Operations on stable moduli spaces

We construct certain operations on stable moduli spaces and use them to compare cohomology of moduli spaces of closed manifolds with tangential structure. We obtain isomorphisms in a stable range provided the p-adic valuation of the Euler characteristics agree, for all primes p not invertible in the coefficients for cohomology.

Now let d = 2n and W be a d-manifold. The connected sum W #(S n × S n ) is then well defined up to (non-canonical) diffeomorphism, as S n × S n admits an orientationreversing diffeomorphism, and we write W #g(S n × S n ) for the g-fold iteration of this operation. Two manifolds W and W are called stably diffeomorphic if W #g(S n × S n ) is diffeomorphic to W #g (S n × S n ) for some g, g ∈ N. For example, any two orientable connected surfaces are stably diffeomorphic, while two non-orientable connected surfaces are stably diffeomorphic if and only if their Euler characteristic have the same parity.
In this paper, we shall ask about the relationship between H * (M(W ); A) and H * (M(W ); A) when W and W are stably diffeomorphic. As a special case, our main result will provide a canonical isomorphism as long as these manifolds are simply-connected and of dimension 2n > 4, and both (−1) n χ(W ) and (−1) n χ(W ) are large compared with i and have the same p-adic valuation.
The precise statement of our main result applies more generally, and before giving it we first explain its natural setting. If W is given an orientation λ, then there is a corresponding moduli space M or (W, λ) classifying smooth fiber bundles with oriented fibers which are oriented diffeomorphic to (W, λ), and a forgetful map M or (W, λ) → M(W ). Then the connected sum W #g(S n × S n ) inherits an orientation, well defined up to oriented diffeomorphism, and we say that (W, λ) is oriented stably diffeomorphic to (W , λ ) provided W #g(S n × S n ) is oriented diffeomorphic to W #g (S n × S n ) for some g, g ∈ N. In this situation, our result will also imply a canonical isomorphism H i (M or (W, λ); Z (p) ) ∼ = H i (M or (W , λ ); Z (p) ), under the same hypotheses.
More generally, for a space equipped with a continuous action of GL d+1 (R), astructure on a d-manifold W is a GL d (R)-equivariant map λ : Fr(TW ) → , or, equivalently, a GL d+1 (R)-equivariant map Fr(ε 1 ⊕ TW ) → . For example, if = {±1} on which GL d+1 (R) acts by multiplication by the sign of the determinant, then a -structure λ : Fr(TW ) → {±1} is the same thing as an orientation: It distinguishes oriented frames from non-oriented ones. Two -structures on the same manifold are homotopic if they are homotopic through equivariant maps, and (W, λ) is -diffeomorphic to (W , λ ) if there exists a diffeomorphism φ : W → W such that λ • Dφ is homotopic to λ . The usual embedding of S n ×S n ⊂ R 2n+1 as the boundary of a thickened S n ×{0} ⊂ R n+1 ×R n gives a trivialization of ε 1 ⊕ T (S n × S n ) and a -structure on W extends to one on W #(S n × S n ), canonically up to -diffeomorphism. For two pairs (W, λ) and (W , λ ) consisting of a manifold and a -structure, we say that they are stably There is a moduli space M (W, λ) parametrizing smooth fiber bundles π : E → X with d-dimensional fibers, and where the fiberwise tangent bundle T π E is equipped with an equivariant map Fr(ε 1 ⊕ T π E) → , such that all fibers of π are -diffeomorphic to (W, λ). Our main result is then as follows.
(iii) χ(W ) and χ(W ) are both non-zero, and v p (χ(W )) = v p (χ(W )) for all primes p which are not invertible in End Z (A).
In Sect. 4, we give an example showing the third condition cannot be relaxed.
The main results of [5][6][7], summarized in [8], provide a map which induces an isomorphism on homology in a range of degrees, when regarded as a map to the path component which it hits. The definition of the codomains is recalled below. However, if χ(W ) = χ(W ), then these two maps land in different path components, and the problem becomes to compare the homology of these two path components.

Remark 1.2
Using the results of Friedrich [4], Theorem 1.1 can be extended to manifolds with virtually polycyclic fundamental groups. In this case, the constant C should be replaced by C + 4 + 2h where h denotes the Hirsch length of the common fundamental group of W and W .

