n ← ֓ R n × R n ֒ → R n × S n , (3.5)where the embeddings are induced by a choice of coordinate chart R n ֒ → S n . Fol-lowing [GRW18, Definition 1.3] a Θ-structure on S n × R n shall be called admissible if it “bounds a disk”, i.e. is (equivariantly) homotopic to a structure that extendsover some embedding S n × R n ֒ → R n . Note that this is automatic if π n (Θ) = 0for some basepoint. A structure on S n × S n \ {∗} is admissible if the restriction toeach piece of the gluing (3.5) is admissible. Definition 3.1.
Assume d = 2 n > W is connected. The genus g ( W, ρ W )of a Θ-manifold ( W, ρ W ) is the maximal number of disjoint embeddings j : S n × S n \ {∗} → W such that j ∗ ρ W is admissible.This appeared as [GRW18, Definition 1.3]. For example, when n = 1 and Θ = π (GL ( R )) = Z × , this is precisely the usual genus of an oriented connected 2-manifold. The admissibility condition may be illustrated by the case Θ = GL ( R ),corresponding to framings on 2-manifolds. The Lie group framing on Σ = S × S satisfies g (Σ , ρ Lie ) = 0, but there exist other framings ρ for which g (Σ , ρ ) = 1.In § g ( W, ρ W ) in terms of more accessible invariants.3.3. Main theorem in rational cohomology.
The main results of [GRW14,GRW18, GRW17] imply that the ring homomorphism homomorphism (3.4) is oftenan isomorphism in a range of degrees when k = Q . We explain the statement. Definition 3.2.
Assume Θ // GL n ( R ) is connected. We say Θ is spherical if S n admits a Θ-structure, i.e. if there exists an equivariant map Fr( T S n ) → Θ.The condition is equivalent to existence of an O(2 n )-equivariant map O(2 n +1) → Θ. This obviously holds if the GL n ( R )-action on Θ admits an extension to an actionof GL n +1 ( R ) which holds in many cases, e.g. Θ = {± } with the orientation action.See [GRW14, Section 5.1] for more information about this condition. ODULI SPACES OF MANIFOLDS 7
Theorem 3.3.
Let d = 2 n > , W be a closed simply-connected d -manifold, and ρ W : Fr( W ) → Θ be a Θ -structure which is n -connected (or, equivalently, such thatthe associated ℓ W : W → B = Θ // GL d ( R ) is n -connected). Equip B with the localsystem Q w as above, and assume that H k + d ( B ; Q w ) is finite dimensional for each k ≥ . Then the ring homomorphism Q [ κ c | c ∈ basis of H >d ( B ; Q w )] −→ H ∗ ( M Θ ( W, ρ W ); Q ) , as in (3.4) is an isomorphism in cohomological degrees ≤ ( g ( W, ρ W ) − / . If inaddition Θ is spherical, then the range may be improved to ≤ ( g ( W, ρ W ) − / . Estimating genus.
To apply Theorem 3.3 we must calculate the invariant g ( W, ρ W ), or at least be able to give useful lower bounds for it. This section explainsa general such lower bound, under the assumptions that d = 2 n >
4, and that thehomotopy orbit space B = Θ // GL d ( R ) is simply-connected. We may then choosean equivariant map Θ → Z × by which any Θ-structure induces an orientation, andwe shall assume given such a map.There is an obvious upper bound for g ( W, ρ W ): it is certainly no larger than thenumber of hyperbolic summands in H n ( W ; Z ) equipped with its intersection form.For odd n this is in turn no more than b n / n it is no more thanmin( b + n , b − n ), where we write b n for the middle Betti number of W and in the evencase, b n = b + n + b − n for its splitting into positive and negative parts. More usefully,[GRW18, Remark 7.16] gives the following lower bound on genus. Theorem 3.4.
Assume d = 2 n > , that the homotopy orbit space B = Θ // GL d ( R ) is simply-connected, and that ℓ W : W → B (or equivalently ρ W : Fr( T W ) → Θ ) is n -connected. Write g a ( W ) = min( b + n , b − n ) for n even and g a ( W ) = b n / for n odd.Then g a ( W ) − c ≤ g ( W, ρ W ) ≤ g a ( W ) , (3.6) with c = 1 + e , where e is the minimal number of generators of the abelian group H n ( B ; Z ) . If n is even or if n ∈ { , } then one may take c = e .Remark . Let us briefly point out that the estimate (3.6) may be expressed usingcharacteristic numbers. Indeed, writing b i = b i ( B ) = b i ( W ) for i = 1 , . . . , n −
1, wehave g a ( W ) = ( − n (cid:0) χ ( W ) / − n − X i =0 ( − i b i (cid:1) − | σ ( W ) | / , (3.7)where χ ( W ) = R W e ( T W ) is the Euler characteristic and σ ( W ) = R W L ( T W ) is thesignature (where we write σ ( W ) = 0 when n is odd).4. General versions of main results
For some purposes, the rational cohomology statement in Theorem 3.3 suffices,but there are several homotopy theoretic strengthenings and variations given in[GRW17], which we now explain.4.1.
Stable homotopy enhancement.
To state a more robust version of Theo-rem 3.3 above, we must first introduce a space Ω ∞ M T
Θ associated to the GL d ( R )-space Θ. The map B = Θ // GL d ( R ) → B GL d ( R ) classifies a d -dimensional realvector bundle over B , and we shall write M T
Θ for the Thom spectrum of its vir-tual inverse and Ω ∞ M T
Θ for the corresponding infinite loop space. The followingresult is a restatement of [GRW17, Corollary 1.7].
Theorem 4.1.
Let d = 2 n > , W be a closed simply-connected d -manifold, and ρ W : Fr( W ) → Θ be an n -connected equivariant map. Then there is a map α : M Θ ( W, ρ W ) −→ Ω ∞ M T Θ , (4.1) SØREN GALATIUS AND OSCAR RANDAL-WILLIAMS inducing an isomorphism in integral homology onto the path component that it hits,in degrees ≤ ( g ( W, ρ W ) − / .If in addition Θ is spherical, then (4.1) induces an isomorphism in homology indegrees ≤ ( g ( W, ρ W ) − / . The statement proved in [GRW17] is in fact stronger: the map α induces anisomorphism in homology with local coefficients in degrees up to ( g ( W, ρ W ) − / acyclic in this range of degrees. (The induced map in π isof course far from an isomorphism.) Remark . Let us also briefly recall why Theorem 3.3 is a consequence of Theo-rem 4.1: under the Thom isomorphism H k + d ( B ; Q w ) ∼ = H k ( M T Θ; Q ) each class c ∈ H k + d ( B ; Q w ) may be represented by a spectrum map M T Θ → Σ k H Q . If wechoose a rational basis B k ⊂ H k + d ( B ; Q w ) ∼ = H k ( M T Θ; Q ) and represent eachbasis element by a spectrum map M T Θ → Σ k H Q , we obtain M T Θ −→ ∞ Y k =1 Y c ∈B k Σ k H Q which induces isomorphisms in rational homology and hence in rationalised homo-topy in positive degrees, at least if each B k is finite in which case the product inthe codomain may be replaced by the wedge. It follows that the induced map ofinfinite loop spaces Ω ∞ M T Θ −→ ∞ Y k =1 Y B k K ( Q , k )induces an equivalence in rationalised homotopy groups in positive degrees, hencein rational cohomology when restricted to any path component of its domain.4.2. MMM-classes in generalised cohomology.
There is a preferred map (4.1)in Theorem 4.1. Following [MT01], in [GRW14, Remark 1.11] we gave an explicitpoint-set model. Here we shall explain the map in a conceptual but somewhatinformal way. It is obtained from the following two ingredients.(i) A smooth d -manifold W and an equivariant map ρ W : Fr( T W ) → Θ inducesa continuous map ℓ W = ρ W // GL d ( R ) : W ≃ Fr(
T W ) // GL d ( R ) −→ B = Θ // GL d ( R )under which the canonical bundle γ on B GL d ( R ) = ∗ // GL d ( R ) is pulled backto T W . By passing to Thom spectra of inverse bundles one gets a map W − T W −→ B − γ = M T Θ , (4.2)in the stable homotopy category.(ii) If W is a closed manifold there is a canonical map S −→ W − T W (4.3)which is Spanier–Whitehead dual to the canonical map W → {∗} under Atiyahduality D ( W + ) ≃ W − T W . The actual map of spectra depends on certainchoices, but in the right setup these choices form a contractible space. (Forexample, one may choose an embedding
W ֒ → R ∞ and get the map (4.3) bythe Pontryagin–Thom collapse construction.)If W is a smooth closed d -manifold and ρ W : Fr( T W ) → Θ is an equivariant map,we may compose (4.3) and (4.2) to get a map of spectra S → M T
Θ, i.e. a pointin Ω ∞ M T
Θ. We shall write α ( W, ρ W ) ∈ Ω ∞ M T
Θ for this point, and writeΩ ∞ [ W,ρ W ] M T Θ ⊂ Ω ∞ M T Θ ODULI SPACES OF MANIFOLDS 9 for the path component containing α ( W, ρ W ). This is the path component that themap (4.1) lands in.The map (4.1) is given by a parametrised version of this construction: given afamily ( π : E → X, ρ : Fr( T π E ) → Θ) as in Definition 2.1 over some base manifold X , it associates a continuous map α : X → Ω ∞ M T
Θ. Said differently, it comesfrom a composition of spectrum maps, the parametrised analogues of (4.3) and (4.2)respectively, Σ ∞ + X −→ E − T π E E − T π E −→ M T Θ . (4.4)In any case, (4.1) is a universal version of this construction. Remark . Applying spectrum homology and the Thom isomorphism to the firstmap in (4.4), we get precisely the fibre integration homomorphism (3.3), while thesecond gives (3.2). This explains the connection to the characteristic classes in § Remark . There is an improvement to Theorem 4.1, in which the domain of (4.1)is replaced by a disconnected space M Θ n containing M Θ ( W, ρ W ) as one of its pathcomponents, cf. § g ( W, ρ W ) ∈ N appearing in Theorem 4.1 is replaced by a function π ( M Θ n ) α ∗ −→ π ( M T Θ) ¯ g Θ −→ Z , whose value on the path component M Θ ( W, ρ W ) is the stable genus , defined in[GRW18, Section 5] and [GRW17, Section 1.3]. It is at least g ( W, ρ W ).If we write χ : π ( M T Θ) → Z and σ : π ( M T Θ) → Z be the homomorphismsarising from the Euler class and (for even n ) the Hirzebruch class, then (3.7) definesa function g a : π ( M T Θ) → N . In terms of this function, the estimate (3.6) alsoholds for ¯ g Θ .4.3. General tangential structures.
