Homomorphisms between mapping class groups

Abstract

Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g\ge 6$ and $Y$ has genus at most $2g-1$; in addition, suppose that $Y$ is not closed if it has genus $2g-1$. Our main result asserts that every nontrivial homomorphism $Map\left(X\right)\to Map\left(Y\right)$ is induced by an embedding, ie a combination of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if $X$ has no boundary then every nontrivial endomorphism $Map\left(X\right)\to Map\left(X\right)$ is in fact an isomorphism.