Operations on infinite loop spaces
The data involved in defining the common target of the maps (1.1) and (1.2) is a GL 2n (R)equivariant fibration u : → with domain which is cofibrant as a GL 2n (R)-space. Letting B denote the Borel construction / /GL 2n (R), MT is then the Thom spectrum of the inverse of the canonical 2n-dimensional vector bundle over B, and ∞ MT is its associated infinite loop space. By functoriality, the group-like topological monoid hAut( ) of GL 2n (R)-equivariant homotopy equivalences f : → acts on the infinite loop space ∞ MT , so the group-like submonoid hAut(u) = {f ∈ hAut( ) | u • f = u} does too.
The target of the maps (1.1) and (1.2) is the Borel construction for this action. In order to prove Theorem 1.1, we shall construct certain operations on the space ∞ MT , in the case where the GL 2n (R)-space is obtained by restriction from a cofibrant GL 2n+1 (R)-space . The space B = / /GL 2n+1 (R) carries a canonical (2n + 1)dimensional vector bundle, and MT denotes its associated Thom spectrum; as above, by functoriality, it carries an action of the monoid hAut( ) of GL 2n+1 (R)-equivariant homotopy equivalences f : → .
A key construction in this paper is a homotopy pullback diagram of infinite loop spaces, equivariant for hAut( ), of the form whose bottom right corner has π 0 ∼ = Z/2 and all higher homotopy groups are 2-power torsion, and the bottom horizontal map induces a surjection on π 1 . It induces an isomorphism whose first coordinate is given by the Euler class and whose second coordinate is given by the stabilization map. To explain this claim and its notation, first note that the 2ndimensional vector bundle over B has an Euler class e ∈ H 2n (B; Z w 1 ), where the coefficients are twisted by the determinant of this vector bundle, and under the Thom isomorphism this gives a class e u −2n ∈ H 0 (MT ; Z). Then χ is the value of this spectrum cohomology class on the Hurewicz image of an element of π 0 MT ; geometrically, it assigns to such an element the Euler characteristic of a manifold representing it. Similarly, the (2n + 1)dimensional vector bundle over B has a 2nth Stiefel-Whitney class w 2n ∈ H 2n (B; Z/2), and under the Thom isomorphism this gives a class w 2n u −2n−1 ∈ H −1 (MT ; Z/2). Then w 2n (x) denotes the value of this spectrum cohomology class on the Hurewicz image of x. For any odd number q, there exists a self-map MT → MT inducing a map

induces an isomorphism in homology with coefficients in any
We shall also prove a version of Theorem 1.3 for q = 2, although it will be marginally weaker in that rather than the map ψ q being defined integrally and inducing an isomorphism with coefficients in any Z[q −1 ]-module, the map ψ 2 will only be defined after localizing the spaces involved away from 2.

Theorem 1.4 In the setup of Theorem 1.3, if χ is even, then there is an hAut( )equivariant weak equivalence of localized spaces
over the identity map of ( ∞−1 MT )[ 1 2 ].
The operations in Theorems 1.3 and 1.4 will arise from self-maps of the lower left corner in (1.3).
The proof of Theorem 1.1 will use these operations to give endomorphisms of the space ( ∞ MT )/ /hAut(u) which mix path components, allowing us to compare the path components hit by the maps (1.1) and (1.2). This strategy is analogous to arguments of Bendersky-Miller [2] and Cantero-Palmer [3] for cohomology of configuration spaces. This strategy has also been used by Krannich [10] an oriented manifold of dimension 2n > 4 and an exotic sphere, in a stable range of degrees when the order of [ ] ∈ 2n is invertible in End Z (A).

