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Combining results of Wahl, Galatius--Madsen--Tillmann--Weiss and Korkmaz onecan identify the homotopy-type of the classifying space of the stablenon-orientable mapping class group $N_\infty$ (after plus-construction). At oddprimes p, the F_p-homology coincides with that of $Q_0(HP^\infty_+)$, but atthe prime 2 the result is less clear. We identify the F_2-homology as a Hopfalgebra in terms of the homology of well-known spaces. As an application wetabulate the integral stable homology of $N_\infty$ in degrees up to six.  As in the oriented case, not all of these cohomology classes have a geometricinterpretation. We determine a polynomial subalgebra of $H^*(N_\infty ; F_2)$consisting of geometrically-defined characteristic classes.

The homology of the stable non-orientable mapping class group