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∞ (after plus-construction). At odd primes p, the Fp-homology coincides with that of Q0(HP∞+), 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.
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