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