Homomorphisms between pure mapping class groups

TL;DR

This paper characterizes homomorphisms between pure mapping class groups of surfaces, showing they are induced by multi-embeddings and classifying all such homomorphisms for certain genus ranges.

Contribution

It establishes that multitwist-preserving maps are induced by multi-embeddings and classifies all homomorphisms between pure mapping class groups for specified genus conditions.

Findings

Multitwist-preserving maps are induced by multi-embeddings.

Complete classification of homomorphisms for genus g ≥ 4 and g' ≤ 6·2^{g-4}.

Provides structural insights into pure mapping class groups.

Abstract

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