Let ${\mathrm{\Gamma }}_{g,1}^m$ be the mapping class group of the orientable surface ${\mathrm{\Sigma }}_{g,1}^m$ of genus $g$ with one parametrized boundary curve and $m$ permutable punctures; when $m=0$ we omit it from the notation. Let ${\beta }_m({\mathrm{\Sigma }}_{g,1})$ be the braid group on $m$ strands of the surface ${\mathrm{\Sigma }}_{g,1}$. We prove that $H_{\ast }({\mathrm{\Gamma }}_{g,1}^m;{\mathbb{Z}}_2)\cong H_{\ast }({\mathrm{\Gamma }}_{g,1};H_{\ast }({\beta }_m({\mathrm{\Sigma }}_{g,1});{\mathbb{Z}}_2))$. The main ingredient is the computation of $H_{\ast }({\beta }_m({\mathrm{\Sigma }}_{g,1});{\mathbb{Z}}_2)$ as a symplectic representation of ${\mathrm{\Gamma }}_{g,1}$.

中文翻译：

---

穿孔映射类组的同源性拆分

让 ${\mathrm{\Gamma }}_{G，\mathrm{1个}}^{米}$ 是可定向曲面的映射类组 ${\mathrm{\Sigma }}_{G，\mathrm{1个}}^{米}$ 属 $G$ 有一条参数化的边界曲线 $米$可变穿刺 什么时候$米=0$我们从符号中省略了它。让${\beta }_{米}（{\mathrm{\Sigma }}_{G，\mathrm{1个}}）$ 成为编织团体 $米$ 表面的股线 ${\mathrm{\Sigma }}_{G，\mathrm{1个}}$。我们证明$H_{\ast }（{\mathrm{\Gamma }}_{G，\mathrm{1个}}^{米};{\mathbb{ž}}_2）\cong H_{\ast }（{\mathrm{\Gamma }}_{G，\mathrm{1个}};H_{\ast }（{\beta }_{米}（{\mathrm{\Sigma }}_{G，\mathrm{1个}}）;{\mathbb{ž}}_2））$。主要成分是$H_{\ast }（{\beta }_{米}（{\mathrm{\Sigma }}_{G，\mathrm{1个}}）;{\mathbb{ž}}_2）$ 作为...的辛表示 ${\mathrm{\Gamma }}_{G，\mathrm{1个}}$。