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Relations in the tautological ring derived from combinatorial classes and hyperelliptic fatgraphs

Published online by Cambridge University Press:  01 March 2008

ALEX JAMES BENE
ALEX JAMES BENE*
Affiliation:
Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, CA 90089-2532, U.S.A. e-mail: bene@usc.edu
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Abstract

A closed formula is obtained for the integral of tautological classes over the locus of hyperelliptic Weier points in the moduli space of curves. As a corollary, a relation between Hodge integrals is obtained.

The calculation utilizes the homeomorphism between the moduli space of curves and the combinatorial moduli space , a PL-orbifold whose cells are enumerated by fatgraphs. This cell decomposition can be used to naturally construct combinatorial PL-cycles whose homology classes are essentially the Poin duals of the Mumford–Morita–Miller classes κa. In this paper we construct another PL-cycle representing the locus of hyperelliptic Weier points and explicitly describe the chain level intersection of this cycle with W1. Using this description of , the duality between Witten cycles Wa and the κa classes, and the Kontsevich--Penner method of integration, scheme of integrating ε classes, the integral is reduced to a weighted sum over graphs and is evaluated by the enumeration of trees.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 2 , March 2008 , pp. 369 - 395
Copyright
Copyright © Cambridge Philosophical Society 2008

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