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Characteristic classes of fiberwise branched surface bundles via arithmetic groups. (English) Zbl 1405.55016
Let \((S,\mathbf{z})\) be a closed oriented surface of genus \(g\geq 2\), \(\mathbf{z}\subset S\) a finite set of points and \(\mathrm{Mod}(S,\mathbf{z})\) be its pure mapping class group. This paper studies the following case: let \(G\) be a finite group of diffeomorphisms of \(S\) and \(\mathbf{z}\) be its fixed point set in \(S\). Moreover, let \(\mathrm{Mod}^{G}(S,\mathbf{z}) \subset \mathrm{Mod}(S,Z)\) denote the subgroup of mapping classes that may be realized by diffeomorphisms of \(S\) that commute with the action of \(G\). There is a representation \(\alpha: \mathrm{Mod}(S,\mathbf{z})\to Sp_{2g}(\mathbb{Z})\) that induces \(\alpha: \mathrm{Mod}^{G}(S,\mathbf{z})\to Sp^{G}_{2g}(\mathbb{Z})\), this last group is the centralizer of \(G\) in \(Sp_{2g}(\mathbb{Z})\). On the other hand, let \(\kappa_{1}\) be the first Miller-Morita-Mumford class in \(H^{2}(\mathrm{Mod}(S,\mathbf{z};\mathbb{Q}))\). For each \(z\in \mathbf{z}\) let \(e_{z}\) denote the Euler class, assume \(G=\langle t\rangle\) is a finite cyclic group with \(t\) a generator. Assume \(\mathbf{z}\) is the disjoint union of the fixed points \(\mathbf{z}_{i}\) of \(z\) and that \(t\) acts on the tangent space \(T_{z}S\) by the appropriate rotation. Let \(\epsilon_{j}=\sum_{z\in\mathbf{z}_{j}} e_{z}.\) The main theorem is as follows:
Theorem. Let \((S,\mathbf{z})\) be as above, \(G\) be a finite cyclic group of order \(m\) acting by orientation preserving diffeomorphisms on \(S\). Assume that the stabilizers of the action are either trivial or \(G\). If the genus of \(S/G\) is at least \(6\), then the image of \(\alpha^{*}: H^{2}(Sp^{2}_{2g}(\mathbb{Z};\mathbb{Q}))\to H^{2}(\mathrm{Mod}^{G}(S,\mathbf{z};\mathbb{Q}))\) is the subspace generated by \(\kappa_{1}\) and \(\epsilon_{j}+\epsilon_{m-j}\) for \(1\leq j <\frac{m}{2}.\) The author has several applications in \(G\)-cobordism, surface group representations and Hirzebruch’s index formula for branched covers.

MSC:
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
20J06 Cohomology of groups
57M99 General low-dimensional topology
58J20 Index theory and related fixed-point theorems on manifolds
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