Proof of Theorem 1.1
We first explain how to deduce Theorem 1.1 from Theorems 1.3 and 1.4.
be a factorization into an n-connected GL 2n+1 (R)equivariant cofibration ρ and a n-co-connected GL 2n+1 (R)-equivariant fibration u, and as above we write for the underlying GL 2n (R)-space of and u for the underlying GL 2n (R)-equivariant map of u. There is then a map . (Note that by considering a GL 2n+1 (R)space rather than a GL 2n (R)-space, the tangential structure is "spherical" by the discussion after [8,Definition 3.2], and so the stability range is as claimed.) Hereḡ(W, λ) is the stable -genus of (W, λ), the largest g ∈ N for which there exists h ∈ N such that W #h(S n × S n ) is -diffeomorphic to W 0 #(g + h)(S n × S n ) for some (W 0 , λ 0 ).
Let (W 0 , λ 0 ) be a manifold stably -diffeomorphic to (W, λ) and minimizing the quantity (−1) n χ(W 0 ). Such a manifold has stable -genus zero and hence for large enough h we It follows that (2.1) is an isomorphism on ith (co)homology as long as If (W , λ ) is stably -diffeomorphic to (W, λ), then the same analysis applies, and there is a map which induces an isomorphism on ith (co)homology onto the path component which it hits, as long as By assumption, we may write for integers a and b all of whose prime factors are invertible in End Z (A). Furthermore, the two Euler characteristics have the same parity, as (de)stabilization changes the Euler characteristic by ±2, so if either a or b is even then both χ(W ) and χ(W ) are even too. By Theorems 1.3 and 1.4, writing , then (after perhaps implicitly localizing away from 2) there are maps By construction, these maps do not change the π −1 MTcomponent: We now analyze the components corresponding to W and W . We for a suitable choice of ρ : Fr(ε 1 ⊕TW ) → lifting λ . Since these two elements of π 0 (MT ) have the same Euler characteristic, it suffices to arrange that they also have the same which is furthermore an h-cobordism. We can therefore extend the -structure given by (W, ρ), stabilized, to a -structure on X lifting the given -structure, and hence obtain a -manifold (W #g (S n × S n ), ρ ) whose underlying -manifold (W #g (S n × S n ), u • ρ ) is the stabilization of (W , λ ). Now the -manifolds (W #g (S n × S n ), ρ ) and (W , ρ )#g (S n × S n ) (2.3) need not be -diffeomorphic, but must differ by an equivalence f : → over (see [6,Lemma 9.2] which induce isomorphisms on homology with coefficients in A. The argument is completed by the following lemma.

Lemma 2.1
The natural map hAut(u) → hAut(u) is a weak equivalence.
Proof Working in the categories of GL 2n (R)-spaces over , or GL 2n+1 (R)-spaces over , we have → has homotopy fiber GL 2n+1 (R)/GL 2n (R) S 2n so is 2n-connected, whereas u : → is n-coconnected, so the restriction map is an equivalence. The claim now follows by restricting to the path-components of homotopy equivalences.

