S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 https://doi.org/10.1007/s40687-020-00207-6 RESEARCH Operations on stable moduli spaces Søren Galatius1* and Oscar Randal-Williams2 *Correspondence: galatius@math.ku.dk 1Department of Mathematics, University of Copenhagen, Copenhagen, Denmark Full list of author information is available at the end of the article Abstract 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. Keywords: Moduli spaces, Diffeomorphism groups, Homological stability, Characteristic classes Mathematics Subject Classification: 55P47, 55R40, 57S05, 57R15, 57R90 1 Introduction An influential theorem of Harer [9] shows that the cohomology of the moduli stack Mg of genus g Riemann surfaces is independent of g in a range of degrees called the stable range, even though there is no direct map between the moduli spaces for different genera. With rational coefficients, the cohomology in the stable range is a polynomial ring, but withmore general coefficients it is best described via infinite loop spaces, as shown by [11– 13]. In earlier papers ([5–7], see also [8] for a survey), we have studied moduli spaces of higher-dimensional manifolds and, in some cases, have again shown that different moduli spaces have isomorphic cohomology in a range of degrees. For n > 1, one can in most cases notmake an integral comparison ofmoduli spaces ofmanifolds related by connected sum with copies of Sn × Sn, at least not by an obvious generalization of the n = 1 case, where a zig-zag of integral homology equivalences can be defined using manifolds with boundary. In this paper, we show that a comparison is possible after all, although not with all coefficient modules. We also give examples showing that assumptions on the coefficients are necessary. 1.1 Comparing moduli spaces of closed manifolds All manifolds in this paper will be smooth, compact, connected, and without boundary. If W denotes such a manifold, then there is a moduli space M(W ) classifying smooth fiber bundles whose fibers are diffeomorphic to W . As a model, we may take M(W ) = BDiff(W ), the classifying space of the diffeomorphismgroupDiff(W ) ofW , equippedwith the C∞ topology. Then for A an abelian group, Hi(M(W );A) is the group of Hi(−;A)- valued characteristic classes of such fiber bundles. 123 © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV 9 Page 2 of 13 S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 Now let d = 2n and W be a d-manifold. The connected sum W#(Sn × Sn) is then well defined up to (non-canonical) diffeomorphism, as Sn × Sn admits an orientation- reversing diffeomorphism, and we write W#g(Sn × Sn) for the g-fold iteration of this operation. Two manifolds W and W ′ are called stably diffeomorphic if W#g(Sn × Sn) is diffeomorphic to W ′#g ′(Sn × Sn) for some g, g ′ ∈ N. For example, any two orientable connected surfaces are stably diffeomorphic, while twonon-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 Hi(M(W );Z(p)) ∼= Hi(M(W ′);Z(p)) as long as these manifolds are simply-connected and of dimension 2n > 4, and both (−1)nχ (W ) and (−1)nχ (W ′) are large comparedwith i andhave the samep-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 corre- sponding moduli space Mor(W, λ) classifying smooth fiber bundles with oriented fibers which are oriented diffeomorphic to (W, λ), and a forgetful map Mor(W, λ) → M(W ). Then the connected sum W#g(Sn × Sn) inherits an orientation, well defined up to ori- ented diffeomorphism, and