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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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