On the cohomology of the mapping class group of the punctured projective plane

Abstract

The mapping class group ${\Gamma }^k(N_g)$ of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of $\mathbb{R}{\mathrm{P}}^2$, we analize the Serre spectral sequence of a fiber bundle $F_k(\mathbb{R}{\mathrm{P}}^2)/{\Sigma }_k\to X_k\to BSO(3)$ where $X_k$ is a $K({\Gamma }^k(\mathbb{R}{\mathrm{P}}^2),1)$ and $F_k(\mathbb{R}{\mathrm{P}}^2)/{\Sigma }_k$ denotes the configuration space of unordered $k$-tuples of distinct points in $\mathbb{R}{\mathrm{P}}^2$. As a consequence, we express the mod-2 cohomology of ${\Gamma }^k(\mathbb{R}{\mathrm{P}}^2)$ in terms of that of $F_k(\mathbb{R}{\mathrm{P}}^2)/{\Sigma }_k$.