The homology of the mapping class group. (English) Zbl 0618.57005

Miller, Edward Y.

J. Differ. Geom. 24, 1-14 (1986).

Let \(S_ g\) be a Riemann surface of genus g, \(\Gamma_ g\) its mapping class group and \(M_ g\) its moduli space. In this paper we show that \(M_ g\), B \(\Gamma\) \({}_ g\), and B Diff\({}^+(S_ g)\) get more and more complicated as the genus g tends to infinity. More precisely, we prove: Theorem 1.1. Let \({\mathbb{Q}}[z_ 2,z_ 4,z_ 6,...]\) denote the polynomial algebra of generators \(z_{2n}\) in dimension 2n, \(n=1,2,3,\cdot \cdot \cdot\). There are classes \(y_ 2,y_ 4,\cdot \cdot \cdot,y_{2n},\cdot \cdot \cdot\) with \(y_{2n}\) in the 2nth cohomology group \(H^{2n}(B Diff^+(S_ g); {\mathbb{Z}})\) such that the homomorphism of algebras sending \(z_{2n}\) to \(y_{2n}\) \[ {\mathbb{Q}}[z_ 2,z_ 4,\cdot \cdot \cdot] \to H^*(M_ g; {\mathbb{Q}})\cong H^*(B Diff^+(S_ g); {\mathbb{Q}}) \] is an injection in dimensions less than (g/3).

MSC:

57N05	Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
30F20	Classification theory of Riemann surfaces
20J05	Homological methods in group theory
57R50	Differential topological aspects of diffeomorphisms

Keywords:

cohomology of mapping class group; Riemann surface; moduli space