Homological stability of diffeomorphism groups (Q2450716)

\textbf{Background.} The motivation for the main result of the paper under review comes from a recent result of [\textit{Søren Galatius} and \textit{Oscar Randal-Williams}, Acta Math. 212, No. 2, 257--377 (2014; Zbl 1377.55012)] identifying the limiting homology of certain sequences of diffeomorphism groups of high-dimensional, highly-connected manifolds. One version of their theorem is as follows, where \(d\geqslant 3\). For an almost parallelizable \((d-1)\)-connected \(2d\)-dimensional closed manifold \(W\) there is an integral homology isomorphism \[ \mathrm{hocolim}_{g\to\infty} B\mathrm{Diff}_D(W\sharp M_g) \longrightarrow \Omega_0^\infty MT^{\theta_{d+1}}(2d), \] where \(M_g\) is the connected sum of \(g\) copies of \(S^d \times S^d\) and the subscript \(D\) indicates that the diffeomorphisms are the identity on a chosen closed disc in \(W\sharp M_g\). The right-hand side is one component of the infinite loop space of a certain Thom spectrum, and its rational cohomology can be easily described, as the authors do in the introduction. The colimit on the left-hand side is taken over the sequence of stabilisation maps \[ B\mathrm{Diff}_D(W\sharp M_g) \longrightarrow B\mathrm{Diff}_D(W\sharp M_{g+1}) \] defined by taking the connected sum with \(M_1\) along \(D\) and extending diffeomorphisms by the identity. \textit{The main result.} Rational homological stability, the property that these maps induce isomorphisms on \(H_k(-;\mathbb{Q})\) in a range of degrees \(k\), would therefore -- if true -- explicitly compute the rational homology of these diffeomorphism groups in this range. The authors' main result is that rational homological stability holds for \(W=S^{2d}\) with a stable range of \(k<\mathrm{min}\{d-2,\frac{1}{2}(g-4)\}\). \textit{Method of proof.} The authors' strategy of proof is different from that of many other proofs of homological stability, in which (for sequences of groups) one constructs highly-connected simplicial complexes for the groups in question to act on, or (for sequences of spaces) one constructs highly-connected semi-simplicial resolutions of the spaces in question. Instead, the authors approach diffeomorphism groups via monoids of homotopy automorphisms (which are amenable to the methods of rational homotopy theory) and block diffeomorphism groups. To pass from homotopy automorphism monoids to block diffeomorphism groups and then to diffeomorphism groups they use respectively the surgery exact sequence and Morlet's lemma of disjunction. \textit{A more detailed overview.} To describe their proof strategy in more detail we first fix some notation. Let \(d\geqslant 3\), let \(M\) be a \((d-1)\)-connected \(2d\)-dimensional closed smooth manifold and write \(N=M\smallsetminus\mathrm{int}(D)\) for a codimension-\(0\) closed disc \(D\subset M\). Also write \(M_g = \sharp^g(S^d \times S^d)\) and \(N_g=M_g\smallsetminus\mathrm{int}(D)\). \textit{Rational homotopy theory.} The authors begin by reviewing rational homotopy theory, and in particular a model for the rational homotopy type of mapping spaces. This model is simplest when the domain is a formal space and the codomain is a coformal space. The notion of a formal space was introduced in [\textit{P. Deligne} et al., Invent. Math. 29, 245--274 (1975; Zbl 0312.55011)] and means that the rational homotopy type of the space is determined by its cohomology algebra. The dual notion of a coformal space was introduced in [\textit{J. Neisendorfer} and \textit{T. Miller}, Ill. J. Math. 22, 565--579 (1978; Zbl 0396.55011)] and means that the rational homotopy type of the space is determined by its homotopy Lie algebra. Applying this to the case of a space \(X\) which is both formal and coformal, the authors describe an explicit chain complex built from the rational homotopy and homology of \(X\) which computes the rational homotopy groups of \(\mathrm{aut}(X)\), the (grouplike) monoid of homotopy self-equivalences of \(X\). When \(H^d(M)\) has rank at least \(3\) then \(M\) is both formal and coformal, by [op.