The requirement in Theorems 3.3 and 4.1that ρ W : Fr( T W ) → Θ be n -connected appears quite restrictive at first sight. Forexample, it usually rules out the interesting special cases Θ = {∗} and Θ = {± } from Remark 2.8, so that the cohomology of B Diff( W ) and B Diff or ( W ) are notimmediately calculated by Theorem 4.1.A more generally useful version of Theorem 4.1, which holds without the con-nectivity assumption, may be deduced by a rather formal homotopy theoretic trick,based on the observation that the map (4.1) is functorial in the GL d ( R )-space Θ.In particular, any map Θ → Θ induces a self-map of Ω ∞ M T
Θ. The followingresult is [GRW17, Corollary 1.9].
Theorem 4.5.
For d = 2 n > and Λ a GL d ( R ) -space, let W be a closed simply-connected smooth d -dimensional manifold, and λ W : Fr( T W ) → Λ be an equivariantmap. Choose an equivariant Moore–Postnikov n -stage λ W : Fr( T W ) ρ W −→ Θ u −→ Λ , (4.5) i.e. a factorisation where u is n -co-connected equivariant fibration and ρ W is an n -connected equivariant cofibration, and write hAut( u ) for the group-like topologicalmonoid consisting of equivariant weak equivalences Θ → Θ over Λ .This topological monoid acts on Ω ∞ M T Θ , and there is a continuous map α : M Λ ( W, λ M ) −→ (Ω ∞ M T Θ) // hAut( u ) . (4.6) which, when regarded as a map onto the path component that it hits, induces anisomorphism in homology with local coefficients in degrees ≤ ( g ( W, λ W ) − / ,and if Θ is spherical it induces an isomorphism in integral homology in degrees ≤ ( g ( W, λ W ) − / . We emphasise that the homotopy orbit space is formed in the category of spaces,not of infinite loop spaces (there is a comparison map (Ω ∞ M T Θ) // hAut( u ) → Ω ∞ (( M T Θ) // hAut( u )) but it is not a weak equivalence and we shall not need itscodomain). Equivariant factorisations (4.5) always exist, and are unique up tocontractible choice.Let us also point out that the group π (hAut( u )) likely acts non-trivially on π ( M T
Θ). The construction of the map (4.6), outlined in § Corollary 4.6.
Let d , Λ , W , λ W , Θ , ρ W , and u be as in Theorem 4.5, and write hAut( u ) [ W,ρ W ] ⊂ hAut( u ) for the submonoid stabilising the element [ W, ρ W ] ∈ π ( M T Θ) defined in § Ω ∞ M T Θ restricts to an action on the path component Ω ∞ [ W,ρ W ] M T Θ defined in § α : M Λ ( W, λ W ) −→ (Ω ∞ [ W,ρ W ] M T Θ) // (hAut( u ) [ W,ρ W ] ) which induces an isomorphism on homology in a range, as in Theorem 4.5. (cid:3) Remark . In both Theorem 4.5 and Corollary 4.6 above, the range is expressedin terms of g ( W, λ W ) but as explained in [GRW17, Lemma 9.4] this is equal to g ( W, ρ W ) when λ W is factored equivariantly as an n -connected map ρ W : Fr( T W ) → Θ followed by an n -co-connected map u : Θ → Λ. Hence the estimates in § e is the minimal number of generators for the abelian group H n (Θ // GL n ( R )).The value of e likely depends on the map λ W : Fr( T W ) → Λ, even if W and Λ arefixed. Remark . The fact that u is n -co-connected implies that hAut( u ) is an ( n − u ):finitely many homotopy groups π , . . . , π n − and finitely many k -invariants.This finiteness is one of the conceptual advantages of our approach to M ( W ) ≃ B Diff( W ), over the more classical method which at first gives a formula for the structure space S ( W ) ≃ G ( W ) / g Diff( W ), where G ( W ) is the monoid of homotopyequivalences from W to itself and g Diff( W ) is the block diffeomorphism group. Inthat method, one subsequently has to study the difference between Diff( W ) and ^ Diff( W ), but also has to take homotopy orbits by the monoid G ( W ) of homotopyautomorphisms of W . While this last step is in some sense “purely homotopytheory”, it is in practice very difficult to get a good handle on G ( W ), even when W is relatively simple and even when one is working up to rational equivalence. Seethe work of Berglund and Madsen [BM13, BM14] for a recent example.4.4. Two fibre sequences.
In [GRW17, Section 9], Theorem 4.5 is deduced fromthe special case given in Theorem 4.1 by a rather formal argument: the homologyequivalence in Theorem 4.1 is natural in the GL n ( R )-space Θ, and hence inducesa homology equivalence by taking homotopy colimit over any diagram in GL n ( R )-spaces; in particular one may form homotopy orbits by hAut( u : Θ → Λ), andTheorem 4.5 is deduced by identifying the resulting map of homotopy orbit spaceswith (4.6). The two fibre sequences arising from these homotopy orbit constructions,see diagram (4.8) below, are important for carrying out calculations in concreteexamples, and hence we recall this story in slightly more detail.
Definition 4.9.
Let u : Θ → Λ be an equivariant n -co-connected fibration. ODULI SPACES OF MANIFOLDS 11 (i) Let M Θ n ⊂ M Θ be the union of those path components M Θ ( W, ρ W ) for which ρ W : Fr( T W ) → Θ is n -connected.(ii) Let M Λ u ⊂ M Λ be the union of those path components M Λ ( W, λ W ) forwhich λ W admits a factorisation through an n -connected equivariant map ρ W : Fr( T W ) → Θ.There is an obvious forgetful map M Θ n → M Λ u given by composing ρ W : Fr( T W ) → Θ with u . The monoid hAut( u ) has the correct homotopy type when Θ is equiv-ariantly cofibrant, which we shall assume. It acts on M Θ n by postcomposing ρ W : Fr( T W ) → Θ with self-maps of Θ. This action commutes with the forgetfulmap, and induces a map ( M Θ n ) // hAut( u ) −→ M Λ u . (4.7)The following lemma is proved by elementary homotopy theoretic methods, cf.[GRW17, Section 9]. Lemma 4.10.
The map (4.7) is a weak equivalence. Hence there is an inducedfibre sequence of the form M Θ n −→ M Λ u −→ B (hAut( u )) . The map M Θ n → Ω ∞ M T
Θ explained in § u ) and induces a map of fibre sequences M Θ n M Λ u B (hAut( u ))Ω ∞ M T
Θ (Ω ∞ M T Θ) // hAut( u ) B (hAut( u )) . (4.8)In the setup of Theorem 4.5, M Θ ( W, ρ W ) is one path component of M Θ n , andsimilarly M Λ ( W, λ W ) is one path component of M Λ u . A slightly stronger versionof Theorem 4.1, which is the statement actually proved in [GRW17, Section 8],shows that the left-most vertical map is acyclic in the range of degrees indicatedin Remark 4.4. Theorem 4.5 is then deduced by a spectral sequence comparisonargument. Remark . This formulation has content even at the level of path components.Suppose that W is a simply-connected 2 n -manifold for 2 n ≥ ρ W : Fr( T W ) → Θis n -connected, and g ( W, ρ W ) ≥
3. If ( W ′ , ρ W ′ ) is another such Θ-manifold and α ( W, ρ W ) = α ( W ′ , ρ W ′ ) ∈ π (Ω ∞ M T
Θ) = π ( M T
Θ)then it follows that (
W, ρ W ) and ( W ′ , ρ W ′ ) lie in the same path-component of M Θ n ,i.e. there is a diffeomorphism from W to W ′ which pulls back ρ W ′ to ρ W up tohomotopy. This recovers a theorem of Kreck [Kre99, Theorem D], though ourrequirement on genus is slightly stronger than Kreck’s.In practice, one usually calculates the cohomology of Ω ∞ [ W,ρ W ] M T
Θ first, andthen uses a spectral sequence to calculate the homology or cohomology of the Borelconstruction in Corollary 4.6, or equivalently (for g ( W, λ W ) ≥
3) one calculatesthe cohomology of M Θ ( W, ρ W ) and then uses the spectral sequence for the fibresequence M Θ ( W, ρ W ) −→ M Λ ( W, λ W ) −→ B (hAut( u ) [ W,ρ W ] ) . (4.9)We shall see examples of such calculations in § § § d ( R ) -spaces versus spaces over B O( d ) . It is well known that the homo-topy theory of spaces with action of GL d ( R ) is equivalent to the homotopy theoryof spaces over B GL d ( R ) ≃ B O( d ), where the weak equivalences are the equivari-ant maps that are weak equivalences of underlying spaces, respectively fibrewisemaps that are weak equivalences of underlying spaces. The translation goes viathe space E GL d ( R ) which simultaneously comes with an action of GL d ( R ) anda map to B GL d ( R ). Explicitly, given a GL d ( R )-space Θ, the Borel construction B = Θ // GL d ( R ) comes with a map B → B GL d ( R ); conversely, given a space B anda map θ : B → B GL d ( R ) the fibre product Θ = E GL d ( R ) × B GL d ( R ) B comes with anaction; these processes are inverse up to (equivariant/fibrewise) weak equivalence,as E GL d ( R ) is contractible.Therefore all of the theorems above that depend on a GL d ( R )-space Θ may bestated in equivalent ways taking as input a space B and a map θ : B → B O( d ).In the papers [GRW14, GRW18, GRW17] we have taken the latter point of view.In this picture, a θ -structure on a manifold W is a (fibrewise linear) vector bundlemap ˆ ℓ W : T W → θ ∗ γ , where γ denotes the universal vector bundle on B GL d ( R ).As in those papers, we shall use the notation M T θ = M T Θ , when θ : B → B O( d ) is the map corresponding to the GL d ( R )-space Θ; i.e., M T θ = B − θ is the Thom spectrum of the virtual inverse of the vector bundle classified by θ : B → B O( d ).We have already seen definitions which may be stated more directly in termsof ( B, θ ) than of (Θ , action), e.g. the characteristic classes κ c from § λ W as in the theorem, set B ′ = Λ // GL d ( R ). Up to contractible choice, λ W induces a map W → B ′ , which one then Moore–Postnikov factors as W −→ B −→ B ′ , into an n -connected cofibration followed by an n -co-connected fibration. In thispicture, hAut( u ) is simply the group-like monoid of those self-maps of B over B ′ that are weak equivalences. For this to have to the correct homotopy type B shouldbe fibrant and cofibrant in the category of spaces over B ′ .In § § Boundary.
