Homomorphisms between mapping class groups

Abstract

Suppose that X and Y are surfaces of finite topological type, where X has genus g >= 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(X ) -> Map(Y) 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(X) -> Map(X) is in fact an isomorphism.