On the moduli space of flat symplectic surface bundles (Q2208493)

In [Geom. Topol. 21, No. 5, 3047--3092 (2017; Zbl 1377.58004)], the author established homological stability for groups of \(C^\infty\)-diffeomorphisms of surfaces as families of discrete groups, and showed that the stable homology of \(C^\infty\)-diffeomorphisms of surfaces as discrete groups is the same as the homology of certain infinite loop spaces related to Haefliger's classifying space of foliations of codimension 2. In this paper, he constructs an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces to the homology of certain infinite loop spaces. The author uses these infinite loop spaces to study characteristic classes of surface bundles whose holonomy groups are area preserving. If \(\Sigma\) is a surface with or without boundary, \(\omega_\Sigma\) is an area form on \(\Sigma\) whose total volume is normalized to be the negative of the Euler number, \(\mathrm{Diff}(\Sigma,\partial)\) and \(\mathrm{Symp}(\Sigma,\partial)\) are respectively the group of orientation preserving diffeomorphisms and the group of \(\omega_\Sigma\)-preserving diffeomorphisms of \(\Sigma\) whose supports are away from the boundary, and the same groups with discrete topology are denoted by \(\mathrm{Diff}^\sigma(\Sigma,\partial)\) and \(\mathrm{Symp}^\sigma(\Sigma,\partial)\) respectively, then the first main result of the paper states that if \(*\le(2g(\Sigma)-2)/3\), then the homology groups \(H_*(\mathrm{Symp}^\sigma(\Sigma,\partial);\mathbb{Z})\) are independent of the genus \(g(\Sigma)\) and the number of boundary components. It is known that the identity component \(\mathrm{Symp}_0(\Sigma,\partial)\) of \(\mathrm{Symp}(\Sigma,\partial)\) is homotopy equivalent to \(\mathrm{Diff}_0(\Sigma,\partial)\) which is contractible for \(g\ge 2\). If a surjective homomorphism \(\mathrm{Flux}:\mathrm{Symp}_0(\Sigma,\partial)\to H^1(\Sigma;\partial;\mathbb{R})\) is the flux homomorphism, then the group of Hamiltonians is defined to be the kernel of Flux, and \(1\to\mathrm{Ham}(\Sigma,\partial)\to\mathrm{Symp}_0(\Sigma,\partial)\overset{\mathrm{Flux}}{\to}H^1(\Sigma,\partial;\mathbb{R})\). The flux homomorphism can be extended to a crossed homomorphism \(\widetilde{\mathrm{Flux}}:\mathrm{Symp}(\Sigma,\partial)\to H^1(\Sigma;\partial;\mathbb{R})\), which is not a group homomorphism, however, its kernel is a subgroup of \(\mathrm{Symp}(\Sigma,\partial)\). This kernel is called an extended Hamiltonian and is denoted by \(\widetilde{\mathrm{Ham}}(\Sigma,\partial)\). The group of extended Hamiltonians is an enlargement of \(\mathrm{Ham}(\Sigma,\partial)\) that intersects all connected components of \(\mathrm{Symp}(\Sigma,\partial)\) and \(1\to\mathrm{Ham}(\Sigma,\partial)\to\widetilde{\mathrm{Ham}}(\Sigma,\partial)\to\mathrm{MCG}(\Sigma,\partial)\to 1\), where \(\mathrm{MCG}(\Sigma,\partial)\) denotes the mapping class group of the surface \(\Sigma\). \textit{D. Kotschick} and \textit{S. Morita} [J. Differ. Geom. 75, No. 2, 273--302 (2007; Zbl 1115.57016)], proved that the homology group of the Hamiltonians is highly nontrivial and it is not stable with respect to the genus. In this paper, the author proves that the homology group \(\widetilde{\mathrm{Ham}}^\sigma(\Sigma,\partial)\) is stable, by showing that if \(\Sigma\) is a surface with at least one boundary component and \(*\le(2g(\Sigma)-2)/3\), then the homology groups \(H_*(\widetilde{\mathrm{Ham}}^\sigma(\Sigma,\partial);\mathbb{Z})\) are independent of the genus \(g(\Sigma)\) and the number of boundary components. Finally, the author studies characteristic classes of surface bundles whose holonomy groups are area preserving which in particular allows to give a homotopy theoretic proof of Kotschick-Morita's theorem.