A further generalisation, also proved in [GRW17, Section 9], al-lows the compact manifolds W to have non-empty boundary. The boundary shouldthen be a closed (2 n − P , which should be equipped with an equivariantmap ρ P : Fr( ε ⊕ T P ) → Θ. The pair (
P, ρ P ) should be fixed and every compact 2 n -manifold in sight should come with a specified diffeomorphism ∂W ∼ = P compatiblewith a structure ρ W : Fr( T W ) → Θ.In terms of classified bundles as in § § F Θ • should be replaced with thefunctor F Θ ,P,ρ P • whose value on a smooth manifold X (without boundary, possiblynon-compact) has 0-simplices the smooth proper maps π : E → X equipped withequivariant maps ρ : Fr( T π E ) → Θ and a diffeomorphism ∂E = X × P such thatthe restriction of ρ to Fr( T π E | ∂E ) = X × Fr(
T P ) is equal to the map arising from ρ P .This kind of bundle also admits a classifying space, denoted N Θ ( P, ρ P ) andcalled the moduli space of null bordisms of ( P, ρ P ). The subspace defined by thecondition that ρ W : Fr( T W ) → Θ be n -connected is denoted N Θ n ( P, ρ P ) and is the moduli space of highly connected null bordisms . The path component of N Θ ( P, ρ P )containing ( W, ρ W ) shall be denoted M Θ ( W, ρ W ), as before. Notice that ( P, ρ P ) is ODULI SPACES OF MANIFOLDS 13 determined by P = ∂W and ρ W by restricting ρ W to T W | P ∼ = ε ⊕ T P . These clas-sifying spaces are introduced in [GRW17, Definition 1.1] using a similar notation.Theorem 4.1 then has the following direct generalisation, also stated as [GRW17,Corollary 1.8 and Section 8.4].
Theorem 4.12.
Let d = 2 n > , let Θ be a GL d ( R ) -space, let P be a closed smooth ( d − -manifold and ρ P : Fr( ε ⊕ T P ) → Θ be a GL d ( R ) -equivariant map. Thenthere is a map (canonical up to homotopy, see below) α : N Θ n ( P, ρ P ) −→ Ω α ( P,ρ P ) , (Ω ∞− M T Θ) , (4.10) where Ω α ( P,ρ P ) , denotes the space of paths starting at a certain point α ( P, ρ P ) ∈ Ω ∞− M T Θ and ending at the basepoint, with the following property.When restricted to the path component containing a particular ( W, ρ W ) , it is ahomology equivalence onto the path component it hits, in degrees up to ( g ( W, ρ W ) − / and possibly with twisted coefficients. If in addition Θ is spherical, then (4.10)induces an isomorphism in homology with constant coefficients in degrees up to ( g ( W, ρ W ) − / . Both the point α ( P, ρ P ) and the map (4.10) are constructed by the procedurein § ∞ M T
Θ, but the map most naturally takes values in thepath space. In the special case P = ∅ we have N Θ n ( ∅ ) = M Θ n , and the map (4.10)is the same as the map appearing in (4.8).As in § u rel P ) = { φ ∈ hAut( u ) | φ ◦ ρ P = ρ P } , (4.11)provided ρ P : Fr( ε ⊕ T P ) → Θ is an equivariant cofibration and u : Θ → Λ is anequivariant n -co-connected fibration, as can be arranged. Homotopy equivalently,factor the induced map W → B ′ = Λ // GL d ( R ) as an n -connected cofibration W → B followed by an n -co-connected fibration B → B ′ , and define hAut( u rel P ) as thehomotopy equivalences of B over B ′ and under P . We formulate the conclusion. Theorem 4.13.
Let n , d , Λ and λ W : Fr( T W ) → Λ be as in Theorem 4.5, butallow W to be a compact manifold with boundary P = ∂W . Let Θ , ρ W , and u beas in Theorem 4.5, and ρ P denote the restriction of ρ W to P . Then there is a map α : M Λ ( W, λ W ) −→ (cid:0) Ω α ( P,ρ P ) , Ω ∞− M T Θ (cid:1) // hAut( u rel P ) (4.12) which, when regarded as a map onto the path component that it hits, induces anisomorphism in homology in a range of degrees, exactly as in Theorem 4.5. This is [GRW17, Theorem 9.5], and the three lines following its proof. Therelevant path component may again be re-written using the orbit-stabiliser theorem,as in Corollary 4.6.
Remark . As in Remark 2.8, in the special case Θ = {∗} a Θ-structure containsno information and we can simply write M ( W ) for M Θ ( W, ρ W ). This space classi-fies smooth fibre bundles with fibres diffeomorphic to W and trivialised boundary,and we have a weak equivalence M ( W ) ≃ B Diff ∂ ( W )where Diff ∂ ( W ) denotes the group of diffeomorphisms of W which fix an openneighbourhood of the boundary, with the C ∞ topology.There are no connectivity assumptions imposed on ρ P : Fr( ε ⊕ T P ) → Θ, but ifit happens to be ( n − u rel P ) is contractible.More generally we have the following. Lemma 4.15.
If the pair ( W, P ) is c -connected for some c ≤ n − , then the monoid hAut( u rel P ) is a (non-empty) ( n − c − -type. In particular, it is contractible if ( W, P ) is ( n − -connected. A familiar special case of this observation is the fact that if a diffeomorphism ofan oriented surface with non-empty boundary is the identity on the boundary, thenthe diffeomorphism is automatically orientation preserving.
Proof.
As before, let us write B = Θ // GL d ( R ) and B ′ = Λ // GL d ( R ). We are thenasking for homotopy automorphisms of B over u : B → B ′ and under ℓ P : P ֒ → B .By adjunction, to give a nullhomotopy of a map f : S k → hAut( u rel P ) is to solvethe relative lifting problem( S k × B ) ∪ S k × P ( D k +1 × P ) BD k +1 × B B B ′ . ˜ f ∪ ( ℓ P ◦ proj) u proj u The map ℓ P : P → B is c -connected because both P ⊂ W and ℓ W : W → B are(the latter is even n -connected). Thus the pair( D k +1 × B, ( S k × B ) ∪ S k × P ( D k +1 × P ))is ( c + k + 1)-connected. But the map u : B → B ′ is n -co-connected, so there areno obstructions to solving this lifting problem if c + k + 1 ≥ n , i.e. k ≥ n − c − u rel P ) is an ( n − c − (cid:3) Remark . Formulating a statement which is valid for manifolds with non-emptyboundary is not purely for the purpose of added generality: it is essential for thestrategy of proof in all three papers [GRW14], [GRW18], [GRW17]. For example,the homological stability results in [GRW18] are proved by a long handle induction argument, in which a compact manifold is decomposed into finitely many handleattachments ; even if one is mainly interested in closed manifolds, this process willcreate boundary. Similarly, an important role in both [GRW14] and [GRW17] isplayed by cobordism categories as studied in [GMTW09], whose morphisms aremanifolds with boundary and composition is gluing along common boundary com-ponents. For example, given a Θ-cobordism (
K, ρ K ) : ( P, ρ P ) ( Q, ρ Q ) there is acontinuous map ( K, ρ K ) ∪ − : N Θ ( P, ρ P ) −→ N Θ ( Q, ρ Q )given by gluing on ( K, ρ K ).4.7. Fundamental group.
The main theorem in either of the three forms givenabove (Theorems 3.3, 4.1, and 4.5) assumed the manifolds W were simply-connected,but in fact it suffices that the fundamental groups π = π ( W, w ) be virtually poly-cyclic , i.e. has a subnormal series with finite or cyclic quotients. In this case the
Hirsch length h ( π ) is the number of infinite cyclic quotients in such a series. Theonly price to pay is that the ranges of homology equivalence become offset by a con-stant depending on h ( π ): the homology isomorphisms Theorems 4.1 and 4.5 hold indegrees ≤ ( g ( W, ρ W ) − ( h ( π ) + 5)) / ≤ ( g ( W, ρ W ) − ( h ( π ) + 6)) / π there is a sense in which the theorems hold in “infinite genus”:certain maps become acyclic after taking a colimit over forming connected sumwith S n × S n infinitely many times. In this form the assumption 2 n > ODULI SPACES OF MANIFOLDS 15
Outlook.
We have attempted to give an overview of the methods developedin [GRW14], [GRW18], [GRW17], with an emphasis on the main results from thereas they may be applied in calculations in practice. This is by no means a survey ofeverything known, let us briefly mention some recent developments and applicationsthat we have not covered:(i) These results—in the form of the calculation described in Theorem 5.1 below—have been used by Weiss [Wei15] to prove that p n = e ∈ H n ( B STop(2 n ); Q )for large enough n . These methods were later used by Kupers [Kup16] toestablish the finite generation of homotopy groups of Diff ∂ ( D d ) for d = 4 , , n -sphere then B Diff( M ) and B Diff( M Z [ k ]-coefficients, where k is the order of Σ is the groupof homotopy spheres.(iv) Progress towards a similar understanding for manifolds of odd dimension hasbeen made by Perlmutter [Per16a, Per16b, Per18], Botvinnik and Perlmutter[BP17] and Hebestreit and Perlmutter [HP16].(v) Progress towards versions for topological and piecewise linear manifolds hasbeen made by Gomez-Lopez [GL16], Kupers [Kup15], and Gomez-Lopez andKupers [GLK18].(vi) Progress towards versions for equivariant smooth manifolds has been made byGalatius and Sz˝ucs [GS18].(vii) There are analogues of many of the theorems above, when the topologicalgroup Diff( W ) is replaced by its underlying discrete group , by Nariman [Nar17a,Nar17b, Nar16].(viii) Progress towards understanding the homotopy equivalences of high genus man-ifolds have been obtained by Berglund and Madsen [BM14]. At present theirresults seem to be qualitatively quite different from the results described here.5. Rational cohomology calculations
We have already advertised the feature that the general theory surveyed in § § § W and a structure λ W : Fr( W ) → Λ, one typicallyfirst estimates the genus of (
W, λ W ). This step is trivial for the manifolds consideredin § § § ρ W : Fr( W ) → Θ andthe space B hAut( u rel P ). This step is mostly resolved by Lemma 4.15 for theexample given in § § § § The manifolds W g, . Recall that we write W g = g ( S n × S n ) for the g -foldconnected sum, and W g, = D n W g ∼ = W g \ int( D n ). These manifolds playa distinguished role in the theory described above, as they are used to measurethe genus of arbitrary 2 n -manifolds: in this sense W g, is the simplest manifold of genus g . In the case 2 n = 2 the solution by Madsen and Weiss [MW07] of theMumford Conjecture gave a description of H ∗ ( M or ( W g, ); Q ) in terms of Miller–Morita–Mumford classes in a stable range of degrees. In this section we wish toexplain how the analogue of Madsen and Weiss’ result in dimensions 2 n ≥ n -stage τ W g, : W g, ℓ Wg, −→ B θ −→ B O(2 n )of a map classifying the tangent bundle of W g, has the cofibration ℓ W g, n -connectedand the fibration θ n -co-connected. As W g, is ( n − τ W g, is nullhomotopic—because W g, admits a framing—we may identify θ with the n -connected cover of B O(2 n ), which we write as θ n : B O(2 n ) h n i θ or n −→ B SO(2 n ) or −→ B O(2 n ) . As the pair ( W g, , ∂W g, ) is ( n − θ or n , ℓ ∂W g, ) is contractible. Thus by Theorem 4.13 there is a map α : M or ( W g, ) ≃ M θ n ( W g, , ℓ W g, ) −→ Ω ∞ M T θ n which is a homology equivalence onto the path component that it hits, in degrees ∗ ≤ g − , as long as 2 n ≥
6. The rational cohomology of a path component ofΩ ∞ M T θ n is calculated as described in Remark 4.2, in terms of H ∗ ( B O(2 n ) h n i ; Q ).To work this out we identify B O(2 n ) h n i = B SO(2 n ) h n i . As H ∗ ( B SO(2 n ); Q ) = Q [ e, p , p , . . . , p n − ]is a free graded-commutative algebra the effect of taking the n -connected cover oncohomology groups is a simple as possible: it simply eliminates all free generatorsof degree ≤ n . Thus H ∗ ( B SO(2 n ) h n i ; Q ) = Q [ e, p ⌈ n +14 ⌉ , . . . , p n − ] . Combining all the above, we obtain the following.
