Homomorphisms between pure mapping class groups

Abstract

Let $S$ and $S^{\prime }$ be orientable finite-type surfaces of genus $g\ge 4$ and $g^{\prime }$, respectively. We prove that every multitwist-preserving map between pure mapping class groups $\text{PMap}(S)\to \text{PMap}(S^{\prime })$ is induced by a multi-embedding. As an application, we classify all homomorphisms $\text{PMap}(S)\to \text{PMap}(S^{\prime })$ for $g\ge 4$ and $g^{\prime }\le 6\cdot 2^{g-4}$.