we say that (W, λ) is oriented stably diffeomorphic to (W ′, λ′) providedW#g(Sn×Sn) is oriented diffeomorphic toW ′#g ′(Sn×Sn) for some g, g ′ ∈ N. In this situation, our result will also imply a canonical isomorphism Hi(Mor(W, λ);Z(p)) ∼= Hi(Mor(W ′, λ′);Z(p)), under the same hypotheses. More generally, for a space  equipped with a continuous action of GLd+1(R), a - structure on a d-manifold W is a GLd(R)-equivariant map λ : Fr(TW ) → , or, equiv- alently, a GLd+1(R)-equivariant map Fr(ε1 ⊕ TW ) → . For example, if  = {±1} on which GLd+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 Sn×Sn ⊂ R2n+1 as the boundary of a thickened Sn×{0} ⊂ Rn+1×Rn gives a trivialization of ε1 ⊕T (Sn × Sn) and a-structure onW extends to one onW#(Sn × Sn), 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 -diffeomorphic ifW#g(Sn × Sn) is -diffeomorphic toW ′#g ′(Sn × Sn) for some g, g ′ ∈ N. There is a moduli spaceM(W, λ) parametrizing smooth fiber bundles π :E → X with d-dimensional fibers, and where the fiberwise tangent bundle TπE is equipped with an equivariant map Fr(ε1 ⊕ TπE) → , such that all fibers of π are -diffeomorphic to (W, λ). Our main result is then as follows. Theorem 1.1 Let be as above, and let λ and λ′ be-structures onW andW ′ such that (W, λ) is stably -diffeomorphic to (W ′, λ′). For an abelian group A, there is a canonical isomorphism Hi(M(W, λ);A) ∼= Hi(M(W ′, λ′);A), S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 Page 3 of 13 9 induced by a zig-zag of maps of spaces, provided (i) d = 2n > 4 and W and W ′ are simply connected, (ii) the integers (−1)nχ (W ) and (−1)nχ (W ′) are both ≥ 4i + C, where C = 6+min{(−1)nχ (W0) | (W0, λ0) stably -diffeomorphic to (W, λ) and (W ′, λ′)}. (iii) χ (W ) and χ (W ′) are both non-zero, and vp(χ (W )) = vp(χ (W ′)) for all primes p which are not invertible in EndZ(A). In Sect. 4, we give an example showing the third condition cannot be relaxed. The main results of [5–7], summarized in [8], provide a map M(W, λ) −→ (∞MT )/hAut(u), (1.1) which induces an isomorphism on homology in a range of degrees, when regarded as a map to the path component which it hits. Similarly there is a map M(W ′, λ′) −→ (∞MT )/hAut(u) (1.2) which induces an isomorphismonhomology in a range of degrees, when regarded as amap 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 mani- folds 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 ofW andW ′. 1.2 Operations on infinite loop spaces The data involved in defining the common target of the maps (1.1) and (1.2) is a GL2n(R)- equivariant fibration u : →  with domain which is cofibrant as a GL2n(R)-space. Letting B denote the Borel construction /GL2n(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 topologicalmonoid hAut( ) of GL2n(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 (∞MT )/hAut(u) 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 GL2n(R)-space is obtained by restriction from a cofi- brant GL2n+1(R)-space . The space B = /GL2n+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 GL2n+1(R)-equivariant homotopy equivalences f : → . 