Theorem 5.1.
For n ≥ , let B denote the set of monomials in the classes e , p n − , p n − , . . . , p ⌈ n +14 ⌉ . Then the map Q [ κ c | c ∈ B , | c | > n ] −→ H ∗ ( M or ( W g, ); Q ) is an isomorphism in degrees ∗ ≤ g − . This is [GRW18, Corollary 1.8]. As we mentioned above, for 2 n = 2 the samestatement (with a slightly different range) was earlier proved by Madsen and Weiss.5.2. The manifolds W g . For 2 n = 2 it is a theorem of Harer [Har85] that themap M or ( W g, ) −→ M or ( W g ) (5.1)given by attaching a disk induces an isomorphism on homology in a stable rangeof degrees. On the other hand M or ( W g ) is the homotopy type of the stack M g ofgenus g Riemann surfaces, and it is in this way that Madsen and Weiss’ topologicalresult determines H ∗ ( M g ; Q ) in a stable range.In dimensions 2 n ≥ H ∗ ( M or ( W g ); Q ) in a stablerange.Continuing to write θ n : B O(2 n ) h n i θ or n −→ B SO(2 n ) or −→ B O(2 n ) ODULI SPACES OF MANIFOLDS 17 as in the previous section, this tangential structure is also the Moore–Postnikov n -stage of the map classifying the tangent bundle of W g . By Theorem 4.5 there isa map α : M or ( W g ) −→ (Ω ∞ M T θ n ) // hAut( θ or n )that induces an isomorphism on homology, onto the path component that it hits,in degrees ∗ ≤ g − as long as 2 n ≥ M or ( W g ) in a range of degreeswe could try to calculate the rational cohomology of the relevant path component of(Ω ∞ M T θ n ) // hAut( θ or n ), but instead we shall use the fibre sequence (4.9). Choosinga θ n -structure ℓ W g on W g , this is a fibre sequence M θ n ( W g , ℓ W g ) −→ M or ( W g ) ξ −→ B (hAut( θ or n ) [ W g ,ℓ Wg ] ) . (5.2)As ℓ W g : W g → B O(2 n ) h n i is n -connected, we may apply Theorem 3.3, giving that Q [ κ c | c ∈ B , | c | > n ] −→ H ∗ ( M θ n ( W g , ℓ W g ); Q )is an isomorphism in degrees ∗ ≤ g − . All the classes κ c are defined on the totalspace M or ( W g ) of the fibration (5.2), which implies that this fibration satisfies theLeray–Hirsch property in the stable range. Thus the Leray–Hirsch map Q [ κ c | c ∈ B , | c | > n ] ⊗ H ∗ ( B (hAut( θ or n ) [ W g ,ℓ Wg ] ); Q ) −→ H ∗ ( M or ( W g ); Q ) (5.3)is an isomorphism in degrees ∗ ≤ g − .To complete this calculation we must calculate the rational cohomology of thespace B (hAut( θ or n ) [ W g ,ℓ Wg ] , and describe the map ξ ∗ : H ∗ ( B (hAut( θ or n ) [ W g ,ℓ Wg ] ); Q ) −→ H ∗ ( M or ( W g ); Q )in terms that we understand.5.2.1. Identifying hAut( θ or n ) . The map θ or n : B O(2 n ) h n i → B SO(2 n ) is a principalfibration for the group-like topological monoid SO[0 , n − B SO(2 n ) → B SO → B (SO[0 , n − ι : SO[0 , n − −→ hAut( θ or n )given by the principal group action. Lemma 5.2.
The map ι is a weak homotopy equivalence.Proof. If we fix a basepoint ∗ ∈ B O(2 n ) h n i , which identifies the fibre through ∗ with SO[0 , n − ev : hAut( θ or n ) −→ SO[0 , n − ϕ ϕ ( ∗ ) , and it is clear that ev ◦ ι is the identity. Thus it is enough to show that ev is aweak homotopy equivalence. Suppose we are given a map ( f, g ) : ( D k +1 , S k ) → (SO[0 , n − , hAut( θ or n )). This determines a relative lifting problem( S k × B O(2 n ) h n i ) ∪ S k ×{∗} ( D k +1 × {∗} ) B O(2 n ) h n i D k +1 × B O(2 n ) h n i B SO(2 n ) g ∪ f θ or n θ or n ◦ π and finding a nullhomotopy of ( f, g ) is the same as solving this relative liftingproblem. The obstructions for doing so lie in the groups e H i − k ( B O(2 n ) h n i ; π i +1 ( B SO(2 n ) , B O(2 n ) h n i )) , but these groups are zero if i − k ≤ n or if i ≥ n , so always vanish. (cid:3) In particular, the submonoid hAut( θ or n ) [ W g ,ℓ Wg ] ≤ hAut( θ or n ) is in fact the wholeof hAut( θ or n ), so we may identify the cohomology of its classifying space with H ∗ ( B hAut( θ or n ); Q ) = H ∗ ( B SO[0 , n ]; Q ) = Q [ p , p , . . . , p ⌊ n ⌋ ] . (5.4)5.2.2. Miller–Morita–Mumford class interpretation.
Combining (5.3) and (5.4) givesa formula for H ∗ ( M or ( W g ); Q ) in a range of degrees. In fact the classes obtainedby pulling back p , p , . . . , p ⌊ n ⌋ along the map ξ : M or ( W g ) −→ B hAut( θ or n )may be re-interpreted as Miller–Morita–Mumford classes. We shall use the followinglemma to explain this. Lemma 5.3.
Let π or : E or ( W g ) → M or ( W g ) denote the the path component ofthe fibration (2.4) modelling the universal oriented W g -bundle, and τ : E or ( W g ) → B SO(2 n ) denote the map classifying the vertical tangent bundle. Then the square E or ( W g ) B SO(2 n ) M or ( W g ) B SO[0 , n ] B SO(2 n )[0 , n ] τπ or ξ ≃ (5.5) commutes up to homotopy.Proof. Let π θ n : E θ n ( W g , ℓ W g ) → M θ n ( W g , ℓ W g ) denote the path component of thefibration modelling the universal W g -bundle with θ n -structure, which as in (2.4)comes with maps τ : E θ n ( W g , ℓ W g ) ℓ −→ B O(2 n ) h n i θ or n −→ B SO(2 n )whose composition classifies the (oriented) vertical tangent bundle. This gives acommutative square E θ n ( W g , ℓ W g ) B O(2 n ) h n iM θ n ( W g , ℓ W g ) ∗ π θn ℓ of hAut( θ or n )-spaces and hAut( θ or n )-equivariant maps. Taking homotopy orbits, andreplacing the spaces at each corner with homotopy equivalent models, we obtainthe homotopy commutative square E or ( W g ) B SO(2 n ) M or ( W g ) B SO[0 , n ] . π or τξ Here the right-hand map is the truncation B SO(2 n ) → B SO(2 n )[0 , n ] followed bythe identification B SO(2 n )[0 , n ] ∼ → B SO[0 , n ], as required. (cid:3)
Now we may calculate as follows: by this lemma we have( π or ) ∗ ξ ∗ ( p i ) = τ ∗ ( p i ) , and so by the projection formula (and commutativity of the cup product) κ ep i = Z π or τ ∗ ( e · p i ) = (cid:18)Z π or τ ∗ e (cid:19) · ξ ∗ ( p i ) = χ ( W g ) · ξ ∗ ( p i )and hence, for χ ( W g ) = 2 + ( − n g = 0, we have ξ ∗ ( p i ) = χ ( W g ) κ ep i .Combined with the previous discussion, we obtain the following. ODULI SPACES OF MANIFOLDS 19
Theorem 5.4.
