The homology of surface diffeomorphism groups and a question of Morita (Q2845069)

This paper concerns the Eilenberg-MacLane homology of surface diffeomorphism groups. As an application of the main theorem, the author affirmatively answers a question posed by \textit{S. Morita} [Proc. Symp. Pure Math. 74, 329--354 (2006; Zbl 1304.57030)] concerning the non-triviality of certain secondary characteristic classes for foliations. As another result naturally related to the study in the main theorem, a version of Harer stability for surface diffeomorphism groups in homology of small degree, is obtained.NEWLINENEWLINELet \(\Sigma_{g}\) denote a closed oriented surface of genus \(g\) and \(\mathrm{Diff}^{\delta}_{+}(\Sigma_{g})\) the group of orientation preserving diffeomorphisms of \(\Sigma_{g}\) with the discrete topology. Morita's question is the following: Is the map \(H_{3}(\mathrm{Diff}^{\delta}_{+}(\Sigma_{g}))\rightarrow\mathbb{R}^{2}\) induced by \(u_{1}c_{1}^{2}\) and \(u_{1}c_{2}\) a surjective for every \(g\)? Here, the map is defined as characteristic classes in \(H^{3}(\mathrm{Diff}^{\delta}_{+}(\Sigma_{g});\mathbb{R})=H^{3}(B\mathrm{Diff}^{\delta}_{+}(\Sigma_{g});\mathbb{R})\) by integrating \(u_{1}c_{1}^{2}, u_{1}c_{2}\in H^{5}(E\mathrm{Diff}^{\delta}_{+}(\Sigma_{g});\mathbb{R})\) along the fibres. The main theorem is as follows: For any inclusion \(D^{2}\subset S^{2}\) the induced homomorphism \(H_{k}(\mathrm{Diff}^{\delta}_{c}(D^{2});\mathbb{Q})\rightarrow H_{k}(\mathrm{Diff}^{\delta}_{+}(S^{2});\mathbb{Q})\) is an isomorphism for \(0\leq k\leq 2\) and surjective for \(k=3\). Here, \(\mathrm{Diff}^{\delta}_{c}(D^{2})\) denotes the group of the diffeomorphisms of \(D^{2}\) each of which support contained in the interior of \(D^{2}\). By this theorem and \textit{O. H. Rasmussen}'s affirmative answer to Morita's question for \(S^{2}\) [Topology 19, 335--349 (1980; Zbl 0443.57021)], the author obtains the affirmative answer for the general case. In fact, since the map \(H_{3}(\mathrm{Diff}^{\delta}_{c}(D^{2}))\rightarrow\mathbb{R}^{2}\) factors through \(H_{3}(\mathrm{Diff}^{\delta}_{+}(S^{2}))\), it is surjective by Rasmussen's result. It also factors through \(H_{3}(\mathrm{Diff}^{\delta}_{+}(\Sigma_{g}))\), which yields the affirmative answer.NEWLINENEWLINEThe author also obtains a version of Harer stability. Let \(\Sigma^{1}_{g}\) denote a compact oriented surface of genus \(g\) with one boundary component and \(\Gamma^{1}_{g}\) the mapping class group of diffeomorphisms with support in the interior of \(\Sigma^{1}_{g}\) modulo isotopy. The author proves the natural inclusion \(\Sigma^{1}_{g}\hookrightarrow\Sigma^{1}_{g+1}\) induces an isomorphism \(H_{k}(\mathrm{Diff}^{\delta}_{c}(\Sigma^{1}_{g});\mathbb{Q})\rightarrow H_{k}(\mathrm{Diff}^{\delta}_{c}(\Sigma^{1}_{g+1});\mathbb{Q})\) for \(k\leq 3\) and \(g\geq 8\). Furthermore, the rank of the image of \(H_{4}(\mathrm{Diff}^{\delta}_{c}(\Sigma^{1}_{g});\mathbb{Q})\) in \(H_{4}(\Gamma^{1}_{g};\mathbb{Q})\) is independent of \(g\) for \(g\geq 8\). An implication of the second half of this theorem is the following: the non-triviality of the image of the second Mumford-Morita-Miller class in \(H^{4}(\Gamma^{1}_{g})\) is independent of \(g\) if \(g\geq 8\).NEWLINENEWLINEThe proofs are given by calculating the spectral sequence converging to \(H_{*}(B\overline{\mathrm{Diff}_{c}(S)})\) whose \(E^{2}\)-term is \(H_{p}(S;H_{q}(B\overline{\mathrm{Diff}_{c}(\mathbb{R}^{2})}))\), where \(S\) denotes a surface, and the Hochschild-Serre spectral sequence associated with the natural short exact sequence \(1\rightarrow\mathrm{Diff}_{c, 0}(\Sigma^{1}_{g})\rightarrow\mathrm{Diff}_{c}(\Sigma^{1}_{g})\rightarrow\Gamma^{1}_{g}\rightarrow 1\).