The mod 2 cohomology of the mapping class group of the Klein bottle with marked points

The purpose of this article is to compute the mod 2 cohomology of Γq(K), the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg-MacLane spaces Xq=K(Γq(K),1) and fiber bundles Fq(K)/Σq→Xq→B(Z2×O(2)), where Fq(K)/Σq denotes the configuration space of unordered q-tuples of distinct points in K and B(Z2×O(2)) is the classifying space of the group Z2×O(2). Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.