Homomorphisms between pure mapping class groups

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 $\mathrm{PMap}\left(S\right)\to \mathrm{PMap}\left(S^{\prime }\right)$ is induced by a multi-embedding. As an application, we classify all homomorphisms $\mathrm{PMap}\left(S\right)\to \mathrm{PMap}\left(S^{\prime }\right)$ for $g\ge 4$ and $g^{\prime }\le 6\cdot 2^{g-4}$.