Constraining mapping class group homomorphisms using finite subgroups

TL;DR

This paper classifies homomorphisms from mapping class groups using finite subgroups, providing new proofs and extending results on triviality of such homomorphisms into homeomorphism groups of spheres.

Contribution

It offers a new proof of a known result on trivial homomorphisms between mapping class groups and shows finiteness of non-trivial homomorphisms into sphere homeomorphism groups.

Findings

Homomorphisms between mapping class groups of closed surfaces are trivial for certain genera.

Only finitely many mapping class groups have non-trivial homomorphisms into $ ext{Homeo}( ext{S}^n)$.

Homomorphisms from $ ext{Mod}(S_g)$ to $ ext{Homeo}( ext{S}^2)$ or $ ext{Homeo}( ext{S}^3)$ are trivial for $g extgreater= 3$.

Abstract

We classify homomorphisms from mapping class groups by using finite subgroups. First, we give a new proof of a result of Aramayona--Souto that homomorphisms between mapping class groups of closed surfaces are trivial for a range of genera. Second, we show that only finitely many mapping class groups of closed surfaces have non-trivial homomorphisms into $\text{Homeo}(\mathbb{S}^n)$ for any $n$. We also prove that every homomorphism from $\text{Mod}(S_g)$ to $\text{Homeo}(\mathbb{S}^2)$ or $\text{Homeo}(\mathbb{S}^3)$ is trivial if $g\ge 3$, extending a result of Franks--Handel.