Operations on stable moduli spaces
Quillen was the first to note the phenomenon of homological stability for groups [unpublished, 1971], and the topological version, due to \textit{G. Segal} [Invent. Math. 21, 213--221 (1973; Zbl 0267.55020)], was applied by Harer to moduli spaces of surfaces in [\textit{J. L. Harer}, Ann. Math. (2) 121, 215--249 (1985; Zbl 0579.57005)]. The authors have studied moduli spaces of higher dimensional manifolds in a series of papers, starting with [\textit{S. Galatius} and \textit{O. Randal-Williams}, Geom. Topol. 14, No. 3, 1243--1302 (2010; Zbl 1205.55007)] and continuing with [\textit{S. Galatius} and \textit{O. Randal-Williams}, J. Am. Math. Soc. 31, No. 1, 215--264 (2018; Zbl 1395.57044); Ann. Math. (2) 186, No. 1, 127--204 (2017; Zbl 1412.57026)]. See also their survey [\textit{O. Randal-Williams} and \textit{S. Galatius}, ``Moduli spaces of manifolds: a user's guide'', in: Handbook of homotopy theory. Boca Raton, FL: CRC Press. 43 p. (2020; Zbl 1476.57058)]. The question addressed in this paper is to what extent the stabilization is topological -- that is, coming from a map between the moduli space constructed by a topological operation on the manifolds under consideration (as is the case with surfaces, where one takes the connected sum with a torus to increase genus). The authors consider the moduli space \(\mathcal{M}(W)\) of fiber bundles with fiber a given smooth closed \(d\)-manifold \(W\) (that is, \(B\mathrm{Diff}(W)\)), when \(d=2n\), under stable diffeomorphims (defined by taking connected sums with copies of \(S^{n}\times S^{n}\)). More generally, they consider moduli spaces \(\mathcal{M}^{\Lambda}(W,\lambda)\) of manifolds \(W\) with \(\Lambda\)-structures \(\lambda:\mathrm{Fr}(TW)\to\Lambda\) on their framed tangent bundles (for a suitable space \(\Lambda\)). Their main theorem states that for any abelian group \(A\), there is a canonical isomorphism \(H^{i}(\mathcal{M}^{\Lambda}(W,\lambda);A)\cong H^{i}(\mathcal{M}^{\Lambda}(W',\lambda');A)\) (induced by a zigzag of maps) whenever \(W\) and \(W'\) are simply connected, \(d=2n>4\), and \(i\) is small compared to the (non-zero) Euler characteristics \(\chi(W)\) and \(\chi(W')\) (which must have the same \(p\)-adic valuation for all primes \(p\) not invertible in \(\mathrm{End}_{\mathbb Z}(A)\)). As an example, they deduce that \(H^{i}(\mathcal{M}(W);{\mathbb Z}_{(p)})\cong H^{i}(\mathcal{M}(W');{\mathbb Z}_{(p)})\) for small \(i\), if \(\chi(W)\) and \(\chi(W')\) have the same \(p\)-adic valuation. In the works cited above the authors had previously produced maps \(\mathcal{M}^{\Lambda}(W,\lambda)\to(\Omega^{\infty}MT\Theta)/\!/\mathrm{hAut}(u)\) which induce isomorphisms in homology in a range, where \(u:\Theta\to\Lambda\) is a cofibrant \(\mathrm{GL}_{d}({\mathbb R})\)-equivariant fibration and \(MT\Theta\) is the Thom spectrum of the inverse of the corresponding \({\mathbb R}^{d}\)-bundle on the Borel construction \(B=\Theta/\!/\mathrm{GL}_{d}({\mathbb R})\). The main theorem here is then proved by constructing certain operations \(\psi^{q}:\Omega_{k}^{\infty}MT\Theta\to\Omega_{qk}^{\infty}MT\Theta\), coming from self-maps of the Thom spectra, for any \(q,k\in{\mathbb Z}\) (with \(2\) inverted when \(q=2\)). Here \(\Omega_{k}^{\infty}MT\Theta\) is a union of certain path components.
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