The homology of the stable non-orientable 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 non-orientable mapping class group $N_{\infty }$ (after plus-construction). At odd primes p, the F_p-homology coincides with that of $Q_0(H{P}_{+}^{\infty })$, 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 $N_{\infty }$ 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 $H^{\ast }(N_{\infty };F_2)$ consisting of geometrically-defined characteristic classes.

15 pages; Section 6 completely revised, otherwise cosmetic changes

References in corpus (1)

• Homological stability for the mapping class groups of non-orientable surfaces

Cited by in corpus (1)

• Splitting Madsen-Tillmann spectra II. The Steinberg idempotents and Whitehead conjecture