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

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 ${\mathcal{M}}_{g,1,p}\to {\mathcal{M}}_{g^{\mathrm{\prime }},1,(b-1)}$, where $g^{\mathrm{\prime }}\ge 0$ and $b\ge 1$. We proof that the constructed homomorphisms $\M_{g,1,p} \to \M_{g’,1,(b-1)}$areinjective\mathrm{.}Oneoftheseembeddingsfor$g = 0$ is classic.