Proof of Theorems 1.and 1.4
The proof of Theorem 1.3 is by an explicit construction of ψ q as a map of spectra. The main ingredient is a certain commutative diagram of spectra, which we first describe informally. It is where s : B → B is the natural map of Borel constructions. The map s is homotopy equivalent to a smooth fiber bundle with fibers S 2n so we have a Becker-Gottlieb transfer t : ∞ B + → ∞ B + , factoring as a pre-transfer p : ∞ B + → MT composed with a map z : MT → ∞ B + induced by the zero section of θ. The spectrum C st is defined to be the homotopy cofiber of st, and both rows are cofiber sequences. It follows that the right square in the diagram is a homotopy pullback, and hence we get the homotopy pullback diagram of infinite loop spaces (1.3) mentioned in the introduction. On spectrum homology the map st induces multiplication by χ(S 2n ) = 2, from which it follows that the homology and hence homotopy groups of C st are 2-power torsion. The space B is path connected, because W is, so π 0 ( ∞ B + ) = H 0 ( ∞ B + ; Z) = Z. Thus π 0 (C st ) = Z/2, and the map ∞ B + → C st is surjective on π 1 because st is injective on π 0 . To produce an endomorphism of ∞ MT satisfying part (ii) of the theorem, it therefore suffices to produce an endomorphism of ∞ B + over C st . For q = 1 + 2k, we may use the map id + kst : ∞ B + → ∞ B + which is obviously over C st , at least in the homotopy category, since C st is the cofiber of the map st. In spectrum homology, st multiplies by χ(S 2n ) = 2 and hence id + kst induces multiplication by 1 + 2k = q on H * ( ∞ B + ; Z) ensuring part (iii) of the theorem. Furthermore, it acts by multiplication by q on π 0 ∞ B + = π 0 Q(B + ) = Z, so indeed sends ∞ χ MT to ∞ qχ MT . It remains to explain how to achieve part (i) of the theorem, that the continuous action of the topological monoid hAut( ) on the space ∞ MT commutes with ψ q . It is not sufficient that ψ q commutes up to homotopy with the action of individual elements of hAut( ), since we want to descend ψ q to the homotopy orbit space. To give a convincing proof, it seems best to spell out a point-set model for the square (1.3).
Proof of Theorem 1.3 As explained above, it remains to give a point-set model for the diagram (1.3) and the self-map id + kst of Q(B + ) over ∞ C st , all of which commute strictly with the action of hAut( ).
We must adopt some conventions. Let us consider GL 2n (R) as lying inside GL 2n+1 (R) using the last 2n coordinates. Let us consider R N −1 as lying inside R N as the subspace of vectors whose last coordinate is 0, and take R ∞ to be the direct limit. To form the Borel constructions, we shall take EGL 2n (R) := Fr 2n (R ∞ ), and similarly take EGL 2n+1 (R) := Fr 2n+1 (R ⊕ R ∞ ). The map Fr 2n (R ∞ ) → Fr 2n+1 (R ⊕ R ∞ ) which adds the basis vector of the first R-summand as the first element of the (2n + 1)-frame is then equivariant for the inclusion GL 2n (R) ⊂ GL 2n+1 (R).
Then we have BGL 2n+1 (R) = Gr 2n+1 (R ⊕ R ∞ ), which we may filter in the usual way by Gr 2n+1 (R ⊕ R N −1 ). Pulling back this filtration along the map θ : B → Gr 2n+1 (R ∞ ), we set ,N for the pullback of the (N − 2n − 1)-dimensional bundle of orthogonal complements. Then MT is the spectrum with N th space given by the Thom space (B N ) θ * N γ ⊥ , so that We similarly define θ N : B N → Gr 2n (R N ), and hence the spectrum MT . There is a map These combine to define a map from MT to the spectrum whose ( The square (1.3) will be assembled from a square of spaces fibered over B N , and we first explain the constructions on fibers. Let V ∈ Gr 2n+1 (R N ) and write S(V ) for the unit sphere of V and S V for the one-point compactification. If x ∈ R N , we shall write π V (x) ∈ V for the orthogonal projection. If x ∈ V \ 0, we shall write π S (x) = x/|x| ∈ S(V ) for the nearest point in the sphere. We will describe certain explicit maps p(V ) : S V → S(V ) ε 1 and z(V ) : S(V ) ε 1 → S(V ) + ∧ S V , and explain how the composition z(V ) • p(V ) gives rise to a model for the Becker-Gottlieb transfer for a linear sphere bundle. (Indeed, we will just unwrap the definition of [1,Sect. 3] in this case.) The map is induced by the Pontryagin-Thom construction applied to the embedding S(V ) ⊂ V . In formulas, we can take, e.g.
The Thom space S(V ) ε 1 is homeomorphic to the quotient S V /S 0 , and under this identification, the map p(V ) is the quotient map. The map is given by the zero section of the tangent bundle of S(V ). In formulas, it sends ( If we compose these two maps and smash with S V ⊥ , we get where the entries in the right column are the mapping cylinders. Since p(V ) induces a homeomorphism S N /S V ⊥ → S(V ) ε ∧ S V ⊥ , it follows from the Puppe sequence that there is a canonical induced homeomorphism C p(V ) ∼ = S 1 ∧ S V ⊥ . Since st(V ) has degree 2, there is a homotopy equivalence from C st(V ) to a mod 2 Moore space, but this is not quite sufficiently canonical for our purposes (since we get a different mod 2 Moore space for each V ). We have proved that for each V ∈ Gr 2n+1 (R N ) there is a canonical commutative diagram which is a pushout and homotopy pushout.
There is a canonical homotopy from the composition of st(V ) : S N → S N and S N → C st(V ) to the constant map. Suspending once, is canonically null homotopic. If k ≥ 0 is an integer, we may use the S 1 coordinate to form the sum of the identity map 1 : S 1 ∧ S N → S 1 ∧ S N and k copies of the map which commutes up to a canonical homotopy. (The canonical nullhomotopy of each st gives a homotopy from 1 + kst to the sum of the identity map and k copies of the constant map; this is in turn canonically homotopic to the identity map.) The homotopy class of the map 1 + kst(V ) : S N → S N is determined by its degree which is 2k + 1, but the actual map depends in a non-trivial way on V ∈ Gr 2n+1 (R N ). All spaces in the diagram "vary continuously in V ," in the sense that they are fibers over V of fiber bundles over Gr 2n+1 (R N ). The commutative diagram (3.2) in the category of spaces over Gr 2n+1 (R N ) may be pulled back along θ N : B N → Gr 2n+1 (R N ) to form a diagram which is again a pushout and homotopy pushout, where C B N st is the mapping cylinder of the map Similarly, the diagrams (3.3) assemble over V to a diagram which commutes up to a canonical homotopy.
Applying N +1 S 1 ∧ (−) to the diagram (3.4) and letting N → ∞, we get a model for (1.3). The monoid hAut( ) acts on the whole diagram (3.4), since it acts on B N over Gr 2n+1 (R N ). This gives a weak equivalence from ∞ MT to the homotopy pullback in (1.3), which is also an hAut( ) equivariant map. The monoid hAut( ) also acts on the diagram (3.5), including the homotopy, and after applying N +1 and taking N → ∞, we obtain a self-map of Q(B + ) which is over ∞ C st up to a specified homotopy. Again this self-map and the specified homotopy commute strictly with the action of hAut( ) since both the map and the homotopy arose from fiberwise constructions over Gr 2n+1 (R N ).
Finally, the self-map of Q(B + ) induces an hAut( )-equivariant self-map of the homotopy pullback of Q(B + ) → ∞ C st ← ∞−1 MT , and we have seen that this pullback is weakly equivalent to ∞ MT by an hAut( )-equivariant map.
Proof of Theorem 1.4 We continue with the notation developed above. The spectrum homology of C st is all 2-torsion, so the localization C st [ 1 2 ] as a spectrum is contractible. However, the localized space ( ∞ C st )[ 1 2 ] is not contractible since it has two components. Instead, there is a spectrum map w 2n : C st → HF 2 which becomes an isomorphism in homology of infinite loop spaces with coefficients in any Z[ 1 2 ]-module. Similarly, the map induces an isomorphism in homology with coefficients in any Z[ 1 2 ]-module, and hence a weak equivalence of localized spaces. The spectrum map 2 : S 0 → S 0 induces a self-map of Q(B + ) commuting with the action of hAut( ) and whose restriction to the even-degree path components commutes with the map to ∞ HF 2 . This self-map can be used in place of 1 + kst to produce ψ 2 .

