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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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