Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces

Details Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces

2010-01-29

arxiv.org/abs/1001.5366v3

Mathematics

Geometric Topology

28 pages, 6 figures. V2: Revised to fit with arXiv:0909.4278. V3: Submitted version

Scientific paper

We give definitions of moduli spaces of framed, r-Spin and Pin surfaces. We apply earlier work of the author to show that each of these moduli spaces exhibits homological stability, and we identify the stable integral homology with that of certain infinite loop spaces in each case. We further show that these moduli spaces each have path components which are Eilenberg--MacLane spaces for the framed, r-Spin and Pin mapping class groups respectively, and hence we also identify the stable group homology of these groups. In particular: the stable framed mapping class group has trivial rational homology, and its abelianisation is Z/24; the rational homology of the stable Pin mapping class groups coincides with that of the non-orientable mapping class group, and their abelianisations are Z/2 for Pin^+ and (Z/2)^3 for Pin^-.

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