Embedding mapping-class groups of orientable surfaces with one boundary component

TL;DR

This paper constructs and proves the injectivity of homomorphisms between mapping-class groups of orientable surfaces with boundary and punctures, extending known embeddings and providing new structural insights.

Contribution

It introduces new injective homomorphisms between mapping-class groups of surfaces with different genus and puncture configurations, generalizing classical embeddings.

Findings

Constructed homomorphisms between mapping-class groups

Proved these homomorphisms are injective

Extended classical embeddings to new surface types

Abstract

Let $S_{g,1,p}$ be an orientable surface of genus $g$ with one boundary component and $p$ punctures. Let $\mathcal{M}_{g,1,p}$ be the mapping-class group of $S_{g,1,p}$ relative to the boundary. We construct homomorphisms $\mathcal{M}_{g,1,p} \to \mathcal{M}_{g',1,(b-1)}$, where $g' \geq 0$ and $b\geq 1$. We proof that the constructed homomorphisms $\M_{g,1,p} \to \M_{g',1,(b-1)}$ are injective. One of these embeddings for $g = 0$ is classic.