English
 
Help Privacy Policy Disclaimer
  Advanced SearchBrowse

Item

ITEM ACTIONSEXPORT

Released

Journal Article

The Euler characteristic of the moduli space of curves.

MPS-Authors
/persons/resource/persons236497

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

External Resource
No external resources are shared
Fulltext (restricted access)
There are currently no full texts shared for your IP range.
Fulltext (public)
There are no public fulltexts stored in PuRe
Supplementary Material (public)
There is no public supplementary material available
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').