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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 N (after plus-construction). At odd primes p, the Fp–homology coincides with that of Q0(+), 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 2000
Primary: 57R20, 55P47
Secondary: 55S12, 55T20
References
Publication
Received: 2 April 2008
Revised: 11 September 2008
Accepted: 12 September 2008
Published: 20 October 2008
Authors
Oscar Randal-Williams
Mathematical Institute
24–29 St Giles’
Oxford
OX1 3LB
UK
http://people.maths.ox.ac.uk/~randal-w/