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Mapping class groups and moduli spaces of curves. (English) Zbl 0914.14013
Kollár, János (ed.) et al., Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9–29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62(pt.2), 97-142 (1997).
The paper under review is an excellent survey on recent results on the structure of the mapping class groups and the geometry of the moduli spaces of smooth complex curves. In particular the authors focus on the contribution from graph cohomology theory and Hodge theory to Mumford’s conjecture that stable cohomology of the mapping class group is generated by the tautological classes.
Let $$\Gamma_{g,r}^n$$ denote the mapping class group of a general compact oriented surface $$S_g$$ of genus $$g$$ with $$n$$ distinguished points and $$n$$ boundary components. Let $${\mathcal M}_{g,r}^n$$ be the moduli space obtained as a quotient orbifold by $$\Gamma_{g,r}^n$$. Notice that $${\mathcal M}_g^n= {\mathcal M}_{g,0}^n$$ is the moduli space of complex genus $$g$$ curves with $$n$$ distinguished points. Let $$T_{g,r}^n$$ be the Torelli group obtained as subgroup of $$\Gamma_{g,r}^n$$ acting trivially on the first homology group of $$S_g$$. The authors mainly review the following.
(a) Harer’s stability theorem and its application to the results of Looijenga and Pikaart on the ring structure and the mixed Hodge structure of the stable cohomology,
(b) the description in low genus case and the conjectures of Mumford, Faber and others on the structure of Chow algebra $$\text{CH}^* (\overline{\mathcal M}_g^n)$$, where $$\overline{\mathcal M}_g^n$$ denotes the Deligne-Mumford-Knudsen compactification of $${\mathcal M}_g^n$$,
(c) a stratification of $${\mathcal M}_g^n$$ using ribbon graph and Strebel’s theorem,
(d) Johnson’s result on the structure of $$\Gamma_{g,r}^n$$,
(e) Hain’s result that there exists a canonical mixed Hodge structure of the relative Mal’tsev Lie algebra associated to the natural homomorphism $$\Gamma_{g,r}^n= \pi_i ({\mathcal M}_{g,r}^n)\to \text{Sp}_g$$,
(f) the result of Kawazumi, Morita and the authors that the elements of $$H^*(\Gamma_\infty^0, \mathbb{Q})$$ constructed by Kontsevich using graph cohomology become polynomials in the tautological classes.
For the entire collection see [Zbl 0882.00033].

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 20F34 Fundamental groups and their automorphisms (group-theoretic aspects) 14H15 Families, moduli of curves (analytic)
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