Combinatorial Miller–Morita–Mumford classes and Witten cycles

Igusa, Kiyoshi

doi:10.2140/agt.2004.4.473

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Combinatorial Miller–Morita–Mumford classes and Witten cycles

Kiyoshi Igusa

Algebraic & Geometric Topology 4 (2004) 473–520

DOI: 10.2140/agt.2004.4.473

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

Volume 4, issue 1 (2004)