Characteristic classes of Klein bottle bundles

The purpose of this article is to compute the cohomology of BDiff(K), the classifying space of the diffeomorphisms of the Klein bottle. We analyze the Serre spectral sequence of the fibration BDiff0(K)→BDiff(K)→B(Z2×Z2), with non-trivial coefficients, and show it collapses. Moreover, we show that BDiff(K) has the homotopy type of BZ2×BO(2). Finally we give direct and concrete constructions for the Eilenberg-MacLane spaces K(Γq(K),1), where Γq(K) denotes the mapping class group of K with marked points π0Diff(K;q).