Algebraic & Geometric Topology

The homology of the stable nonorientable mapping class group

Oscar Randal-Williams

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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.

Article information

Source
Algebr. Geom. Topol., Volume 8, Number 3 (2008), 1811-1832.

Dates
Received: 2 April 2008
Revised: 11 September 2008
Accepted: 12 September 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796908

Digital Object Identifier
doi:10.2140/agt.2008.8.1811

Mathematical Reviews number (MathSciNet)
MR2448874

Zentralblatt MATH identifier
1211.55007

Subjects
Primary: 57R20: Characteristic classes and numbers 55P47: Infinite loop spaces
Secondary: 55S12: Dyer-Lashof operations 55T20: Eilenberg-Moore spectral sequences [See also 57T35]

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

Citation

Randal-Williams, Oscar. The homology of the stable nonorientable mapping class group. Algebr. Geom. Topol. 8 (2008), no. 3, 1811--1832. doi:10.2140/agt.2008.8.1811. https://projecteuclid.org/euclid.agt/1513796908


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