Mathematics – Algebraic Topology
Scientific paper
2008-03-26
Algebraic & Geometric Topology 8 (2008) 1811 - 1832
Mathematics
Algebraic Topology
15 pages; Section 6 completely revised, otherwise cosmetic changes
Scientific paper
10.2140/agt.2008.8.1989
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 $Q_0(HP^\infty_+)$, 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 $H^*(N_\infty ; F_2)$ consisting of geometrically-defined characteristic classes.
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