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Homological stability for the mapping class groups of non-orientable surfaces

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We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum built from the canonical bundle over the Grassmannians of 2-planes in ℝn+2. In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i – this is the non-oriented analogue of the Mumford conjecture.

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  1. Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL, 60637, USA

    Nathalie Wahl

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  1. Nathalie Wahl
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Wahl, N. Homological stability for the mapping class groups of non-orientable surfaces. Invent. math. 171, 389–424 (2008). https://doi.org/10.1007/s00222-007-0085-7

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  • DOI: https://doi.org/10.1007/s00222-007-0085-7

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