Nagoya Mathematical Journal

Nilpotency and triviality of mod $p$ Morita-Mumford classes of mapping class groups of surfaces

Toshiyuki Akita

Full-text: Open access

Abstract

This paper is concerned with mod $p$ Morita-Mumford classes $e_{n}^{(p)} \in H^{2n}(\Gamma_{g}, \mathbb{F}_{p})$ of the mapping class group $\Gamma_{g}$ of a closed oriented surface of genus $g \geq 2$, especially triviality and nontriviality of them. It is proved that $e_{n}^{(p)}$ is nilpotent if $n \equiv -1 \pmod{p-1}$, while the stable mod $p$ Morita-Mumford class $e_{n}^{(p)} \in H^{2n}(\Gamma_{\bullet}, \mathbb{F}_{p})$ is proved to be nontrivial and not nilpotent if $n \not\equiv -1 \pmod{p-1}$. With these results in mind, we conjecture that $e_{n}^{(p)}$ vanishes whenever $n \equiv -1 \pmod{p-1}$, and obtain a few pieces of supporting evidence.

Article information

Source
Nagoya Math. J., Volume 165 (2002), 1-22.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631695

Mathematical Reviews number (MathSciNet)
MR1892095

Zentralblatt MATH identifier
1041.55011

Subjects
Primary: 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20]
Secondary: 14H37: Automorphisms 20J06: Cohomology of groups 57N05: Topology of $E^2$ , 2-manifolds 57R20: Characteristic classes and numbers

Citation

Akita, Toshiyuki. Nilpotency and triviality of mod $p$ Morita-Mumford classes of mapping class groups of surfaces. Nagoya Math. J. 165 (2002), 1--22. https://projecteuclid.org/euclid.nmj/1114631695


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