Advanced Studies in Pure Mathematics

On the stable cohomology algebra of extended mapping class groups for surfaces

Nariya Kawazumi

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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 $\Gamma_{g,1}$.

Article information

Source
Groups of Diffeomorphisms: In honor of Shigeyuki Morita on the occasion of his 60th birthday, R. Penner, D. Kotschick, T. Tsuboi, N. Kawazumi, T. Kitano and Y. Mitsumatsu, eds. (Tokyo: Mathematical Society of Japan, 2008), 383-400

Dates
Received: 5 May 2007
Revised: 28 October 2007
First available in Project Euclid: 28 November 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1543447490

Digital Object Identifier
doi:10.2969/aspm/05210383

Mathematical Reviews number (MathSciNet)
MR2509717

Zentralblatt MATH identifier
1185.57012

Subjects
Primary: 57R20: Characteristic classes and numbers
Secondary: 14H15: Families, moduli (analytic) [See also 30F10, 32G15] 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 57M20: Two-dimensional complexes 57M50: Geometric structures on low-dimensional manifolds

Citation

Kawazumi, Nariya. On the stable cohomology algebra of extended mapping class groups for surfaces. Groups of Diffeomorphisms: In honor of Shigeyuki Morita on the occasion of his 60th birthday, 383--400, Mathematical Society of Japan, Tokyo, Japan, 2008. doi:10.2969/aspm/05210383. https://projecteuclid.org/euclid.aspm/1543447490


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