Abstract
let $\Gamma$ be the mapping class group of an orientable surface $F$ of genus $g$ with $r$ boundary components and $s$ punctures. The main theorem of this paper is that for $g\ge 3k -1$, $H_k(\Gamma)$ is independent of $g$ and $r$ as long as $r>0$, for $g\ge 3k$, $H_k(\Gamma;\mathbf{Q})$ is independent of $g$ and $r$ for every $r$, and for $g \ge 3k + 1$, $H_k(\Gamma)$ is independent of $g$ and $r$ for every $r$.