9 Page 4 of 13 S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 A key construction in this paper is a homotopy pullback diagram of infinite loop spaces, equivariant for hAut( ), of the form ∞MT ∞−1MT Q(B+) ∞Cst , (1.3) whose bottom right corner has π0 ∼= Z/2 and all higher homotopy groups are 2-power torsion, and the bottom horizontal map induces a surjection on π1. It induces an isomor- phism π0MT ∼=−→ {(χ , x) ∈ Z × π−1MT | χ mod 2 = w2n(x)}, (1.4) 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 2n- dimensional vector bundle overBhas anEuler class e ∈ H2n(B;Zw1 ),where the coefficients are twisted by the determinant of this vector bundle, and under the Thom isomorphism this gives a class e u−2n ∈ H0(MT ;Z). Then χ is the value of this spectrum cohomol- ogy class on theHurewicz image of an element ofπ0MT ; 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 w2n ∈ H2n(B;Z/2), and under the Thom isomorphism this gives a class w2n u−2n−1 ∈ H−1(MT ;Z/2). Then w2n(x) denotes the value of this spectrum cohomology class on the Hurewicz image of x. Theorem 1.3 For χ ∈ Z, write ∞χ MT for the inverse image of χ under the map ∞MT → Z induced by the class e u−2n ∈ H0(MT ;Z), i.e., the union of the path components of the form (χ , ?) under the bijection (1.4). For any odd number q, there exists a self-map MT → MT inducing a map ψq :∞χ MT −→ ∞qχMT such that (i) ψq commutes (strictly) with the action of hAut( ), (ii) ψq is over the identity map of ∞−1MT , (iii) ψq induces an isomorphism in homology with coefficients in any Z[q−1]-module. 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 ψq being defined integrally and inducing an isomor- phism with coefficients in any Z[q−1]-module, the map ψ2 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 ψ2 : (∞χ MT ) [ 1 2 ] −→ (∞2χMT ) [ 1 2 ] over the identity map of (∞−1MT )[ 12 ]. S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 Page 5 of 13 9 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) whichmix 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] to show that Hi(Mor(W, λ);A) ∼= Hi(Mor(W# , λ);A) for (W, λ) 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 EndZ(A). 2 Proof of Theorem 1.1 We first explain how to deduce Theorem 1.1 from Theorems 1.3 and 1.4. Let λ : Fr(ε1 ⊕ TW ) ρ→ u→  be a factorization into an n-connected GL2n+1(R)- equivariant cofibration ρ and a n-co-connected GL2n+1(R)-equivariant fibration u, and as above we write for the underlying GL2n(R)-space of and u for the underlying GL2n(R)-equivariant map of u. There is then a map M(W, λ) −→ (∞MT )/hAut(u), (2.1) which by [6, Corollary 1.9] induces an isomorphism on ith (co)homology onto the path component which it hits, as long as i ≤ g(W,λ)−32 . (Note that by considering a GL2n+1(R)- space  rather than a GL2n(R)-space, the tangential structure is “spherical” by the discussion after [8, Definition 3.2], and so the stability range is as claimed.) Here g¯(W, λ) is the stable -genus of (W, λ), the largest g ∈ N for which there exists h ∈ N such that W#h(Sn × Sn) is -diffeomorphic toW0#(g + h)(Sn × Sn) for some (W0, λ0). Let (W0, λ0) be amanifold stably-diffeomorphic to (W, λ) andminimizing the quantity (−1)nχ (W0). Such a manifold has stable -genus zero and hence for large enough h we must have thatW#h(Sn × Sn) is -diffeomorphic toW0#(h + g(W, λ))(Sn × Sn), so g(W, λ) = (−1)n(χ (W ) − χ (W0))/2. It follows that (2.1) is an isomorphism on ith (co)homology as long as (−1)nχ (W ) ≥ 4i + (6 + (−1)nχ (W0)) . If (W ′, λ′) is stably -diffeomorphic to (W, λ), then the same analysis applies, and there is a map M(W ′, λ′) −→ (∞MT )/hAut(u) (2.2) which induces an isomorphism on ith (co)homology onto the path component which it hits, as long as (−1)nχ (W ′) ≥ 4i + (6 + (−1)nχ (W0)) . By assumption, we may write a · χ (W ) = b · χ (W ′) 9 Page 6 of 13 S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 