## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- 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

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

#### 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

**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