Algebraic & Geometric Topology 8 (2008) 1811–1832

DOI: 10.2140/agt.2008.8.1811

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 N_∞ (after plus-construction). At odd primes p, the F_p–homology coincides with that of Q₀(H P₊^∞), but at the prime 2 the result is less clear. We identify the F₂–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_∞ 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^*(N_∞;F₂) 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

Primary: 55P47, 57R20

Secondary: 55S12, 55T20

References

Publication

Received: 2 April 2008
Revised: 11 September 2008
Accepted: 12 September 2008
Published: 20 October 2008

Authors