for integers a and b all of whose prime factors are invertible in EndZ(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. ByTheorems 1.3 and 1.4, writingψx = ψx/2v2(x) ◦(ψ2)v2(x), then (after perhaps implicitly localizing away from 2) there are maps ∞ χ (W )MT ψa ∞aχ (W )MT ∞ χ (W ′)MT ψb ∞bχ (W ′)MT which are hAut( )-equivariant and induce isomorphisms on A-homology, as A is a Z[a−1, b−1]-module. By construction, these maps do not change the π−1MT - component: We now analyze the components corresponding toW andW ′. We now claim that ψa([W, ρ]) = ψb([W ′, ρ′]) ∈ π0(∞MT ) for a suitable choice of ρ′ : Fr(ε1⊕TW ′) → liftingλ′. Since these twoelements ofπ0(MT ) have the sameEuler characteristic, it suffices to arrange that they also have the same π−1MT -component. The stable -diffeomorphism from (W, λ) to (W ′, λ′) gives a -cobordism X :W#g(Sn × Sn)  W ′#g ′(Sn × Sn), 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 ′(Sn × Sn), ρ′′) whose underlying -manifold (W ′#g ′(Sn × Sn), u ◦ ρ′′) is the stabilization of (W ′, λ′). Now the -manifolds (W ′#g ′(Sn × Sn), ρ′′) and (W ′, ρ′)#g ′(Sn × Sn) (2.3) need not be -diffeomorphic, but must differ by an equivalence f : → over (see [6, Lemma 9.2]). However, the -structure ρ′ on W ′ is merely a choice of lift of λ′ along u, and by rechoosing it to be f ◦ ρ′, we may then suppose that the manifolds (2.3) are indeed -diffeomorphic. With this choice, we therefore have the desired [W, ρ] = [W ′, ρ′] ∈ π−1MT , using the -cobordism X and the fact that this cobordism theory is insensitive to stabi- lization by standard Sn × Sn’s. Denoting by [[W, λ]] ⊂ π0MT the π0hAut(u)-orbit of [W, ρ], and similarly [[W ′, λ′]], and using the forgetful homomorphism hAut(u) → hAut( ) to let the monoid hAut(u) act on ∞MT , we therefore have a zig-zag of maps ( ∞[[W,λ]]MT ) /hAut(u) −→ · ←− ( ∞[[W ′ ,λ′]]MT ) /hAut(u), (2.4) which induce isomorphisms on homology with coefficients in A. The argument is com- pleted by the following lemma. Lemma 2.1 The natural map hAut(u) → hAut(u) is a weak equivalence. S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 Page 7 of 13 9 Proof Working in the categories of GL2n(R)-spaces over, or GL2n+1(R)-spaces over, we have map/GL2n(R)( , ) = map / GL2n+1(R)(GL2n+1(R) ×GL2n(R) , ) but the natural GL2n+1(R)-equivariant map GL2n+1(R) ×GL2n(R) → has homo- topy fiber GL2n+1(R)/GL2n(R)  S2n so is 2n-connected, whereas u : →  is n-co- connected, so the restriction map map/GL2n+1(R)( , ) −→ map / GL2n+1(R)(GL2n+1(R) ×GL2n(R) , ) is an equivalence. The claim now follows by restricting to the path-components of homo- topy equivalences. unionsq Remark 2.2 This argument also gives a conclusion about homology with certain local coefficients. The maps (2.1) and (2.2) are in fact acyclic in a range of degrees [6, Corollary 1.9], and the maps ψq are acyclic with Z[q−1]-module coefficients (as they are infinite loop maps which induce isomorphisms on homology with these coefficients) so remain so after taking homotopy orbits by hAut(u). So if A is a system of local coefficients on the middle space of the zig-zag (2.4), with typical fiber A and having vp(χ (W )) = vp(χ (W ′)) for all primes pwhich are not invertible in EndZ(A), then there is also an isomorphism Hi(M(W, λ);A) ∼= Hi(M(W ′, λ′);A) in a range of degrees. 