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

## TL;DR

This paper investigates the cohomology of the mapping class group of punctured non-orientable surfaces, especially the real projective plane, using homotopy theory and spectral sequences to relate it to configuration spaces.

## Contribution

It introduces a non-orientable version of the Birman exact sequence and analyzes the Serre spectral sequence for the real projective plane case, linking cohomology of the group to configuration spaces.

## Key findings

- Derived a non-orientable Birman exact sequence.
- Expressed mod-2 cohomology of the group in terms of configuration space cohomology.
- Analyzed the Serre spectral sequence for the associated fiber bundle.

## 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 $\mathbb R {\rm P}^2$, we analize the Serre spectral sequence of a fiber bundle $F_k(\mathbb R {\rm P}^2)/\Sigma_k \to X_k \to BSO(3)$ where $X_k$ is a $K(\Gamma^k(\mathbb R {\rm P}^2),1)$ and $F_k(\mathbb R {\rm P}^2)/\Sigma_k$ denotes the configuration space of unordered $k$-tuples of distinct points in $\mathbb R {\rm P}^2$. As a consequence, we express the mod-2 cohomology of $\Gamma^k(\mathbb R {\rm P}^2)$ in terms of that of $F_k(\mathbb R {\rm P}^2)/\Sigma_k$.