The homology of the stable non-orientable mapping class group

Combining results of Wahl, Galatius–Madsen–Tillmann–Weiss and Korkmaz one can identify the homotopy-type of the classifying space of the stable non-orientable mapping class group $N_{\mathrm{\infty }}$ (after plus-construction). At odd primes p, the F_p-homology coincides with that of $Q_0(H{P}_{+}^{\mathrm{\infty }})$, but at the prime 2 the result is less clear. We identify the F_2-homology as a Hopf algebra in terms of the homology of well-known spaces. As an application we tabulate the integral stable homology of $N_{\mathrm{\infty }}$ in degrees up to six. As in the oriented case, not all of these cohomology classes have a geometric interpretation. We determine a polynomial subalgebra of $H^{\ast }(N_{\mathrm{\infty }};F_2)$ consisting of geometrically-defined characteristic classes.