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 ${\mathrm{\Gamma }}^q(\mathbb{K})$, the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces $X_q=K({\mathrm{\Gamma }}^q(\mathbb{K}),1)$ and fiber bundles $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q\to X_q\to B({\mathbb{Z}}_2\times O(2))$, where $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q$ denotes the configuration space of unordered q-tuples of distinct points in $\mathbb{K}$ and $B({\mathbb{Z}}_2\times O(2))$ is the classifying space of the group ${\mathbb{Z}}_2\times O(2)$. Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.     

Comparing motives of smooth algebraic varieties

Given a perfect field of exponential characteristic e, the Cor-, $K_0^{\oplus }$-, $K_0$- and ${\mathbb{K}}_0$-motives of smooth algebraic varieties with $\mathbb{Z}[1/e]$-coefficients are shown to be locally quasi-isomorphic to each other. Moreover, it is proved that their triangulated categories of motives with $\mathbb{Z}[1/e]$-coefficients are equivalent. An application is given for the bivariant motivic spectral sequence.