Geometry & Topology, Vol. 8 (2004) Paper no. 26, pages 969--1012.

Increasing trees and Kontsevich cycles

Kiyoshi Igusa, Michael Kleber

Abstract. It is known that the combinatorial classes in the cohomology of the mapping class group of punctures surfaces defined by Witten and Kontsevich are polynomials in the adjusted Miller-Morita-Mumford classes. The leading coefficient was computed in [Kiyoshi Igusa: Algebr. Geom. Topol. 4 (2004) 473-520]. The next coefficient was computed in [Kiyoshi Igusa: arXiv:math.AT/0303157, to appear in Topology]. The present paper gives a recursive formula for all of the coefficients. The main combinatorial tool is a generating function for a new statistic on the set of increasing trees on 2n+1 vertices. As we already explained in the last paper cited this verifies all of the formulas conjectured by Arbarello and Cornalba [J. Alg. Geom. 5 (1996) 705--749]. Mondello [arXiv:math.AT/0303207, to appear in IMRN] has obtained similar results using different methods.

Keywords. Ribbon graphs, graph cohomology, mapping class group, Sterling numbers, hypergeometric series, Miller-Morita-Mumford classes, tautological classes

AMS subject classification. Primary: 55R40. Secondary: 05C05.

DOI: 10.2140/gt.2004.8.969

E-print: arXiv:math.AT/0303353

Submitted to GT on 30 March 2003. Paper accepted 11 June 2004. Paper published 8 July 2004.

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Kiyoshi Igusa, Michael Kleber
Department of Mathematics, Brandeis University
Waltham, MA 02454-9110, USA

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