The homology of the stable nonorientable mapping class group

Randal-Williams, Oscar

doi:10.2140/agt.2008.8.1811

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The homology of the stable nonorientable mapping class group

Oscar Randal-Williams

Algebraic & Geometric Topology 8 (2008) 1811–1832

DOI: 10.2140/agt.2008.8.1811

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 $N_{\infty}$ (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.

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)