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The second homology group of the mapping class group of an orientable surface. (English) Zbl 0533.57003
In this paper the second homology group \(H_ 2(\Gamma)\) of the mapping class group \(\Gamma\) of an orientable surface is computed. Let F be an oriented surface of genus g with r boundary components and n distinguished points. The mapping class group \(\Gamma =\Gamma(F)\) of F is \(\pi_ 0(Diff^+ F)\) where \(Diff^+ F\) is the topological group of orientation preserving diffeomorphisms of F which fix the n points and restrict to the identity on \(\partial F\). Theorem: \(H_ 2(\Gamma)={\mathbb{Z}}^{n+1}\) if \(g\geq 5\), \(r+n>0\); and \(H_ 2(\Gamma)={\mathbb{Z}}\oplus {\mathbb{Z}}/(2g-2)\) if \(g\geq 5\), \(r=n=0.\)
The proof is long and involved. Using maximal systems of isotopy classes of nonintersecting simple closed curves on the surface as vertices, A. Hatcher and W. Thurston [Topology 19, 221-237 (1980; Zbl 0447.57005)] constructed a complex on which \(\Gamma\) operates (from this a presentation of \(\Gamma\) can be derived). In the proof of the main theorem of the present paper a simplified version Y of this complex is constructed (which can be used to give a simpler presentation of \(\Gamma\) [cf. B. Wajnryb (see the preceding review)]). ”A well-known spectral sequence technique then allows us to find \(H_ 2(\Gamma)\) in terms of \(H_ 2(Y/\Gamma)\) and the lower homology groups of the stabilizers of the cells of Y.”
The theorem can be interpreted in terms of bordism classes of fiber bundles \(F\to W^ 4\to T\) over closed surfaces T. It answers a conjecture of Mumford that the Picard group Pi\(c({\mathcal M}) (\cong H^ 2(\Gamma))\) of the moduli space of genus \(g\geq 5\) has rank one. As noted by Mumford, it also gives a proof of the ”rational version of the Francetta conjecture”.
Reviewer: B.Zimmermann

MSC:
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57R50 Differential topological aspects of diffeomorphisms
30F20 Classification theory of Riemann surfaces
20J05 Homological methods in group theory
14D20 Algebraic moduli problems, moduli of vector bundles
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