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Mod 2 homology of the stable spin mapping class group. (English) Zbl 1102.55005
Author’s abstract: Using the main result of I. Madsen and M. Weiss [The stable moduli space of Riemann surfaces: Mumford’s conjecture, arXiv:math.AT/0212321 (2002)], we compute the mod 2 homology of spin mapping class groups in the stable range. In earlier work [Topology 43, No. 5, 1105–1132 (2004; Zbl 1074.57013)], we computed the stable mod \(p\) homology of the oriented mapping class group, and the methods and results here are very similar. The forgetful map from the spin mapping class group to the oriented mapping class groups induces a homology isomorphism for odd \(p\) but for \(p=2\) it is far from being an isomorphism. We include a general discussion of tangential structures on 2-manifolds and their mapping class groups and then specialise to spin structures.

MSC:
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
55P47 Infinite loop spaces
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
57M50 General geometric structures on low-dimensional manifolds
57M99 General low-dimensional topology
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References:
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