For n ≥ , let B denote the set of monomials in the classes e , p n − , p n − , . . . , p ⌈ n +14 ⌉ , and C denote the set of the remaining Pontryagin classes p , p , . . . , p ⌊ n ⌋ . Then the map Q [ κ c | c ∈ ( B ⊔ e · C ) , | c | > n ] −→ H ∗ ( M or ( W g ); Q ) is an isomorphism in degrees ∗ ≤ ( g − / . For 2 n = 2 the same statement (with a slightly different range) holds by thetheorem of Madsen and Weiss, and in this case the set C is empty and the result isthe same as that of Theorem 5.1. For 2 n ≥ θ or n ) ≃ SO[0 , n −
6≃ ∗ and so it follows from the discussion in this section that the map (5.1) is not anisomorphism on integral cohomology in any range of degrees (though for 2 n = 6it is still an isomorphism on rational cohomology in a stable range, as SO[0 , ≃ K ( Z / ,
1) is rationally acyclic; in Theorem 5.4 this corresponds to the fact that C is empty in this case). Remark . In the statement of Theorem 5.4 we do not assert that κ c = 0 for mono-mials c
6∈ B ⊔ e · C . Indeed a further consequence of the homotopy commutativity of(5.5) is the following description of κ c for a general monomial c = e i · p j · · · p j n − n − in terms of the generators of Theorem 5.4: κ c = (cid:18) κ e · p χ ( W g ) (cid:19) j · (cid:18) κ e · p χ ( W g ) (cid:19) j · · · (cid:18) κ e · p k χ ( W g ) (cid:19) j k · κ (cid:16) e i · p kk +1 k +1 ··· p jn − n − (cid:17) , where we write k = ⌊ n ⌋ . This follows immediately from the observation that p i ( T π ) = π ∗ ξ ∗ ( p i ) = π ∗ ( κ e · pi χ ( W g ) ) for i ≤ k .5.3. Hypersurfaces in CP . If V ⊂ CP r +1 is a smooth hypersurface, determinedby a homogeneous complex polynomial of degree d , then it is an observation ofThom that its diffeomorphism type depends only on the degree d , and not on theparticular polynomial: we call the resulting 2 r -manifold V d . As we shall explainin § V d ⊂ CP of degree d , and determine a formula for therational cohomology of M or ( V d ) in a range of degrees. Let us start with outliningthe steps again, and state the conclusions in this example.(i) Determine the genus of V d : it turns out to be ( d − d + 10 d − d + 4).(ii) Determine the Moore–Postnikov 3-stage V d ℓ Vd −−→ B d θ d −→ B SO(6) of a mapclassifying the oriented tangent bundle of V d .(iii) Calculate the ring H ∗ ( M θ d ( V d , ℓ V d ); Q ) in the stable range. It turns out tobe the Q -algebra A = Q [ κ t n c | c ∈ B , n ≥ | c | + 2 n > , (5.6)where B is the set of monomials in classes p , p , and e of degree | p | = 4, | p | = 8, and | e | = 6, and t is a class of degree 2.(iv) Use the spectral sequence arising from Corollary 4.6 to determine the coho-mology of M or ( V d ) from that of M θ d ( V d , ℓ V d ) in a stable range. The result isa short exact sequence0 −→ H ∗ ( M or ( V d ); Q ) −→ A d −→ A −→ , (5.7) where d : A → A is the unique derivation satisfying d ( κ t n c ) = nκ t n − c . (Theresult is a scalar when | t n − c | = 6; this scalar is a characteristic number of V d and therefore the derivation d depends on the degree d .)5.3.1. Algebraic topology of V d . By the Lefschetz hyperplane theorem the inclusion i : V d → CP is 3-connected. This first implies that V d is simply-connected. Writing H ∗ ( CP ; Z ) = Z [ x ] / ( x ) for x = c ( O (1)) and t = i ∗ ( x ), we have H ( V d ; Z ) = Z H ( V d ; Z ) = 0 H ( V d ; Z ) = Z { t } and hence by Poincar´e duality we have H ( V d ; Z ) = Z { s } H ( V d ; Z ) = 0 H ( V d ; Z ) = Z { u } where h [ V d ] , u i = 1 and s · t = u . We also have h i ∗ [ V d ] , x i = d , obtained byintersecting V d with a generic CP ⊂ CP , giving t = d · u and hence t = d · s . Bydefinition, V d is the zero locus of a transverse section of O ( d ) → CP , so its normalbundle in CP is i ∗ ( O ( d )) and hence as complex vector bundles we have T V d ⊕ i ∗ ( O ( d )) ⊕ C = i ∗ ( T CP ) ⊕ C = i ∗ ( O (1)) ⊕ so taking total Chern classes yields c ( T V d ) = i ∗ (cid:16) (1+ x ) (1+ dx ) (cid:17) . We can therefore extract c ( T V d ) = (cid:18)(cid:18) (cid:19) − (cid:18) (cid:19) d + (cid:18) (cid:19) d − d (cid:19) t and so compute the Euler characteristic of V d as h [ V d ] , c ( T V d ) i to be χ ( V d ) = d · (10 − d + 5 d − d ) . We therefore find that H ( V d ; Z ) is free of rank 4 − χ ( V d ) = d − d +10 d − d +4,which finishes our calculation of the cohomology of V d .For later use we record two further characteristic classes of V d , namely w ( T V d ) = (5 − d ) t mod 2 p ( T V d ) = (5 − d ) t , obtained from the identities w ≡ c mod 2 and p = c − c among characteristicclasses of complex vector bundles.5.3.2. Genus of V d . The genus of V d may be estimated from below by the methodsof § W has adecomposition W ∼ = M g ( S × S ) with H ( M ; Z ) = 0, and so the genus of sucha W is given by half its third Betti number. Thus we have g ( V d ) = 12 ( d − d + 10 d − d + 4) . Similar formulae are known for higher dimensional smooth complex complete inter-section varieties, see e.g. [Woo75, Mor75, Bro79].
Remark . More generally, if L is an ample line bundle over a smooth projectivecomplex manifold M of complex dimension n +1 then for all d ≫ U d arising as the zeroes of generic holomorphic sections of L ⊗ d . Writing x := c ( L ), as L is ample we have N := R M x n +1 = 0. Writing i : U d ֒ → M for the inclusion, and using that i ∗ [ U d ] is Poincar´e dual to e ( L ⊗ d ) = dx ,it follows that R U d i ∗ ( x ) n = d · N = 0. The analogue of the calculation above givesthat c ( T U d ) = i ∗ ( c ( T M )(1+ dx ) ) and hence we have χ ( U d ) = ( − n d n +1 N + O ( d n ) and so b n = d n +1 N + O ( d n ). If n is odd then by the discussion in § g ( U d ) = 12 d n +1 N + O ( d n ) . ODULI SPACES OF MANIFOLDS 21 If n is even, then the analogous calculation with the total Hirzebruch L -class givesthat L ( T U d ) = i ∗ ( L ( T M ) dx/ tanh( dx ) ) so σ ( U d ) = n +2 (2 n +2 − B n +2 ( n +2)! d n +1 N + O ( d n ), where B i denote the Bernoulli numbers. Hence, by the discussion in § g ( U d ) = 12 (cid:18) − n +2 (2 n +2 − | B n +2 | ( n + 2)! (cid:19) d n +1 N + O ( d n ) . The term n +2 (2 n +2 − | B n +2 | ( n +2)! does not matter much for large n : the fact that theTaylor series for tanh( z ) has convergence radius π/ /π ) n + ε as n → ∞ , for any ε >
0; in particular it quicklybecomes much smaller than 1. In the relevant cases n ≥ / Moore–Postnikov 3-stage of V d . Let us write τ : V d ℓ Vd −→ B d θ d −→ B SO(6)for the Moore–Postnikov 3-stage of a map τ classifying the oriented tangent bundleof V d , so ℓ V d is 3-connected and θ d is 3-co-connected. From this we easily calculatethe homotopy groups of B d , as0 = π ( V d ) ∼ −→ π ( B d ) Z = π ( V d ) ∼ −→ π ( B d ) π ( B d ) ∼ −→ π ( B SO(6)) = 0 π i ( B d ) ∼ −→ π i ( B SO(6)) for all i ≥ . To understand the map θ d on homotopy groups, it remains to understand thecomposition τ ∗ : Z = π ( V d ) ∼ −→ π ( B d ) −→ π ( B SO(6)) = Z / . (5.8)The latter group is detected by the Stiefel–Whitney class w , so this map is non-zero if and only if the class w ( T V d ) ∈ H ( V d ; Z / ∼ = Hom( π ( V d ) , Z /
2) is non-zero.We have seen that w ( T V d ) = (5 − d ) t , so (5.8) is surjective if and only if d is even.Let us abuse notation by writing t ∈ H ( B d ; Z ) for the unique class which pullsback to t along ℓ V d . If d is even, then t satisfies t ≡ w ( θ ∗ d γ ) mod 2. Thus thereis a Spin c -structure on the bundle θ ∗ d γ with c = t , and choosing one provides acommutative diagram B d B Spin c (6) B SO(6) . θ d f It may be directly checked using the above calculations that the map f induces anisomorphism on all homotopy groups, so is a weak equivalence (over B SO(6)).If d is odd then we have w ( θ ∗ d γ ) = 0, so we may choose a Spin-structure on thebundle θ ∗ d γ , which provides a commutative diagram B d B Spin(6) B SO(6) . θ d h It may be directly checked using the above calculations that the map h × t : B d → B Spin(6) × K ( Z ,
2) induces an isomorphism on all homotopy groups, so is a weakequivalence (over B SO(6)).
In either case, the map θ d × t : B d −→ B SO(6) × K ( Z , B SO(6)), so we have H ∗ ( B d ; Q ) = Q [ t, p , p , e ] . Writing as in the previous examples B for the set of monomials in p , p , and e , byTheorem 3.3 the map Q [ κ t i c | c ∈ B , i ≥ , | c | + 2 i > −→ H ∗ ( M θ d ( V d , ℓ V d ); Q )is an isomorphism in degrees ∗ ≤ d − d +10 d − d +44 , establishing (5.6).5.3.4. Change of tangential structure.
We wish to use the above to compute therational cohomology of M or ( V d ) in a range of degrees, so must analyse the forgetfulmap M θ d ( V d , ℓ V d ) → M or ( V d ). We shall do this in two stages, given by the mapsof tangential structures B d B SO(6) × K ( Z , B SO(6) B SO(6) . θ d u = θ d × t µ µ Id The space of θ d -structures on V d refining the µ -structure u ◦ ℓ V d is homotopyequivalent to the space of lifts B d V d B SO(6) × K ( Z , , uu ◦ ℓ Vd and π (hAut( u )) acts on the set of homotopy classes of such lifts. If G ≤ hAut( u )is the submonoid of those path components that preserve the θ d -structure ℓ V d upto diffeomorphisms of V d preserving the µ -structure u ◦ ℓ V d , then there is a fibrationsequence M θ d ( V d , ℓ V d ) −→ M µ ( V d , u ◦ ℓ V d ) −→ BG.
By the discussion in Remark 4.11, G = hAut( u ) [ V d ,ℓ Vd ] as long as g ( V d , ℓ V d ) ≥ u is a rational homotopy equivalence, and itis immediate from this that π i (hAut( u )) ⊗ Q = 0 for i >
0, so G has no higherrational homotopy groups.We claim that π ( G ) is also trivial, and in fact we shall show that π (hAut( u ))is trivial (so G = hAut( u )). To see this, let φ ∈ hAut( u ), and we must then showthat the following lifting problem admits a solution: ∂ [0 , × B d B d [0 , × B d B d B SO(6) × K ( Z , . Id ⊔ φ uproj u By consideration of the cases B d = B Spin c (6) and B d = B Spin(6) × K ( Z , u is a K ( Z / , H ([0 , × B d , ∂ [0 , × B d ; Z / ∼ = H ( B d ; Z /
2) = 0 . It follows that BG is simply-connected and has trivial higher rational homotopygroups, so M θ d ( V d , ℓ V d ) → M µ ( V d , u ◦ ℓ V d ) is a rational homotopy equivalence. ODULI SPACES OF MANIFOLDS 23
Analogously to the above, if H ≤ hAut( µ ) is the submonoid of those pathcomponents that preserve the µ -structure u ◦ ℓ = τ × t up to orientation-preservingdiffeomorphism of V d , then there is a fibration sequence M µ ( V d , u ◦ ℓ V d ) −→ M or ( V d ) −→ BH.
Again, by Remark 4.11, H = hAut( µ ) [ V d ,u ◦ ℓ Vd ] as long as g ( V d , u ◦ ℓ V d ) ≥
3, andthis fibration sequence is an instance of (4.9). As the fibration µ is trivial, we havehAut( µ ) ≃ map( B SO(6) , hAut( K ( Z , ≃ Z × ⋉ map( B SO(6) , K ( Z , ≃ Z × ⋉ K ( Z , . The non-trivial path component of this monoid acts on H ( B SO(6) × K ( Z , Z ) = Z { t } as −
1, but any orientation-preserving diffeomorphism of V d fixes t ∈ H ( V d ; Z )so acts as +1 on H ( V d ; Z ) = Z { t } . Thus the non-trivial path component of hAut( µ )does not lie in H , so H ≃ K ( Z , M µ ( V d , u ◦ ℓ V d ) −→ M or ( V d ) −→ K ( Z , . The Serre spectral sequence for this fibration, in rational cohomology, has twocolumns and so a single possible non-zero differential. In the stable range, usingthe above, it has the form E ∗ , ∗ = Λ[ ι ] ⊗ Q [ κ t i c | c ∈ B , i ≥ , | c | + 2 i >
6] = ⇒ H ∗ ( M or ( V d ); Q ) . It remains to determine the d -differential, which by the Leibniz rule is done by thefollowing lemma. Lemma 5.7.
