Combining results of Wahl, Galatius–Madsen–Tillmann–Weiss and Korkmaz, one can identify the homotopy type of the classifying space of the stable nonorientable mapping class group (after plus-construction). At odd primes , the –homology coincides with that of , but at the prime 2 the result is less clear. We identify the –homology as a Hopf algebra in terms of the homology of well-known spaces. As an application we tabulate the integral stable homology of 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 consisting of geometrically-defined characteristic classes.
"The homology of the stable nonorientable mapping class group." Algebr. Geom. Topol. 8 (3) 1811 - 1832, 2008. https://doi.org/10.2140/agt.2008.8.1811