Operations on stable moduli spaces

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 $\mathcal{M}(W)$ classifying smooth fibre bundles whose fibres are diffeomorphic to $W$. As a model we may take $\mathcal{M}(W)=B\mathrm{Diff}(W)$, the classifying space of the diffeomorphism group $\mathrm{Diff}(W)$ of $W$, equipped with the $C^{\infty}$ topology. Then for $A$ an abelian group $H^{i}(\mathcal{M}(W);A)$ is the group of $H^{i}(-;A)$-valued characteristic classes of such fibre bundles.

Now let $d=2n$ and $W$ be a $d$-manifold. The connected sum $W\#(S^{n}\times S^{n})$ is then well defined up to (non-canonical) diffeomorphism, as $S^{n}\times S^{n}$ admits an orientation-reversing diffeomorphism, and we write $W\#g(S^{n}\times S^{n})$ for the $g$-fold iteration of this operation. Two manifolds $W$ and $W^{\prime}$ are called stably diffeomorphic if $W\#g(S^{n}\times S^{n})$ is diffeomorphic to $W^{\prime}\#g^{\prime}(S^{n}\times S^{n})$ for some $g,g^{\prime}\in\mathbb{N}$. 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 $H^{*}(\mathcal{M}(W);A)$ and $H^{*}(\mathcal{M}(W^{\prime});A)$ when $W$ and $W^{\prime}$ 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 $(-1)^{n}\chi(W)$ and $(-1)^{n}\chi(W^{\prime})$ 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 $\lambda$ then there is a corresponding moduli space $\mathcal{M}^{\mathrm{or}}(W,\lambda)$ classifying smooth fibre bundles with oriented fibres which are oriented diffeomorphic to $(W,\lambda)$, and a forgetful map $\mathcal{M}^{\mathrm{or}}(W,\lambda)\to\mathcal{M}(W)$. Then the connected sum $W\#g(S^{n}\times S^{n})$ inherits an orientation, well defined up to oriented diffeomorphism, and we say that $(W,\lambda)$ is oriented stably diffeomorphic to $(W^{\prime},\lambda^{\prime})$ provided $W\#g(S^{n}\times S^{n})$ is oriented diffeomorphic to $W^{\prime}\#g^{\prime}(S^{n}\times S^{n})$ for some $g,g^{\prime}\in\mathbb{N}$. In this situation our result will also imply a canonical isomorphism $H^{i}(\mathcal{M}^{\mathrm{or}}(W,\lambda);\mathbb{Z}_{(p)})\cong H^{i}(\mathcal{M}^{\mathrm{or}}(W^{\prime},\lambda^{\prime});\mathbb{Z}_{(p)})$, under the same hypotheses.

More generally, for a space $\Lambda$ equipped with a continuous action of $\mathrm{GL}_{d+1}(\mathbb{R})$ a $\Lambda$-structure on a $d$-manifold $W$ is a $\mathrm{GL}_{d}(\mathbb{R})$-equivariant map $\lambda\mathrel{\mathop{\mathchar 58\relax}}\mathrm{Fr}(TW)\to\Lambda$, or, equivalently, a $\mathrm{GL}_{d+1}(\mathbb{R})$-equivariant map $\mathrm{Fr}(\varepsilon^{1}\oplus TW)\to\Lambda$. For example, if $\Lambda=\{\pm 1\}$ on which $\mathrm{GL}_{d+1}(\mathbb{R})$ acts by multiplication by the sign of the determinant, then a $\Lambda$-structure $\lambda\mathrel{\mathop{\mathchar 58\relax}}\mathrm{Fr}(TW)\to\{\pm 1\}$ is the same thing as an orientation: it distinguishes oriented frames from non-oriented ones. Two $\Lambda$-structures on the same manifold are homotopic if they are homotopic through equivariant maps, and $(W,\lambda)$ is $\Lambda$-diffeomorphic to $(W^{\prime},\lambda^{\prime})$ if there exists a diffeomorphism $\phi\mathrel{\mathop{\mathchar 58\relax}}W\to W^{\prime}$ such that $\lambda\circ D\phi$ is homotopic to $\lambda^{\prime}$. The usual embedding of $S^{n}\times S^{n}\subset\mathbb{R}^{2n+1}$ as the boundary of a thickened $S^{n}\times\{0\}\subset\mathbb{R}^{n+1}\times\mathbb{R}^{n}$ gives a trivialisation of $\varepsilon^{1}\oplus T(S^{n}\times S^{n})$ and a $\Lambda$-structure on $W$ extends to one on $W\#(S^{n}\times S^{n})$, canonically up to $\Lambda$-diffeomorphism. For two pairs $(W,\lambda)$ and $(W^{\prime},\lambda^{\prime})$ consisting of a manifold and a $\Lambda$-structure, we say that they are stably $\Lambda$-diffeomorphic if $W\#g(S^{n}\times S^{n})$ is $\Lambda$-diffeomorphic to $W^{\prime}\#g^{\prime}(S^{n}\times S^{n})$ for some $g,g^{\prime}\in\mathbb{N}$.

