zbMATH — the first resource for mathematics

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

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
Full Text: DOI EuDML
[1] Birman, J.: Braids, links, and mapping class groups. Ann. of Math. Studies 82, Princeton Univ. Press, Princeton 1975 · Zbl 0305.57013
[2] Dehn, M.: Die Gruppe der Abbildungsklassen. Acta Math.69, 135-206 (1938) · Zbl 0019.25301
[3] Earle, C.J., Eells, J.: The diffeomorphism group of a compact Riemann surface. Bull. AMS73, 557-559 (1967) · Zbl 0196.09402
[4] Harer, J.: Pencils of Curves on 4-Manifolds. Ph.D. Thesis, University of California, Berkeley 1979
[5] Hatcher, A., Thurston, W.: A presentation for the mapping class group of a closed orientable surface. Top.19, 221-237 (1980) · Zbl 0447.57005
[6] Meyer, W.: Die Signatur von Flächenbündeln. Math. Ann.201, 239-264 (1973) · Zbl 0245.55017
[7] Mumford, D.: Abelian quotients of the Teichmuller modular group. Journal d’Analyse Mathematique18, 227-244 (1967) · Zbl 0173.22903
[8] Neumann, W.: Signature related invariants of manifolds. I. Monodromy and ?-Invariants. Top.18, 239-264 (1973)
[9] Powell, J.: Two theorems on the mapping class group of a surface. Proc. Amer. Math. Soc.68, No. 3, 347-350 (1978) · Zbl 0391.57009
[10] Wajnryb, B.: A simple presentation for the mapping class group of an orientable surface. Preprint 1982 · Zbl 0533.57002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.