We have d ( κ t n c ) = ι ⊗ n · κ t n − c .Proof. We have d ( κ t n c ) = ι ⊗ x for some x . The action map a : K ( Z , × M µ ( V d , u ◦ ℓ V d ) −→ M µ ( V d , u ◦ ℓ V d )classifies the following data: the V d -bundle π : K ( Z , × E µ ( V d , u ◦ ℓ V d ) → K ( Z , × M µ ( V d , u ◦ ℓ V d )pulled back by projection to the second factor, equipped with the µ -structure K ( Z , × E µ ( V d , u ◦ ℓ V d ) τ × ˜ t −→ B SO(6) × K ( Z , τ is given by projection to E µ ( V d , u ◦ ℓ V d ) and its vertical tangent bundle,and ˜ t = ι ⊗ ⊗ t .The class x is related to this action by the formula a ∗ ( κ t n c ) = 1 ⊗ κ t n c + ι ⊗ x + · · · . Using the description above we calculate a ∗ ( κ t n c ) as π ! (( ι ⊗ ⊗ t ) n · τ ∗ ( c )) = π ! n X i =0 (cid:18) ni (cid:19) ι i ⊗ ( t n − i · τ ∗ c ) ! and the K¨unneth factor in H ( K ( Z , Z ) ⊗ H | κ tnc |− ( M µ ( V d , u ◦ ℓ V d ); Z ) is ι ⊗ ( n · κ t n − c ). It follows that x = n · κ t n − c , as required. (cid:3) It follows from this lemma that the differential d is a surjection from the firstcolumn to the third column, so that H ∗ ( M or ( V d ); Q ) ∼ = Ker( d (cid:8) Q [ κ t n c | c ∈ B , i ≥ , | c | + 2 n > ∗ ≤ d − d +10 d − d +44 , establishing (5.7).It may at first appear that this ring does not depend on d , but this formula isto be understood carefully. If | κ t n c | = 2 then d ( κ t n c ) ∈ Q is a scalar, and must be evaluated: this is a boundary condition for the derivation d , and is a characteristicnumber of V d . The κ t n c of degree 2 are given by the t i c of degree 8, so are p , p , te , t p , and t , and these have d ( κ p ) = 0 d ( κ p ) = 0 d ( κ te ) = κ e = χ ( V d ) = d · (10 − d + 5 d − d ) d ( κ t p ) = 2 κ tp = 2 d (5 − d ) d ( κ t ) = 4 κ t = 4 · d. For the penultimate one we use the calculation of the first Pontryagin class of V d .As an example, H ( M or ( V d ); Q ) is 4-dimensional and is spanned by the classes κ p , κ p , κ te − − d + 5 d − d κ t , and κ t p − − d κ t . Remark . In the Serre spectral sequence for the fibration π : E or ( V d ) → M or ( V d )modelling the universal oriented V g -bundle as in (2.4), the class t ∈ H ( V d ; Q ) = E , must be a permanent cycle. (This may be seen as the Euler class of thevertical tangent bundle T π E or ( V d ) restricts to e ( T V d ) ∈ H ( V g ; Q ), so this must bea permanent cycle, and this is a non-zero multiple of t . As d ( t ) = 3 t · d ( t ), if d ( t ) = 0 then t would not by a permanent cycle, a contradiction.) Thus thereexists a class ¯ t ∈ H ( M or ( V d ); Q ) restricting to t ∈ H ( V d ; Q ). We may thereforeconstruct the class κ ¯ t n c := π ! (¯ t n c ) ∈ H ∗ ( M or ( V d ); Q ).However, the class ¯ t is not uniquely determined by the above discussion: if δt ∈ H ( M or ( V d ); Q ) is any class then ¯¯ t = ¯ t + π ∗ ( δt ) is another possible choice, andwe then have κ ¯¯ t n c = π ! ((¯ t + π ∗ ( δt )) n c ) = κ ¯ t n c + ( δt )( n · κ ¯ t n − c ) + ( δt ) · · · , a potentially different cohomology class.By the formula in Lemma 5.7, we may think of the derivation d as being ∂∂t .From this point of view the polynomials in the classes κ t n c that lie in the kernel of d = ∂∂t are precisely those that are independent of the choice of ¯ t when evaluatedin H ∗ ( M or ( V d ); Q ) as described above.5.4. Another
Spin c (6) example. For a simply-connected manifold W of dimen-sion 2 n ≥
6, the formula of Theorem 4.5 for the homology of M or ( W ) in a rangeof degrees seems at first glance as though it only depends on the equivariant ho-motopy type of the GL n ( R )-space Θ having an n -co-connected equivariant map u : Θ → Z × and an n -connected equivariant map ρ W : Fr( T W ) → Θ. However,the codomain of the map (4.6) is the disconnected space (Ω ∞ M T Θ) // hAut( u ), andthe different path-components of this space can have different cohomology, evenrationally. In this section we give an example of this behaviour. Construction 5.9.
Let V → S be the unique non-trivial 5-dimensional real vectorbundle, and M = S ( V ) be its sphere bundle; it is an S -bundle over S with thesame homology as S × S . If we write π : M → S for the bundle projection, thenthere is an isomorphism T M ∼ = π ∗ ( V ) ⊕ ε . In particular, the Spin c -structure on V given by a generator of H ( S ; Z ) gives one on M (which is Spin c -nullbordant),and the corresponding map ℓ M : M → B Spin c (6) is 3-connected. This induces aSpin c -structure on M g := M g ( S × S ) such that ℓ M g : M g → B Spin c (6) is also3-connected. ODULI SPACES OF MANIFOLDS 25
Let θ : B Spin c (6) → B SO(6). As in the last section, if K ≤ hAut( θ ) is thesubmonoid of those path components that stabilise the θ -structure ℓ M g up to dif-feomorphism of M g , then there is a fibration sequence M θ ( M g , ℓ M g ) −→ M or ( M g ) −→ BK.
Lemma 5.10.
We have hAut( θ ) ≃ Z × ⋉ K ( Z , .Proof. The fibration θ : B Spin c (6) → B SO(6) has fibre K ( Z , K ( Z ,
2) on B Spin c (6) fibrewise over B SO(6). Furthermore,writing Spin c (6) = Spin(6) × Z × U(1) we see that complex conjugation on the U(1)factor gives an involution c of B Spin c (6) over B SO(6). Together these give a map Z × ⋉ K ( Z , → hAut( θ ) which can be shown to be an equivalence by obstructiontheory just as in § (cid:3) Lemma 5.11.
We have K = hAut( θ ) . Let us give two proofs of this lemma, one in terms of the manifolds themselves,and one using the infinite loop spaces of the relevant Thom spectrum.
Proof.
The proof of the previous lemma shows that if ℓ and ℓ ′ are two θ -structureson M g then there is a unique obstruction to them being homotopic, namely ℓ ∗ ( t ) − ( ℓ ′ ) ∗ ( t ) ∈ H ( M g ; Z ) . We therefore see that ℓ M g and c ◦ ℓ M g , where c is the involution of B Spin c (6)over B SO(6), are not fibrewise homotopic as the obstruction is 2 ℓ ∗ M g ( t ) = 0 ∈ H ( M g ; Z ).However, pulling back the vector bundle V → S along a diffeomorphism of S of degree − π ( B SO(5)) = Z / − M → M which acts as − H ( M ; Z ) and as +1 on H ( M ; Z ), so is orientation-reversing.Composing this with the fibrewise antipodal map of π : M → S gives a diffeo-morphism ϕ : M → M which acts as − H ( M ; Z ) and H ( M ; Z ), so isorientation-preserving: we may then isotope it to fix a disc, and hence extend it toa diffeomorphism ϕ g : M g → M g acting as − H ( M ; Z ) and on H ( M ; Z ).Now the θ -structures ℓ M g ◦ Dϕ g and c ◦ ℓ M g on M g are homotopic, as( ℓ M g ◦ Dϕ g ) ∗ ( t ) = ϕ ∗ g ( ℓ ∗ M g ( t )) = − ℓ ∗ M g ( t ) = ℓ ∗ M g ( − t ) = ( c ◦ ℓ M g ) ∗ ( t ) . This shows that c ∈ hAut( θ ) lies in the submonoid K , as it preserves the θ -structure ℓ M g up to a diffeomorphism of M g . (cid:3) Alternative proof.
