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

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 ${\mathcal{M}}_{g,1,p}\to {\mathcal{M}}_{g^{\prime },1,(b-1)}$, where $g^{\prime }\ge 0$ and $b\ge 1$. 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.