Volume 8, issue 2 (2004)

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Increasing trees and Kontsevich cycles

Kiyoshi Igusa and Michael Kleber

Geometry & Topology 8 (2004) 969–1012

arXiv: math.AT/0303353


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 first two coefficients were computed by the first author in earlier papers. 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 this verifies all of the formulas conjectured by Arbarello and Cornalba. Mondello has obtained similar results using different methods.

ribbon graphs, graph cohomology, mapping class group, Sterling numbers, hypergeometric series, Miller–Morita–Mumford classes, tautological classes
Mathematical Subject Classification 2000
Primary: 55R40
Secondary: 05C05
Forward citations
Received: 30 March 2003
Accepted: 11 June 2004
Published: 8 July 2004
Proposed: Shigeyuki Morita
Seconded: Ralph Cohen, Martin Bridson
Kiyoshi Igusa
Department of Mathematics
Brandeis University
Massachusetts 02454-9110
Michael Kleber
Department of Mathematics
Brandeis University
Massachusetts 02454-9110