By the discussion in Remark 4.11, the submonoid K ≤ hAut( θ )agrees with hAut( θ ) [ M g ,ℓ Mg ] as long as g ≥
3, so is the stabiliser of α ( M g , ℓ M g ) ∈ π ( M T
Spin c (6)).Thomifying the map B Spin c (6) → B Spin c gives a fibre sequence of spectra F −→ M T
Spin c (6) −→ Σ − M Spin c and it is easy to check that F is connective and has π ( F ) ∼ = Z . We therefore havean exact sequence π ( F ) ∼ = Z −→ π ( M T
Spin c (6)) −→ π ( M Spin c ) = Ω Spin c , and the left-hand map can be seen to send a generator to α ( S , ℓ S ), where ℓ S isthe unique Spin c (6)-structure on S compatible with its orientation.As the Spin c (6)-manifold M is constructed as the sphere bundle of a Spin c vectorbundle, its class is trivial in Ω Spin c as it bounds the associated disc bundle; similarlyfor M g = M g ( S × S ). Thus α ( M g , ℓ M g ) is a multiple of α ( S , ℓ S ) (by taking Euler characteristic we see that it is 2 − g times it) and so is fixed by hAut( θ ), asthe Spin c (6)-structure on S is unique given its orientation. (cid:3) We may therefore develop the following diagram of fibration sequences M θ ( M g , ℓ M g ) X g K ( Z , M θ ( M g , ℓ M g ) M or ( M g ) B ( Z × ⋉ K ( Z , ∗ B Z × B Z × , whose middle row is the fibration sequence, with lower middle arrow defined tomake the bottom right-hand square commute, and X g as its homotopy fibre, andtop right-hard square homotopy cartesian.The calculation of the previous section applies to the top row, showing that H ∗ ( X g ; Q ) = Ker( d (cid:8) Q [ κ t n c | c ∈ B , i ≥ , | c | + 2 n > d ( κ te ) = κ e = χ ( M g ) = 4 − gd ( κ t p ) = 2 κ tp = 0 d ( κ t ) = 4 κ t = 0 . However, now the Serre spectral sequence for the middle column gives the calcula-tion H ∗ ( M or ( M g ); Q ) = H ∗ ( X g ; Q ) Z × = Ker( d (cid:8) Q [ κ t n c | c ∈ B , i ≥ , | c | + 2 n > Z × in a stable range, where the invariants are taken with respect to the involution t
7→ − t .Let us explain something of the structure of this ring in low degrees. In particular,we shall see that unlike the previous examples it not a free graded-commutativealgebra, even in the stable range where our formulae apply. Before taking Z × -invariants, in degree 2 the kernel is spanned by { κ p , κ p , κ t p , κ t } , and theseclasses are all fixed by the involution, givingdim Q H ( M or ( M g ); Q ) = 4 . In degree 4 the kernel of d is 16 dimensional, spanned by the 10-dimensional vectorspace Sym ( Q { κ p , κ p , κ t p , κ t } ) along with the classes κ te κ p − (4 − g ) κ tp κ te κ p − (4 − g ) κ tp (4 − g ) κ t p − κ t p κ te (4 − g ) κ t − κ t κ te κ p e κ te − (4 − g ) κ t e . Of these, the last two classes are invariant under the involution while first four are anti- invariant, and hence dim Q H ( M or ( M g ); Q ) = 12 . In higher degrees, we find that even though, for example, the class κ te κ p − (4 − g ) κ tp is not invariant under the involution, its square is invariant and therefore ODULI SPACES OF MANIFOLDS 27 defines a class in H ( M or ( M g ); Q ). Similarly with products of any two classes thatare anti-invariant and in the kernel of d . In degree 16 we find the relation(( κ te κ p − (4 − g ) κ tp )( κ te κ p − (4 − g ) κ tp )) = ( κ te κ p − (4 − g ) κ tp ) ( κ te κ p − (4 − g ) κ tp ) among squares of classes of degree 8, showing that the ring is not free.6. Abelianisations of mapping class groups
The theory described above may in principle be used for calculations in integralhomology and cohomology, though this is of course far more difficult. In practicesuch calculations are restricted to low dimensions, and have a different flavour tothose described in §
5. Here one must obtain information about the low-dimensionalhomology of Ω ∞ M T
Θ, which is roughly the same as the low-dimensional homotopyof Ω ∞ M T
Θ, which is the homotopy of the spectrum
M T
Θ in small positive degrees.But the spectrum
M T
Θ is non-connective, so computing its π i is comparable tocomputing π i +2 n of a connective spectrum (in the alternative proof of Lemma 5.11we have already engaged with this a bit, though we avoided having to actuallycompute).As an example of the kinds of calculations that one is required to make, and togive some ideas of the kinds of techniques that can be used to tackle them, in thissection we shall survey the calculation in [GRW16] of H ( M ( W g, ); Z ), and thendescribe analogous calculations for certain non-simply connected 6-manifolds.Recall that for a manifold W , possibly with boundary, its mapping class group is Γ ∂ ( W ) := π (Diff ∂ ( W )) . Equivalently, it is the fundamental group of B Diff ∂ ( W ), so by the Hurewicz theoremwe may identify its abelianisation asΓ ∂ ( W ) ab ∼ = H ( B Diff ∂ ( W ); Z ) ∼ = H ( M ( W ); Z ) . The manifolds W g, . We return to the 2 n -manifolds W g, of § M ( W g, ) ≃ M θ n ( W g, , ℓ W g, ) −→ Ω ∞ M T θ n that for 2 n ≥ ≤ g − onto the path compo-nent that it hits. In particular, as long as g ≥ ∂ ( W g, ) ab ∼ = H ( M ( W g, ); Z ) ∼ = H (Ω ∞ M T θ n ; Z ) ∼ = π ( M T θ n ) , using that all path components of Ω ∞ M T θ n are homotopy equivalent, that theHurewicz map is an isomorphism (as this space is a loop space), and that π of thespace Ω ∞ M T θ n is the same as that of the spectrum M T θ n .In [GRW16] we attempted to calculate this group, at least in terms of otherstandard groups arising in geometric topology. Here we shall summarise the resultsand general strategy of that paper, though we refer there for more details.To state the main result, consider the bordism theory Ω h n i∗ associated to thefibration B O h n i → B O given by the n -connected cover, and represented by thespectrum M O h n i (cf. [Sto68, p. 51]). The natural map B O(2 n ) h n i → B O h n i covering the stabilisation map B O(2 n ) → B O provides a map of spectra s : M T θ n −→ Σ − n M O h n i , and on π this gives a homomorphism s ∗ : π ( M T θ n ) → Ω h n i n +1 . The group π ( M T θ n ) is determined in terms of this as follows. Theorem 6.1.
There is an isomorphism s ∗ ⊕ f : π ( M T θ n ) −→ Ω h n i n +1 ⊕ ( Z / if n is even if n is 1, 3, or 7 Z / elsefor a certain homomorphism f . Furthermore, the groups Ω h n i n +1 are related to the stable homotopy groups ofspheres as follows: there is a homomorphism ρ ′ : Cok( J ) n +1 −→ Ω h n i n +1 given by considering a stably framed manifold as a manifold with B O h n i -structure,which is surjective and whose kernel is generated by the class of a certain homotopysphere Σ n +1 Q . In several cases it follows from work of Stolz that the class of Σ Q in Cok( J ) is trivial—so ρ ′ is an isomorphism—but this is not known in general.Combining the above with known calculations of Ω h i∗ = Ω h i∗ = Ω Spin ∗ and Ω h i∗ =Ω String ∗ gives the following. n π ( M T θ n ) 0 ( Z / Z / Z / Z / ⊕ Z / Z / F denote the homotopy fibre of the map of spectra s : M T θ n → Σ − n M O h n i , and construct a map Σ − n SO / SO(2 n ) → F which canbe shown to be n -connected, for example by computing its effect on homology. Onthe other hand SO / SO(2 n ) is (2 n − π n +1 (SO / SO(2 n )) −→ π s n +1 (SO / SO(2 n )) ∼ = π s (Σ − n SO / SO(2 n ))is an isomorphism for n ≥
2, and similarly for one homotopy group lower. Itfollows from a calculation of Paechter [Pae56] that π n +1 (SO / SO(2 n )) is ( Z / if n is even and is Z / n is odd, and also that π n (SO / SO(2 n )) ∼ = Z . Putting theabove together, we find an exact sequenceΩ h n i n +2 ∂ −→ ( ( Z / if n is even Z / n is odd −→ π ( M T θ n ) −→ Ω h n i n +1 −→ Z . (6.1)The rightmost map is zero (as its domain is easily seen to be a torsion group). Inthe cases n ∈ { , , } it can be shown that the images of CP , HP , and OP under the leftmost map are non-zero modulo 2, so the leftmost map is surjective.In the remaining cases one must show that the leftmost map is zero, and that theresulting short exact sequence is split, via a homomorphism f as in the statementof Theorem 6.1.At this point is is convenient to use the isomorphism Γ ∂ ( W g, ) ab ∼ = π ( M T θ n ) forsome g ≫
0. The action of Γ ∂ ( W g, ) on H n ( W g, ; Z ) respects the ( − n -symmetricintersection form λ , and if n = 1, 3, or 7 then it also respects a certain quadraticrefinement µ of this bilinear form. This yields a homomorphismΓ ∂ ( W g, ) −→ Aut( H n ( W g, ; Z ) , λ, µ ) . These automorphism groups have been studied by other authors, and their abelian-isations have been identified (for g ≫
0) as ( Z / if n if even or Z / n is odd. ODULI SPACES OF MANIFOLDS 29
A careful analysis of the maps involved shows that the resulting homomorphism f : π ( M T θ n ) ∼ = Γ ∂ ( W g, ) ab −→ Aut( H n ( W g, ; Z ) , λ, µ ) ab ∼ = ( ( Z / if n is even Z / n is oddsplits the short exact sequence arising from (6.1), as required. Remark . The Pontryagin dual of the finite abelian group H ( M ( W g, ); Z ) cal-culated here is the torsion subgroup of H ( M ( W g, ); Z ). The torsion free quotientof the latter group has been analysed in detail by Krannich and Reinhold [KR18].The (unknown, at present) order of the element [Σ Q ] ∈ Cok( J ) n +1 arises theretoo.6.2. Some non-simply-connected 6-manifolds.
For the example discussed inthe previous section the theory described above is not the only way to calculateΓ ∂ ( W g, ) ab , because Kreck [Kre79] has described the groups Γ ∂ ( W g, ) up to twoextension problems, and Krannich [Kra19] has recently resolved these extensionscompletely for n odd, and determined enough about them to calculate Γ ∂ ( W g, ) ab for all n ≥ g ≥
1. However, for even slightly more complicated mani-folds such an alternative approach is not available, and we suggest that the theorydescribed above is the best way to approach the calculation of Γ ∂ ( W ) ab . In thissection we illustrate this with an example which seems inaccessible by other means.Let G be a virtually polycyclic group, and consider a compact 6-manifold W such that(i) a map τ W classifying the tangent bundle of W admits a lift ℓ W along θ : B Spin(6) × BG pr −→ B Spin(6) −→ B O(6)such that ℓ W : W → B Spin(6) × BG is 3-connected, and(ii) ( W, ∂W ) is 2-connected.Such manifolds exist for any virtually polycyclic G : these groups satisfy Wall’s[Wal65] finiteness condition ( F ) by [Rat83, p.183], and so also ( F ), so W may betaken to be a regular neighbourhood of an embedding of a finite 3-skeleton of BG into R . As further examples of such manifolds, one may take W = ( M × D ) g ( S × S ) (6.2)where M is a closed oriented 3-manifold that is irreducible (so that π ( M ) = 0)and has virtually polycyclic fundamental group.Recall from § Theorem 6.3.
Suppose that G is virtually polycyclic of Hirsch length h , and W isa 6-manifold satisfying (i) and (ii) above, of genus g ( W ) ≥ h . Then there is ashort exact sequence −→ G ab −→ Γ ∂ ( W ) ab −→ ko ( BG ) −→ which is (non-canonically) split. This short exact sequence was first established by Friedrich [Fri18], who alsoshowed that it is split after inverting 2. We shall give a different argument to hers,which gives the splitting at the prime 2 as well.The following three examples concern the manifolds W of construction (6.2) with M a 3-manifold having finite fundamental group, and g ≥ Example 6.4.
Let M = L p,q be the ( p, q )th lens space, with fundamental group G = Z /p with p prime. Then we haveΓ ∂ ( W ) ab ∼ = Z / ⊕ Z / p = 2 Z / ⊕ Z / p = 3( Z /p ) if p ≥ . The required calculation of ko ( B Z /p ) may be extracted from [BG10, Example7.3.1] for p = 2, and from the Atiyah–Hirzebruch spectral sequence, the fact thatko ∗ ( B Z /p )[ ] is a summand of ku ∗ ( B Z /p )[ ], and [BG03, Remark 3.4.6] for odd p . Example 6.5.
Let M be the spherical 3-manifold with fundamental group G = Q . Then we have Γ ∂ ( W ) ab ∼ = ( Z / ⊕ ( Z / ⊕ Z / . The required calculation of ko ( BQ ) may be extracted from [BG10, p. 138]. Example 6.6.
