Constraining mapping class group homomorphisms using finite subgroups

We classify homomorphisms from mapping class groups by using arguments involving finite subgroups. First, we give a new proof of a result of Aramayona–Souto that all homomorphisms between certain mapping class groups of closed surfaces are trivial. Second, we show that only finitely many mapping class groups of closed surfaces have nontrivial homomorphisms to \(\text {Homeo}(\mathbb {S}^n)\) $\text{Homeo}({\mathbb{S}}^n)$ for any n, where \(\mathbb {S}^n\) ${\mathbb{S}}^n$ is the n-sphere. We also effectivize this result for small values of n; for instance, we prove that every homomorphism from \(\text {Mod}(S_g)\) $\text{Mod}(S_g)$ to \(\text {Homeo}(\mathbb {S}^2)\) $\text{Homeo}({\mathbb{S}}^2)$ or \(\text {Homeo}(\mathbb {S}^3)\) $\text{Homeo}({\mathbb{S}}^3)$ is trivial if \(g\ge 3\) $g\ge 3$ , extending a result of Franks–Handel.