3 Proof of Theorems 1.3 and 1.4 The proof of Theorem 1.3 is by an explicit construction of ψq as a map of spectra. The main ingredient is a certain commutative diagram of spectra, which we first describe informally. It is ∞B+ p MT sz S1 ∧ MT ∞B+ st ∞B+ Cst where s :B → B is the natural map of Borel constructions. The map s is homotopy equivalent to a smooth fiber bundle with fibers S2n 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 Cst 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 diagramof infinite loop spaces (1.3)mentioned in the introduction.On spectrum homology the map st induces multiplication by χ (S2n) = 2, fromwhich it follows that the homology and hence homotopy groups of Cst are 2-power torsion. The space B is path connected, because W is, so π0( ∞B+) = H0( ∞B+;Z) = Z. Thus π0(Cst ) = Z/2, and the map ∞B+ → Cst is surjective on π1 because st is injective on π0. To produce an endomorphism of ∞MT satisfying part (ii) of the theorem, it there- fore suffices to produce an endomorphism of ∞B+ over Cst . For q = 1 + 2k , we may use the map id + kst : ∞B+ → ∞B+ which is obviously over Cst , at least in the homotopy category, since Cst is the cofiber of the map st. In spectrum homology, st 9 Page 8 of 13 S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 multiplies by χ (S2n) = 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 π0 ∞B+ = π0Q(B+) = Z, so indeed sends ∞χ MT to ∞qχMT . 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 ψq . It is not sufficient that ψq commutes up to homotopy with the action of individual elements of hAut( ), since we want to descendψq 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 ∞Cst , all of which commute strictly with the action of hAut( ). We must adopt some conventions. Let us consider GL2n(R) as lying inside GL2n+1(R) using the last 2n coordinates. Let us consider RN−1 as lying inside RN as the sub- space of vectors whose last coordinate is 0, and take R∞ to be the direct limit. To form the Borel constructions, we shall take EGL2n(R) := Fr2n(R∞), and similarly take EGL2n+1(R) := Fr2n+1(R ⊕ R∞). The map Fr2n(R∞) → Fr2n+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 GL2n(R) ⊂ GL2n+1(R). Then we have BGL2n+1(R) = Gr2n+1(R⊕R∞), which we may filter in the usual way by Gr2n+1(R⊕RN−1). Pulling back this filtration along the map θ :B → Gr2n+1(R∞), we set BN := (θ )−1(Gr2n+1(R⊕RN−1)). There is an induced map θN :BN → Gr2n+1(R⊕RN−1) and we shall write θ∗Nγ ⊥ = θ∗Nγ ⊥2n+1,N for the pullback of the (N − 2n − 1)-dimensional bundle of orthogonal complements. ThenMT is the spectrum withN th space given by the Thom space (BN )θ ∗ N γ ⊥ , so that ∞−1MT = colim N→∞  N−1(BN )θ ∗ N γ ⊥ . We similarly define θN :BN → Gr2n(RN ), and hence the spectrumMT . There is a map Gr2n(RN−1) ↪→ Gr2n+1(R ⊕ RN−1), (3.1) given by direct sum with the 1-dimensional vector space given by the first R-summand, which induces a map BN−1 → BN . The map (3.1) is 2n-connected, but is covered by an (N − 2)-connected map Gr2n(RN−1) → S(γ2n+1,N ) and hence gives a (N − 2)-connected mapBN−1 → S(θ∗Nγ2n+1,N ). Passing to Thom spaces, this gives a (2N−2n−2)-connected map S1 ∧ (BN−1)(θN |BN−1 )∗γ⊥2n,N−1 −→ S(θ∗Nγ2n+1,N )θ ∗ N γ ⊥ 2n+1,N . These combine to define a map from MT to the spectrum whose (N − 1)st space is S(θ∗Nγ2n+1,N )θ ∗ N γ ⊥ 2n+1,N , and this map is a weak equivalence. This map is also hAut( )- equivariant. (This weak equivalence does not come with a spectrum map in the other direction, let alone an equivariant one.) The square (1.3) will be assembled from a square of spaces fibered over BN , and we first explain the constructions onfibers. LetV ∈ Gr2n+1(RN ) andwriteS(V ) for theunit sphere of V and SV for the one-point compactification. If x ∈ RN , we shall write πV (x) ∈ V for S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 Page 9 of 13 9 the orthogonal projection. If x ∈ V \ 0, we shall write πS(x) = x/|x| ∈ S(V ) for the nearest point in the sphere. We will describe certain explicit maps p(V ) : SV → S(V )ε1 and z(V ) : S(V )ε1 → S(V )+ ∧ SV , 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 p(V ) : SV −→ S(V )ε1 , is induced by the Pontryagin–Thom construction applied to the embedding S(V ) ⊂ V . In formulas, we can take, e.g. p(V )(x) = (πS(x), log |x|) ∈ S(V )+ ∧ S1 = S(V )ε1 when x = 0,∞ ∈ SV . The Thom space S(V )ε1 is homeomorphic to the quotient SV /S0, and under this identification, the map p(V ) is the quotient map. The map z(V ) : S(V )ε1 −→ S(V )TS(V )⊕ε1 = S(V )+ ∧ SV is given by the zero section of the tangent bundle of S(V ). In formulas, it sends (x, t) ∈ S(V ) × R ⊂ S(V )ε1 to (x, tx) ∈ S(V ) × V ⊂ S(V )+ ∧ SV . If we compose these two maps and smash with SV⊥ , we get SN = SV ∧ SV⊥ p(V )∧id−−−−→ S(V )ε1 ∧ SV⊥ z(V )∧id−−−−→ S(V )+ ∧ SV ∧ SV⊥ = S(V )+ ∧ SN . Finally, we write s(V ) : S(V )+ ∧ SN → SN for the map induced by collapsing S(V ) to a point. Then the composition b(V ) = s(V ) ◦ (z(V ) ∧ id) ◦ (p(V ) ∧ id) : SN −→ SN is a continuousmapof degreeχ (S2n) = 2 (by thePoincaré–Hopf theorem, see [1,Theorem 2.4]), depending continuously on the point V ∈ Gr2n+1(RN ). The resulting continuous map b : Gr2n+1(RN ) → NSN in the limit gives a map BGL2n+1(R) → QS0 which is a model for the Becker–Gottlieb transfer of the sphere bundle over the spaceBGL2n+1(R)  BO(2n + 1). Now consider the diagram SN p(V ) S(V )ε1 ∧ SV⊥ sz Cp(V ) SN st(V ) SN Cst(V ), where the entries in the right column are the mapping cylinders. Since p(V ) induces a homeomorphism SN /SV⊥ → S(V )ε ∧SV⊥ , it follows from the Puppe sequence that there is a canonical induced homeomorphism Cp(V ) ∼= S1 ∧ SV⊥ . Since st(V ) has degree 2, there is a homotopy equivalence from Cst(V ) to a mod 2Moore space, but this is not quite 9 Page 10 of 13 S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 sufficiently canonical for our purposes (since we get a different mod 2 Moore space for eachV ).We have proved that for eachV ∈ Gr2n+1(RN ) there is a canonical commutative diagram S(V )ε1 ∧ SV⊥ sz S1 ∧ SV⊥ SN Cst(V ), (3.2) which is a pushout and homotopy pushout. There is a canonical homotopy from the composition of st(V ) : SN → SN and SN → Cst(V ) to the constant map. Suspending once, S1 ∧ SN → S1 ∧ SN → S1 ∧ Cst(V ) is canonically null homotopic. If k ≥ 0 is an integer, we may use the S1 coordinate to form the sum of the identity map 1 : S1 ∧ SN → S1 ∧ SN and k copies of the map st(V ) : S1 ∧ SN → S1 ∧ SN . We obtain a diagram S1 ∧ SN 1+kst(V ) S1 ∧ Cst(V ) S1 ∧ SN S1 ∧ Cst(V ), (3.3) 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 ) : SN → SN is determined by its degree which is 2k + 1, but the actual map depends in a non-trivial way on V ∈ Gr2n+1(RN ). All spaces in the diagram “vary continuously in V ,” in the sense that they are fibers over V of fiber bundles over Gr2n+1(RN ). The commutative diagram (3.2) in the category of spaces over Gr2n+1(RN ) may be pulled back along θN : BN → Gr2n+1(RN ) to form a diagram S(θ∗Nγ )ε 1⊕θ∗N γ⊥ sz S1 ∧ Bθ ∗ N γ ⊥ N SN ∧ (BN )+ CBNst , (3.4) which is again a pushout and homotopy pushout, where CBNst is the mapping cylinder of the map SN ∧ (BN )+ → SN ∧ (BN )+ given on (v, x) ∈ SN × BN by st(v, x) = (st(f (x))v, x). Similarly, the diagrams (3.3) assemble over V to a diagram S1 ∧ SN ∧ (BN )+ 1+kst S1 ∧ CBNst S1 ∧ SN ∧ (BN )+ S1 ∧ CBNst , (3.5) which commutes up to a canonical homotopy. S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 Page 11 of 13 9 Applying N+1S1 ∧ (−) 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 BN over Gr2n+1(RN ). 