Algebraic and Geometric Topology 4 (2004), paper no. 23, pages 473-520.

Combinatorial Miller-Morita-Mumford classes and Witten cycles

Kiyoshi Igusa

Abstract. We obtain a combinatorial formula for the Miller-Morita-Mumford classes for the mapping class group of punctured surfaces and prove Witten's conjecture that they are proportional to the dual to the Witten cycles. The proportionality constant is shown to be exactly as conjectured by Arbarello and Cornalba [J. Alg. Geom. 5 (1996) 705-749]. We also verify their conjectured formula for the leading coefficient of the polynomial expressing the Kontsevich cycles in terms of the Miller-Morita-Mumford classes.

Keywords. Mapping class group, fat graphs, ribbon graphs, Miller-Morita-Mumford classes, tautological classes, Witten conjecture, Stasheff associahedra

AMS subject classification. Primary: 57N05. Secondary: 55R40, 57M15.

DOI: 10.2140/agt.2004.4.473

E-print: arXiv:math.GT/0207042

Submitted: 18 December 2003. (Revised: 26 May 2004.) Accepted: 6 July 2004. Published: 8 July 2004.

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Kiyoshi Igusa
Department of Mathematics, Brandeis University
Waltham, MA 02454-9110, USA
Email: igusa@brandeis.edu

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