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

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.