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 ∞Cst 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 Gr2n+1(RN ). Finally, the self-map of Q(B+) induces an hAut( )-equivariant self-map of the homo- topy pullback of Q(B+) → ∞Cst ← ∞−1MT , 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 Cst is all 2-torsion, so the localization Cst [ 12 ] as a spectrum is contractible. However, the localized space (∞Cst )[ 12 ] is not contractible since it has two components. Instead, there is a spectrum map w2n :Cst → HF2 which becomes an isomorphism in homology of infinite loop spaces with coefficients in any Z[ 12 ]-module. Similarly, the map ∞MT −→ Q(B+) ×∞HF2 ∞−1MT induces an isomorphism in homology with coefficients in any Z[ 12 ]-module, and hence a weak equivalence of localized spaces. The spectrum map 2 : S0 → S0 induces a self-map ofQ(B+) commuting with the action of hAut( ) and whose restriction to the even-degree path components commutes with the map to∞HF2. This self-map can be used in place of 1 + kst to produce ψ2. 4 An example In this section, we will give an example to show that in Theorem 1.1 it is indeed necessary to take homologywith certain primes inverted.Wewill take as an example the 6-manifolds Vd given by a smooth degree d hypersurface in CP4, 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 Vd,g : = Vd#g(S3 × S3) obtained by connect-sum of Vd with g copies of S3 × S3, which contain g(Vd,g ) = g + 12 (d4 − 5d3 + 10d2 − 10d + 4) copies of S3 × S3. Theorem 4.1 Let p ≥ 7 be a prime number, and suppose that g(Vd,g ) ≥ 9. Then H3(Mor(Vd,g );Z(p)) ∼= Z(p)/ gcd(d, g). The formulaχ = χ (Vd,g ) = d(10−10d+5d2−d3)−2g implies that gcd(d, g) = gcd(d,χ ), so the theorem may also be written H3(Mor(Vd,g );Z(p)) ∼= Z/pmin(vp(d),vp(χ ))Z. 9 Page 12 of 13 S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 Hence the moduli spaces for the oriented stably diffeomorphic manifolds Vd,g and Vd,g ′ have isomorphic H3(−;Z(p)) if and only if vp(χ (Vd,g )) = vp(χ (Vd,g ′ )), provided those p-adic valuations are at most vp(d). Proof of Theorem 4.1 In [8, Sect. 5.3], we computed the Q-cohomology of Mor(Vd,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 Mor(Vd,g ), because Vd,g and Vd have the same Moore–Postnikov 3-stage, and because any orientation preserving diffeomorphism of Vd,g must also act trivially on H2(Vd,g ;Z). The only difference is that the formula for the d3-differential now involves characteristic numbers of Vd,g , which can be calculated to give d3(κp2 ) = 0, d3(κp21 ) = 0, d3(κte) = κe = χ (Vd,g ) = d(10 − 10d + 5d2 − d3) − 2g, d3(κt2p1 ) = 2κtp1 = 2d(5 − d2), d3(κt4 ) = 4κt3 = 4d. Secondly, the Q-cohomology calculation yields an analogous Z(p)-cohomology calcula- tion for large enough primes p. Specifically the spectrumMTθd is (−6)-connected, so by the Atiyah–Hirzebruch spectral sequence the Hurewicz map πi(MTθd)(p) −→ Hi(MTθd ;Z(p)) ∼= Hi+6(Bd ;Z(p)) 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∗(Bd ;Z(p)) = H∗(BSO(6) × K (Z, 2);Z(p)) = Z(p)[p1, p2, e, t]. Thus we have π1(∞0 MTθd)(p) = 0, π2(∞0 