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The Euler characteristic of the moduli space of curves.

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Zagier,  Don
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Harer, J., & Zagier, D. (1986). The Euler characteristic of the moduli space of curves. Inventiones Mathematicae, 85, 457-485.


Cite as: http://hdl.handle.net/21.11116/0000-0004-3966-A
Abstract
Let Γ\sp 1\sb g, g≥ 1, be the mapping class group consisting of all isotopy classes of base-point and orientation preserving homeomorphisms of a closed, oriented surface of genus g. Main theorem: \par χ(Γ\sp 1\sb g)=ζ (1-2g), where ζ(s) is the Riemann zeta function, and χ(Γ\sp 1\sb g)=[Γ\sp 1\sb g:Γ]\sp- 1χ (E/Γ) for any torsion free subgroup Γ of finite index in Γ\sp 1\sb g. E is a contractible space on which Γ acts freely and properly discontinuously. χ(E/Γ) is the usual Euler characteristic. \par For every positive integer n, consider a fixed 2n-gon with sides S\sb 1,...,S\sb2n. Then denote by ε\sb g(n) the number of ways of grouping S\sb 1,...,S\sb2n into n pairs making a surface of genus g under suitable identification of sides and denote by λ\sb g(n) the number of such groupings which do not contain the special two types of configurations. The authors prove a formula for χ(Γ\sp 1\sb g) expressed by λ\sb g(n) in theorem 1 and a formula for ε\sb g(n) as (essentially) the coefficient of x\sp2g in (x/\tanh (x))\spn+1 in theorem 2 and combine them to deduce the main theorem aforementioned. \par The authors write that the proof of theorem 2 is rather indirect, but the idea of their 'indirect' proof is very interesting: the main point is the integral formula for C(n,k)=\sum\sb0≤ g≤ nε\sb g(n)k\spn+1-2g given in \S 4. - In \S 6, the Euler characteristic of Γ\sp 1\sb g is given (theorems 4 and 4').