The homology of the stable nonorientable mapping class group

Randal-Williams, Oscar¹

¹Mathematical Institute, 24–29 St Giles’, Oxford, OX1 3LB, UK

Algebraic and Geometric Topology, Tome 8 (2008) no. 3, pp. 1811-1832

Résumé

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 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.

Détail

DOI : 10.2140/agt.2008.8.1811

Keywords: mapping class group, characteristic class, surface bundle, nonorientable surface, Dyer–Lashof operation, Eilenberg–Moore spectral sequence

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