ON THE COHOMOLOGY OF THE MAPPING CLASS GROUP OF THE PUNCTURED PROJECTIVE PLANE

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 RP2, we analyze the Serre spectral sequence of a fiber bundle F-k (RP2) / Sigma(k) -> X-k -> BSO(3) where X-k is a K (Gamma(k)(RP2), 1) and Gamma(k) (RP2) / Sigma(k) denotes the configuration space of unordered k-tuples of distinct points in RP2. As a consequence, we express the mod-2 cohomology of Gamma(k) (RP2) in terms of that of F-k (RP2) / Sigma(k).