An example
In this section, we will give an example to show that in Theorem 1.1 it is indeed necessary to take homology with certain primes inverted. We will take as an example the 6-manifolds V d given by a smooth degree d hypersurface in CP 4 , which we have studied in detail in [8,Sect. 5.3]. Any unattributed claims about these manifolds may be found there. We will also consider their stabilizations obtained by connect-sum of V d with g copies of S 3 × S 3 , which contain The formula χ = χ(V d,g ) = d(10−10d +5d 2 −d 3 )−2g implies that gcd(d, g) = gcd(d, χ), so the theorem may also be written Hence the moduli spaces for the oriented stably diffeomorphic manifolds V d,g and V d,g have isomorphic H 3 (−; Z (p) ) if and only if v p (χ(V d,g )) = v p (χ(V d,g )), provided those p-adic valuations are at most v p (d).
Proof of Theorem 4.1 In [8, Sect. 5.3], we computed the Q-cohomology of M or (V d,g ) in a stable range. We will refer to details of the notation from that discussion, which differs slightly from the notation used earlier in this note. Firstly, the Q-cohomology calculation goes through without significant changes for M or (V d,g ), because V d,g and V d have the same Moore-Postnikov 3-stage, and because any orientation preserving diffeomorphism of V d,g must also act trivially on H 2 (V d,g ; Z).
The only difference is that the formula for the d 3 -differential now involves characteristic numbers of V d,g , which can be calculated to give Secondly, the Q-cohomology calculation yields an analogous Z (p) -cohomology calculation for large enough primes p. Specifically the spectrum MT θ d is (−6)-connected, so by the Atiyah-Hirzebruch spectral sequence the Hurewicz map is an isomorphism as long as i < 2p − 3 − 6, so as long as i ≤ 5 since we have assumed that p ≥ 7. As p is odd we have H * (B d ; Z (p) ) = H * (BSO(6) × K (Z, 2); Z (p) ) = Z (p) [p 1 , p 2 , e, t].