MTθd)(p) ∼= Z5(p) with the isomorphism given by the tautological classes κp2 , κp21 , κte, κt2p1 , κt4 , and π3( ∞ 0 MTθd)(p) = 0. Therefore Hi(Mθd (Vd,g , Vd,g );Z(p)) = ⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ Z(p) i = 0 0 i = 1 Z(p){κp2 , κp21 , κte, κt2p1 , κt4 } i = 2 0 i = 3. The submonoid G ≤ hAut(u) of those path components which stabilize [Vd,g , Vd,g ] is path connected, and as the map u :Bd → BSO(6) × K (Z, 2) is a Z(p)-homology equiv- alence, since p is odd, we also have that πi(G) ⊗ Z(p) = 0 for i > 0. Thus the map Mθd (Vd,g , Vd,g ) → Mμ(Vd,g , u ◦ Vd,g ) is a Z(p)-homology equivalence. It remains to study the Serre spectral sequence for the fibration sequence Mμ(Vd,g , u ◦ Vd,g ) −→ Mor(Vd,g ) −→ K (Z, 3), which in low degrees has a single differential d3 :E0,23 = Z(p){κp2 , κp21 , κte, κt2p1 , κt4 } −→ E 3,0 3 = H3(K (Z, 3);Z(p)) = Z(p) S. Galatius, O. Randal-Williams Res Math Sci (2020) 7:9 Page 13 of 13 9 given by the formula above, so H3(Mor(Vd,g );Z(p)) is given by the cokernel of this differ- ential. The claim now follows by the identity of ideals (4d, 2d(5 − d2), d(10 − 10d + 5d2 − d3) − 2g) = (d, g) of Z(p), using again that p is odd. Acknowledgements SG was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 682922), the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92), and the EliteForsk Prize. ORW was supported by EPSRC grant EP/M027783/1, the ERC under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 756444), and a Philip Leverhulme Prize from the Leverhulme Trust. Author details 1Department of Mathematics, University of Copenhagen, Copenhagen, Denmark, 2Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK. Received: 21 December 2019 Accepted: 23 March 2020 References 1. Becker, J.C., Gottlieb, D.H.: The transfer map and fiber bundles. Topology 14, 1–12 (1975) 2. Bendersky, M., Miller, J.: Localization and homological stability of configuration spaces. Q. J. Math. 65(3), 807–815 (2014) 3. Cantero, F., Palmer, M.: On homological stability for configuration spaces on closed backgroundmanifolds. Doc. Math. 20, 753–805 (2015) 4. Friedrich, N.: Homological stability of automorphism groups of quadratic modules and manifolds. Doc. Math. 22, 1729–1774 (2017) 5. Galatius, S., Randal-Williams, O.: Stable moduli spaces of high-dimensional manifolds. Acta Math. 212(2), 257–377 (2014) 6. Galatius, S., Randal-Williams, O.: Homological stability for moduli spaces of high dimensional manifolds. II. Ann. Math. (2) 186(1), 127–204 (2017) 7. Galatius, S., Randal-Williams, O.: Homological stability for moduli spaces of high dimensional manifolds. I. J. Am. Math. Soc. 31(1), 215–264 (2018) 8. Galatius, S., Randal-Williams, O.: Moduli spaces of manifolds: a user’s guide. In: Handbook of Homotopy Theory, pp. 445–487. Chapman & Hall/CRC, CRC Press, Boca Raton, FL (2019) 9. Harer, J.L.: Stability of the homology of the mapping class groups of orientable surfaces. Ann. Math. (2) 121(2), 215–249 (1985) 10. Krannich, M.: On characteristic classes of exotic manifold bundles, Mathematische Annalen (2019), arXiv:1802.02609 11. Madsen, I., Tillmann, U.: The stable mapping class group and Q(CP∞+ ). Invent. Math. 145(3), 509–544 (2001) 12. Madsen, I., Weiss, M.: The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. Math. (2) 165(3), 843–941 (2007) 13. Tillmann, U.: On the homotopy of the stable mapping class group. Invent. Math. 130(2), 257–275 (1997) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.