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VOL. 52 | 2008 On the stable cohomology algebra of extended mapping class groups for surfaces
Nariya Kawazumi

Editor(s) Robert Penner, Dieter Kotschick, Takashi Tsuboi, Nariya Kawazumi, Teruaki Kitano, Yoshihiko Mitsumatsu

## Abstract

Let $\Sigma_{g,1}$ be an oriented compact surface of genus $g$ with 1 boundary component, and $\Gamma_{g,1}$ the mapping class group of $\Sigma_{g,1}$. We determine the stable cohomology group of $\Gamma_{g,1}$ with coefficients in $H^1 (\Sigma_{g ,1} ; \mathbb{Z})^{\otimes n}$, $n \ge 1$, explicitly modulo the stable cohomology group with trivial coefficients. As a corollary the rational stable cohomology algebra of the semi-direct product $\Gamma_{g,1} \ltimes H_1 (\Sigma_{g,1} ; \mathbb{Z})$ (which we call the extended mapping class group) is proved to be freely generated by the generalized Morita-Mumford classes $\widetilde{m_{i,j}}$'s $(i \ge 0,\, j \ge 1,\, i+j \ge 2)$ [11] over the rational stable cohomology algebra of the group $\Gamma_{g,1}$.

## Information

Published: 1 January 2008
First available in Project Euclid: 28 November 2018

zbMATH: 1185.57012
MathSciNet: MR2509717

Digital Object Identifier: 10.2969/aspm/05210383

Subjects:
Primary: 57R20
Secondary: 14H15, 32G15, 57M20, 57M50

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Vol. 52 • 1 January 2008