The homology of the stable nonorientable mapping class group

Abstract

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 ${\mathcal{N}}_{\infty }$ (after plus-construction). At odd primes $p$, the ${\mathbb{F}}_p$–homology coincides with that of $Q_0\left({ℍ\mathbb{P}}_{+}^{\infty }\right)$, but at the prime 2 the result is less clear. We identify the ${\mathbb{F}}_2$–homology as a Hopf algebra in terms of the homology of well-known spaces. As an application we tabulate the integral stable homology of ${\mathcal{N}}_{\infty }$ 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 $H^{\ast }\left({\mathcal{N}}_{\infty };F_2\right)$ consisting of geometrically-defined characteristic classes.