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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Combinatorial Miller–Morita–Mumford classes and Witten cycles

Kiyoshi Igusa

Algebraic & Geometric Topology 4 (2004) 473–520

arXiv: math.GT/0207042

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, tautological classes, Miller–Morita–Mumford classes, Witten conjecture, Stasheff associahedra
Mathematical Subject Classification 2000
Primary: 57N05
Secondary: 55R40, 57M15
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Publication
Received: 18 December 2003
Revised: 26 May 2004
Accepted: 6 July 2004
Published: 8 July 2004
Authors
Kiyoshi Igusa
Department of Mathematics
Brandeis University
Waltham MA 02454-9110
USA