\ cit.], and the authors use their model to compute the rational homotopy groups (based at the identity) of \(\mathrm{aut}(M)\) -- as well as the monoid of based self-equivalences \(\mathrm{aut}_*(M)\) and the monoid \(\mathrm{aut}_{\partial D}(N)\) of self-equivalences equal to the identity on \(\partial D\). For the purposes of proving homological stability, the important corollary of this is that \(\mathrm{aut}_{\partial D}(N)\) is rationally \((d-2)\)-connected. The authors also prove that the group \(\pi_0(\mathrm{aut}_{\partial D}(N))\) is commensurable with the automorphism group of a certain quadratic module depending on the manifold \(M\). In particular when \(M=M_g\) the group \(\pi_0(\mathrm{aut}_{\partial D}(N_g))\) is commensurable with either \(O_{g,g}(\mathbb{Z})\) or \(Sp_{2g}(\mathbb{Z})\) depending on the parity of \(d\). \textit{Block diffeomorphisms.} Now abbreviate \(\mathrm{D}=\mathrm{Diff}_{\partial D}(N)\). This may be considered as a \(\Delta\)-group by taking its singular complex, and as such it includes into the larger \(\Delta\)-group of \textit{block diffeomorphisms} \(\widetilde{\mathrm{D}}=\widetilde{\mathrm{Diff}}_{\partial D}(N)\). Similarly, \(\mathrm{aut}=\mathrm{aut}_{\partial D}(N)\) may be considered as a \(\Delta\)-monoid, and includes into the \(\Delta\)-monoid \(\widetilde{\mathrm{A}}=\widetilde{\mathrm{Aut}}_{\partial D}(N)\) of \textit{block homotopy self-equivalences}. The inclusion \(\mathrm{aut}\to\widetilde{\mathrm{A}}\) is a homotopy equivalence, so the inclusion \(\mathrm{D}\to\mathrm{aut}\) factors up to homotopy as \(\mathrm{D}\to\widetilde{\mathrm{D}}\to\widetilde{\mathrm{A}}\simeq\mathrm{aut}\). Write \(\widetilde{J}\) for the inclusion \(\widetilde{\mathrm{D}}\to\widetilde{\mathrm{A}}\), and add a subscript \({-}_g\) whenever we assume that \(N=N_g\). The subgroup \(\mathrm{im}(\pi_0 \widetilde{J})\) of \(\pi_1 B\widetilde{\mathrm{A}}\) determines a covering space \(\bar{B}\widetilde{\mathrm{A}} \to B\widetilde{\mathrm{A}}\) and there is a homotopy fibre sequence \[ \phantom{(*)}\qquad\qquad\qquad\qquad (\widetilde{\mathrm{A}}/\widetilde{\mathrm{D}})_{(1)} \longrightarrow B\widetilde{\mathrm{D}} \longrightarrow \bar{B}\widetilde{\mathrm{A}},\qquad\qquad\qquad\qquad (*) \] where \({-}_{(1)}\) denotes the path-component of the identity. The idea is now to understand \(\bar{B}\widetilde{\mathrm{A}}\) using the previous rational homotopy theoretic results and to understand \((\widetilde{\mathrm{A}}/\widetilde{\mathrm{D}})_{(1)}\) using surgery theory. \textit{The base space of \((*)\).} First, from above we know that \(\widetilde{\mathrm{A}}\) is rationally \((d-2)\)-connected, so the map \(B\widetilde{\mathrm{A}} \to B\pi_0 \widetilde{\mathrm{A}}\) is rationally \(d\)-connected. This therefore lifts to a map \(\bar{B}\widetilde{\mathrm{A}}\to B(\mathrm{im}(\pi_0 \widetilde{J}))\) which is rationally \(d\)-connected. Second, using their results about \(\pi_0(\mathrm{aut})\) the authors prove that the image of \(\pi_0 \widetilde{J}\) has finite index in \(\pi_0 \widetilde{\mathrm{A}}\). Finally, we also know from above that \(\pi_0(\mathrm{aut}_g)\) is commensurable with either \(O_{g,g}(\mathbb{Z})\) or \(Sp_{2g}(\mathbb{Z})\). Together, this identifies the rational homotopy \((d-1)\)-type of \(\bar{B}\widetilde{\mathrm{A}}_g\): \[ \bar{B}\widetilde{\mathrm{A}}_g \;\simeq_{\leqslant d-1,\mathbb{\scriptstyle{Q}}}\; B(\mathrm{im}(\pi_0 \widetilde{J}_g)) \;\simeq_{\mathbb{\scriptstyle{Q}}}\; B\pi_0 \widetilde{\mathrm{A}}_g \;\simeq\; B\pi_0 \mathrm{aut}_g \;\simeq_{\mathbb{\scriptstyle{Q}}}\; BO_{g,g}(\mathbb{Z}) \text{ or } BSp_{2g}(\mathbb{Z}). \] \textit{Surgery theory.