There is a moduli space $\mathcal{M}^{\Lambda}(W,\lambda)$ parametrising smooth fibre bundles $\pi\mathrel{\mathop{\mathchar 58\relax}}E\to X$ with $d$-dimensional fibres, and where the fibrewise tangent bundle $T_{\pi}E$ is equipped with an equivariant map $\mathrm{Fr}(\varepsilon^{1}\oplus T_{\pi}E)\to\Lambda$, such that all fibres of $\pi$ are $\Lambda$-diffeomorphic to $(W,\lambda)$. Our main result is then as follows.

In Section 4 we give an example showing the third condition cannot be relaxed.

The main results of [GRW14, GRW18, GRW17], summarised in [GRW19], provide a map

which is 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

which is 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 $\chi(W)\neq\chi(W^{\prime})$ then these two maps land in different path components, and the problem becomes to compare the homology of these two path components.

The data involved in defining the common target of the maps (1.1) and (1.2) is a $\mathrm{GL}_{2n}(\mathbb{R})$-equivariant fibration $u\mathrel{\mathop{\mathchar 58\relax}}\Theta\to\Lambda$ with domain which is cofibrant as a $\mathrm{GL}_{2n}(\mathbb{R})$-space. Letting $B$ denote the Borel construction $\Theta/\!\!/\mathrm{GL}_{2n}(\mathbb{R})$, $MT\Theta$ is then the Thom spectrum of the inverse of the canonical $2n$-dimensional vector bundle over $B$, and $\Omega^{\infty}MT\Theta$ is its associated infinite loop space. By functoriality the group-like topological monoid $\mathrm{hAut}(\Theta)$ of $\mathrm{GL}_{2n}(\mathbb{R})$-equivariant homotopy equivalences $f\mathrel{\mathop{\mathchar 58\relax}}\Theta\to\Theta$ acts on the infinite loop space $\Omega^{\infty}MT\Theta$, so the group-like submonoid $\mathrm{hAut}(u)=\{f\in\mathrm{hAut}(\Theta)\,|\,u\circ f=u\}$ 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 $\Omega^{\infty}MT\Theta$, in the case where the $\mathrm{GL}_{2n}(\mathbb{R})$-space $\Theta$ is obtained by restriction from a cofibrant $\mathrm{GL}_{2n+1}(\mathbb{R})$-space $\overline{\Theta}$. The space $\overline{B}=\overline{\Theta}/\!\!/\mathrm{GL}_{2n+1}(\mathbb{R})$ carries a canonical $(2n+1)$-dimensional vector bundle, and $MT\overline{\Theta}$ denotes its associated Thom spectrum; as above, by functoriality it carries an action of the monoid $\mathrm{hAut}(\overline{\Theta})$ of $\mathrm{GL}_{2n+1}(\mathbb{R})$-equivariant homotopy equivalences $f\mathrel{\mathop{\mathchar 58\relax}}\overline{\Theta}\to\overline{\Theta}$.

A key construction in this paper is a homotopy pullback diagram of infinite loop spaces, equivariant for $\mathrm{hAut}(\overline{\Theta})$, of the form

whose bottom right corner has $\pi_{0}\cong\mathbb{Z}/2$ and all higher homotopy groups are $2$-power torsion, and the bottom horizontal map induces a surjection on $\pi_{1}$. It induces an isomorphism

whose first coordinate is given by the Euler class and whose second coordinate is given by the stabilisation map. To explain this claim and its notation, first note that the $2n$-dimensional vector bundle over $B$ has an Euler class $e\in H^{2n}(B;\mathbb{Z}^{w_{1}})$, where the coefficients are twisted by the determinant of this vector bundle, and under the Thom isomorphism this gives a class $e\smile u_{-2n}\in H^{0}(MT\Theta;\mathbb{Z})$. Then $\chi$ is the value of this spectrum cohomology class on the Hurewicz image of an element of $\pi_{0}MT\Theta$; geometrically, it assigns to such an element the Euler characteristic of a manifold representing it. Similarly, the $(2n+1)$-dimensional vector bundle over $\overline{B}$ has a $2n$th Stiefel–Whitney class $w_{2n}\in H^{2n}(\overline{B};\mathbb{Z}/2)$, and under the Thom isomorphism this gives a class $w_{2n}\smile u_{-2n-1}\in H^{-1}(MT\overline{\Theta};\mathbb{Z}/2)$. Then $w_{2n}(x)$ denotes the value of this spectrum cohomology class on the Hurewicz image of $x$.

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 $\psi^{q}$ being defined integrally and inducing an isomorphism with coefficients in any $\mathbb{Z}[q^{-1}]$-module, the map $\psi^{2}$ will only be defined after localising the spaces involved away from $2$.

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 $(\Omega^{\infty}MT\Theta)/\!\!/\mathrm{hAut}(u)$ 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 [BM14] and Cantero–Palmer [CP15] for cohomology of configuration spaces. This strategy has also been used by Krannich [Kra19] to show that $H^{i}(\mathcal{M}^{\mathrm{or}}(W,\lambda);A)\cong H^{i}(\mathcal{M}^{\mathrm{or}}(W\#\Sigma,\lambda);A)$ for $(W,\lambda)$ an oriented manifold of dimension $2n>4$ and $\Sigma$ an exotic sphere, in a stable range of degrees when the order of $[\Sigma]\in\Theta_{2n}$ is invertible in $\mathrm{End}_{\mathbb{Z}}(A)$.