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.
Notes on file formats
Kiyoshi Igusa, Michael Kleber
Department of Mathematics, Brandeis University
Waltham, MA 02454-9110, USA
Email: igusa@brandeis.edu, kleber@brandeis.edu
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