The second homology group of the mapping class group of an orientable surface.

*(English)*Zbl 0533.57003In 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”.

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 |

##### Keywords:

second homology group; mapping class group; orientable surface; maximal systems of isotopy classes of nonintersecting simple closed curves; bordism classes of fiber bundles; Picard group; moduli space; Francetta conjecture**OpenURL**

##### References:

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