Let M = Σ be the Poincar´e homology 3-sphere, with fundamentalgroup G the binary icosahedral group, isomorphic to SL ( F ). Then g we haveΓ ∂ ( W ) ab ∼ = ( Z / ⊕ Z / ⊕ Z / . As Σ is a homology sphere, H ( BG ; Z ) ∼ = H (Σ ; Z ) = 0 so we must justcalculate ko ( BG ). The order of G ∼ = SL ( F ) is 120 = 2 · ·
5, so we shallcalculate the localisations ko ( BG ) ( p ) for p ∈ { , , } . Recall that the cohomologyring of BG is H ∗ ( BG ; Z ) = Z [ z ] / (120 · z ) with | z | = 4, which may be computedfrom the fibration sequence S → Σ → BG .When p is odd, G has cyclic Sylow p -subgroup, so by transfer ko ( BG ) ( p ) isa summand of ko ( B Z /p ) ( p ) , which we have explained in Example 6.4 is Z / p = 3 and ( Z / for p = 5. Comparing this with the Atiyah–Hirzebruch spectralsequence computing ko ∗ ( BG ) ( p ) gives the claimed answer.When p = 2, G has Sylow 2-subgroup Q , so by transfer ko ( BG ) (2) is a sum-mand of ko ( BQ ) (2) , which we have explained in Example 6.4 is ( Z / ⊕ Z /
64. Bya theorem of Mitchell and Priddy [MP84, Theorem D] there is a stable splitting of BQ as BG (2) ∨ X ∨ X for some spectrum X , from which it follows that ko ( BG ) (2) is either Z /
64 or ( Z / ⊕ Z /
64; we may see that the first case occurs from theAtiyah–Hirzebruch spectral sequence computing ko ∗ ( BG ) (2) .The rest of this section is concerned with the proof of Theorem 6.3.6.2.1. Reduction to homotopy theory.
We have assumed that (
W, ∂W ) is 2-connected,so by Lemma 4.15 we have that hAut( θ, ∂W ) ≃ ∗ . Thus by Theorem 4.1 and thediscussion in § α : M ( W ) ≃ M θ ( W ) −→ Ω ∞ M T θ = Ω ∞ ( M T
Spin(6) ∧ BG + )which induces an isomorphism on homology onto the path component that it hitsin degrees ∗ ≤ g ( W ) − h − , so in particular as long as g ( W ) ≥ h it induces anisomorphism on first homology. As described above the first homology of M ( W ) isthe abelianisation of the mapping class group Γ ∂ ( W ), so to establish Theorem 6.3we must establish a short exact sequence0 −→ G ab −→ π ( M T
Spin(6) ∧ BG + ) −→ ko ( BG ) −→ ODULI SPACES OF MANIFOLDS 31
The exact sequence.
Specialising the construction in § n = 6, weshowed that there is a cofibration sequence of spectra F −→ M T
Spin(6) −→ Σ − M Spin , (6.4)that F is connective, and that π ( F ) ∼ = Z and π ( F ) ∼ = Z /
4. The Atiyah–Hirzebruchspectral sequence for F ∧ BG + gives isomorphisms π ( F ∧ BG + ) ∼ = Z π ( F ∧ BG + ) ∼ = Z / ⊕ H ( BG ; Z )where the splitting in the latter is induced by the retraction S → BG + → S ofpointed spaces. Smashing the cofibration sequence (6.4) with BG + , the associatedlong exact sequence of homotopy groups has the formΩ Spin8 ( BG ) ∂ −→ Z / ⊕ H ( BG ; Z ) −→ π ( M T
Spin(6) ∧ BG + ) −→ Ω Spin7 ( BG ) −→ Z and by naturality the long exact sequence for G = { e } splits off of this one. As π ( M T
Spin(6)) = 0 by Theorem 6.1, and Ω
Spin7 = 0 by [Mil63], by the discussionin § e Ω Spin8 ( BG ) ∂ −→ H ( BG ; Z ) −→ π ( M T
Spin(6) ∧ BG + ) −→ Ω Spin7 ( BG ) −→ . The Atiyah–Bott–Shapiro map M Spin → ko is 8-connected, so the induced mapΩ Spin7 ( BG ) → ko ( BG ) is an isomorphism. To obtain the claimed short exactsequence we shall therefore prove the following. Lemma 6.7. If G is finitely-generated then the connecting map ∂ : e Ω Spin8 ( BG ) → H ( BG ; Z ) is trivial.Proof. If this connecting map were non-trivial for some finitely-generated G , thenbecause every non-trivial element of H ( BG ; Z ) = G ab remains non-trivial undersome homomorphism G ab → Z /p k with p prime and k ≥
1, by naturality thisconnecting map would be non-trivial for G = Z /p k . So it suffices to show that themap is trivial in this case.As M Spin → ko is 8-connected, the map e Ω Spin8 ( BG ) → f ko ( BG ) is an isomor-phism. The result then follows as f ko ( B Z /p k ) = 0, by a trivial application of theAtiyah–Hirzebruch spectral sequence for p odd and by [BGS97, Theorem 2.4] for p = 2. (cid:3) Simplifying the splitting problem.
Let x : F → H Z be the 0-th Postnikovtruncation of F . The composition k : Σ − M Spin ∂ −→ Σ F Σ x −→ Σ H Z represents, under the Thom isomorphism, some element κ ∈ H ( B Spin; Z ) whichwe identify as follows. Lemma 6.8.
We have κ = β ( w ) .Proof. We have H ( B Spin; Z /
2) = Z / { w } , but w of course vanishes when re-stricted to B Spin(6). The long exact sequence on Z / H ( M T
Spin(6); Z / ←− H (Σ − M Spin; Z /
2) = Z / { w · u − } ←− H (Σ F ; Z / x is 1-connected, the map Σ x is 2-connected, so we have an isomorphism(Σ x ) ∗ : Z / { Σ ι } = H (Σ H Z ; Z / ∼ −→ H (Σ F ; Z / . It follows that κ reduces to w = 0 modulo 2. The integral cohomology of B Spinis known to only have torsion of order 2 (this may be deduced from [Kon86]), sothe Bockstein sequence for B Spin shows that H ( B Spin; Z ) = Z /
2, which must therefore be generated by βw as this reduces modulo 2 to Sq ( w ) = w . Thus κ = β ( w ). (cid:3) As w · u − = Sq ( u − ), we may write the map k as the composition k : Σ − M Spin u − −→ Σ − H Z / Sq −→ H Z / β −→ Σ H Z and so we may form a spectrum E fitting into the diagram M T
Spin(6) Σ − M Spin Σ FE Σ − H Z / H Z ∂u − Σ xβ Sq in which the rows are homotopy cofibre sequences. Smashing with BG + and con-sidering the map of long exact sequences givesΩ Spin8 ( BG ) H ( BG ; Z / Z / ⊕ H ( BG ; Z ) H ( BG ; Z ) π ( M T
Spin(6) ∧ BG + ) π ( E ∧ BG + )Ω Spin7 ( BG ) H ( BG ; Z / H ( BG ; Z ) ( β Sq ) ∗ =0pr σ G ( β Sq ) ∗ =0 where the columns are exact and the two indicated maps are zero by instability ofthe (co)homology operation β Sq in these degrees. It therefore suffices to show thatthe right-hand short exact sequence has a dashed splitting σ G as indicated. (Notethat the commutativity of the top square gives another proof of Lemma 6.7.)6.2.4. Reducing the splitting problem to cyclic groups.
The abelianisation homomor-phism a : G → G ab induces a map on the short exact sequences of the form0 H ( BG ; Z ) π ( E ∧ BG + ) H ( BG ; Z /
2) 00 H ( BG ab ; Z ) π ( E ∧ BG ab + ) H ( BG ab ; Z /
2) 0 = a ∗ a ∗ σ Gab so if we can show that the short exact sequence is split for G ab , say via the dashedhomomorphism σ G ab , then the sequence for G is also split, via σ G := σ G ab ◦ a ∗ .This reduces us to studying abelian groups.On the other hand, if such short exact sequences are split for each cyclic groupof prime power order or Z , then we may write G ab = C ⊕ · · · ⊕ C n for cyclic groups C i of prime power order or Z and combine their splittings to obtain one for G ab (though of course it depends on the choice of expression for G ab as a sum of cyclicsubgroups, so is not canonical). We have therefore reduced the question of splittingthe short exact sequences (6.3) to the case of such cyclic groups. ODULI SPACES OF MANIFOLDS 33
The splitting for G = Z /p k or Z . If G = Z or G = Z /p k with p odd then H ( BG ; Z /
2) = 0 and so the short exact sequence becomes0 −→ G −→ π ( E ∧ BG + ) −→ −→ G = Z / k we must go to more trouble: in this case the short exact sequencebecomes 0 −→ Z / k −→ π ( E ∧ B Z / k + ) −→ Z / −→ π ( E ∧ B Z / k + ) ∼ = Z / k +1 . Consider first smashing thecofibre sequence defining E with S/
2, giving the cofibration sequence S/ ∧ E −→ S/ ∧ Σ − H Z / ≃ Σ − H Z / ∨ Σ − H Z / Sq ∨ Sq −→ Σ H Z / ≃ S/ ∧ H Z . Now further smashing this with B Z / k + and considering the long exact sequenceon homotopy groups gives an exact sequence H ( B Z / k ; Z / ⊕ H ( B Z / k ; Z / H ( B Z / k ; Z / π ( S/ ∧ E ∧ B Z / k + ) H ( B Z / k ; Z / ⊕ H ( B Z / k ; Z / H ( B Z / k ; Z / Sq ∗ ⊕ Sq ∗ Sq ∗ ⊕ Sq ∗ and so π ( S/ ∧ E ∧ B Z / k + ) has cardinality 8, because the homology operationsSq ∗ and Sq ∗ are zero in these degrees by instability. On the other hand computingwith the long exact sequence as above gives an exact sequence0 −→ Z = H ( B Z / k ; Z ) −→ π ( E ∧ B Z / k + ) −→ H ( B Z / k ; Z /
2) = Z / −→ B Z / k + → S . ThusKer(2 · − : π ( E ∧ B Z / k + ) → π ( E ∧ B Z / k + )) = Z / π ( S/ ∧ E ∧ B Z / k + ) has cardinality 8, it follows thatCoker(2 · − : π ( E ∧ B Z / k + ) → π ( E ∧ B Z / k + ))has cardinality 4. Thus π ( E ∧ B Z / k + ) cannot be cyclic, so (6.5) is split. Acknowledgements.
The authors would like to thank M. Krannich for usefulcomments on a draft of this paper. S. Galatius was partially supported by theEuropean Research Council (ERC) under the European Union’s Horizon 2020 re-search and innovation programme (grant agreement No. 682922), and by NSF grantDMS-1405001. O. Randal-Williams was partially supported by the ERC under theEuropean Union’s Horizon 2020 research and innovation programme (grant agree-ment No. 756444).
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E-mail address : [email protected]@dpmms.cam.ac.uk