#### Volume 8, issue 3 (2008)

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The homology of the stable nonorientable mapping class group

### Oscar Randal-Williams

Algebraic & Geometric Topology 8 (2008) 1811–1832
##### 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 ${\mathsc{N}}_{\infty }$ (after plus-construction). At odd primes $p$, the ${\mathbb{F}}_{p}$–homology coincides with that of ${Q}_{0}\left({ℍ\phantom{\rule{0.3em}{0ex}}ℙ}_{+}^{\infty }\right)$, 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 ${\mathsc{N}}_{\infty }$ 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 ${H}^{\ast }\left({\mathsc{N}}_{\infty };{F}_{2}\right)$ consisting of geometrically-defined characteristic classes.

##### Keywords
mapping class group, characteristic class, surface bundle, nonorientable surface, Dyer–Lashof operation, Eilenberg–Moore spectral sequence
##### Mathematical Subject Classification 2000
Primary: 57R20, 55P47
Secondary: 55S12, 55T20