Characteristic classes of fiberwise branched surface bundles via arithmetic groups
type: Item
instance of: scholarly article
title: Characteristic classes of fiberwise branched surface bundles via arithmetic groups
zbMATH Open document ID: 1405.55016
author: Bena Tshishiku
published in: Michigan Mathematical Journal
publication date: 2018-10-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1606.07119
full work available at URL: https://projecteuclid.org/euclid.mmj/1516330969
review text: 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.
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 58J20
zbMATH DE Number: 6965588
zbMATH Keywords: Miller-Morita-Mumford characteristic classes
zbMATH Keywords: characteristic classes of bundles.
reviewed by: Daniel Juan Pineda
MaRDI profile type: Publication
OpenAlex ID: W2964089376
arXiv ID: 1606.07119
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DOI: 10.1307/MMJ/1516330969
description: scientific article