Abelianization of Symmetric Mapping Class Groups

Abstract

Let $\tilde{S}\to S$ be an unbranched regular $p$-fold cyclic cover of a closed orientable surface $S$ of genus $g$. Two natural groups are associated to this cover. The first is the centralizer $\mathrm{Mod}(\tilde{S},\sigma )$ of a chosen generator $\sigma$ of the deck transformation group in $\mathrm{Mod}(\tilde{S})$. The second is the finite-index subgroup $\mathrm{Mod}(S,[\beta ])$ of $\mathrm{Mod}(S)$ consisting of mapping classes that fix the nonzero class $[\beta ]\in H_1(S;\mathbb{Z}/p\mathbb{Z})$ corresponding to the cover. For $p=2$, Sato computed the abelianizations of these groups. We compute their abelianizations for every odd prime $p$ and show that they exhibit a splitting phenomenon different from the case $p=2$. We also construct explicit abelianization maps using Morita's crossed homomorphism and the Prym representation.

arXiv comment21 pages, 1 figure. Replaced the technical computation of the Johnson homomorphism in the final step of the proof of Theorem 1.1 with an explicit construction of the abelianization map