} The surgery exact sequence for a compact oriented smooth manifold \(X\) was reformulated by Quinn as the exact sequence on homotopy groups of a certain homotopy fibre sequence whose fibre is the surgery structure space. The authors show that the identity component of the surgery structure space is weakly equivalent to \((\widetilde{\mathrm{Aut}}_{\partial X}(X)/\widetilde{\mathrm{Diff}}_{\partial X}(X))_{(1)}\). Hence in the case \(X=N_g\) one obtains a homotopy fibre sequence \[ (\widetilde{\mathrm{A}}_g/\widetilde{\mathrm{D}}_g)_{(1)} \longrightarrow \mathrm{Map}_*(M_g,G/O)_{(1)} \longrightarrow \mathbb{L}(M_g)_{(1)}. \] Using this sequence the authors calculate the rational homology of \((\widetilde{\mathrm{A}}_g/\widetilde{\mathrm{D}}_g)_{(1)}\) and its structure as a module over \(\mathrm{im}(\pi_0 \widetilde{J}) = \pi_1\bar{B}\widetilde{\mathrm{A}}\) arising from \((*)\). \textit{Stability for block diffeomorphisms.} The stabilisation map induces a map of homotopy fibre sequences \((*)\) and therefore a map of the associated Serre spectral sequences for rational homology. The above results give a complete algebraic description of the \(E^2\) page, in the range \(p\leqslant d-1\), in terms of twisted homology groups of \(O_{g,g}(\mathbb{Z})\) or \(Sp_{2g}(\mathbb{Z})\). These groups satisfy homological stability, with twisted as well as untwisted coefficients, by \textit{R. Charney} [J. Pure Appl. Algebra 44, 107--125 (1987; Zbl 0615.20024)]. Thus the map of spectral sequences induced by the stabilisation map is an isomorphism on the \(E^2\) page in a range of degrees, and by the comparison theorem [\textit{E. C. Zeeman}, Proc. Camb. Philos. Soc. 53, 57--62 (1957; Zbl 0077.36601)] the same is true for the map in the limit. This proves rational homological stability for the groups \(\widetilde{\mathrm{D}}_g = \widetilde{\mathrm{Diff}}_{\partial D}(N_g) \cong \widetilde{\mathrm{Diff}}_{D}(M_g)\). \textit{Passing to diffeomorphisms.} The final step is to pass from block diffeomorphisms to diffeomorphisms using Morlet's lemma of disjunction. Applied to the manifold \(N_g\) this says that \(\mathrm{D}_g / \mathrm{D}_0 \to \widetilde{\mathrm{D}}_g / \widetilde{\mathrm{D}}_0\) is \((2d-6)\)-connected, which by some diagram yoga implies that \(\mathrm{D}_{g+1} / \mathrm{D}_g \to \widetilde{\mathrm{D}}_{g+1} / \widetilde{\mathrm{D}}_g\) is also \((2d-6)\)-connected. Hence (within this range) rational homological stability for \(\mathrm{D}_g\) follows from rational homological stability for \(\widetilde{\mathrm{D}}_g\). \textit{Related results.} A strengthening of the main theorem has subsequently been proved in [\textit{S. Galatius} and \textit{O. Randal-Williams}, ``Homological stability for moduli spaces of high dimensional manifolds'', arXiv:1203.6830 (2012)]: still assuming that \(d\geqslant 3\), homological stability is true for \(B\mathrm{Diff}_D(M_g)\) with integral coefficients in the range \(k\leqslant \frac12(g-4)\). This has been generalised further in [\textit{S. Galatius} and \textit{O. Randal-Williams}, ``Homological stability for moduli spaces of high dimensional manifolds.\ {I}'', \url{arXiv:1403.2334} (2014)] to \(B\mathrm{Diff}_D(W\sharp M_g)\) whenever \(W\) is a \(2d\)-dimensional simply-connected closed manifold. In particular this applies to all \(W\) for which the limiting homology is known. There is a sequel [\textit{A. Berglund} and \textit{Ib~Madsen}, ``Rational homotopy theory of automorphisms of highly connected manifolds'', \url{arXiv:1401.4096} (2014)] to the paper under review in which the authors improve their stability result for block diffeomorphism groups and prove an analogous result for homotopy automorphism groups. Precisely, they prove rational homological stability for \(B\mathrm{aut}_{\partial D}(N_g)\) and for \(B\widetilde{\mathrm{Diff}}_{D}(M_g)\) in the range \(k<\frac12(g-4)\). Moreover they calculate the stable rational cohomology of \(B\mathrm{aut}_{\partial D}(N_g)\), which turns out, when \(d\) is odd, to contain the unstable rational homology of all outer automorphism groups of free groups.