The homology of the stable nonorientable mapping class group

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 $N_{\infty}$ (after plus-construction). At odd primes $p$ , the $F_{p}$ –homology coincides with that of $Q_{0} ({ℍ ℙ}_{+}^{\infty})$ , 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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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/

Volume 8, issue 3 (2008)

Oscar Randal-Williams

Abstract