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Stability of the homology of the moduli spaces of Riemann surfaces with spin structure. (English) Zbl 0715.57004
It is shown that the homology of the spin moduli spaces \({\mathcal M}_ g[\epsilon]\) of Riemann surfaces of genus g with spin structure of Arf invariant \(\epsilon\in {\mathbb{Z}}/2{\mathbb{Z}}\) (resp. of the corresponding spin mapping class groups) is stable, i.e. independent of g and \(\epsilon\) for sufficiently large g. As the author notes, the interest in these moduli spaces comes from fermionic string theory. For a second paper the computation of the first (integer coefficients) and second (rational coefficients) homology, and thus of the Picard group, of the spin moduli spaces is announced. All of this generalizes results and methods (constructing simplicial complexes from configuration of simple closed curves on a surface on which the mapping class groups act, then applying spectral sequence arguments) of two of the author’s previous papers in which he obtained analogous results for the ordinary mapping class groups resp. moduli spaces.
Reviewer: B.Zimmermann

MSC:
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M99 General low-dimensional topology
30F60 Teichm├╝ller theory for Riemann surfaces
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
57R50 Differential topological aspects of diffeomorphisms
14H10 Families, moduli of curves (algebraic)
14H55 Riemann surfaces; Weierstrass points; gap sequences
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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