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Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

Published online by Cambridge University Press:  01 May 2008

S. MORITA  and
R. C. PENNER
S. MORITA*
Affiliation:
Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan. e-mail: morita@ms.u-tokyo.ac.jp
R. C. PENNER
Affiliation:
Departments of Mathematics and Physics/Astronomy, University of Southern California, Los Angeles, CA 90089, U.S.A. e-mail: rpenner@math.usc.edu
*
†Partially supported by JSPS Grant 16204005.
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Abstract

Infinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given, and finite presentations of the classical Torelli groups acting trivially on homology modulo N are derived for all N. Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit canonical 1-cocycle of the Teichmüller space. The main tool for these results is the known mapping class group invariant ideal cell decomposition of the Teichmüller space.

This new 1-cocycle is mapping class group equivariant, so various contractions of its powers yield various combinatorial (co)cycles of the moduli space of curves, which are also new. Our combinatorial construction can be related to former works of Kawazumi and the first-named author with the consequence that the algebra generated by the cohomology classes represented by the new cocycles is precisely the tautological algebra of the moduli space.

There is finally a discussion of prospects for similarly finding cocycle lifts of the higher Johnson homomorphisms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 3 , May 2008 , pp. 651 - 671
Copyright
Copyright © Cambridge Philosophical Society 2008

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