Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Equivalence relations for homology cylinders and the core of the Casson invariant

Authors: Gwénaël Massuyeau and Jean–Baptiste Meilhan
Journal: Trans. Amer. Math. Soc. 365 (2013), 5431-5502
MSC (2010): Primary 57M27, 57N10, 20F38
Published electronically: February 25, 2013
MathSciNet review: 3074379
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Abstract: Let $ \Sigma $ be a compact oriented surface of genus $ g$ with one boundary component. Homology cylinders over $ \Sigma $ form a monoid $ \mathcal {IC}$ into which the Torelli group $ \mathcal {I}$ of $ \Sigma $ embeds by the mapping cylinder construction. Two homology cylinders $ M$ and $ M'$ are said to be $ Y_k$-equivalent if $ M'$ is obtained from $ M$ by ``twisting'' an arbitrary surface $ S\subset M$ with a homeomorphism belonging to the $ k$-th term of the lower central series of the Torelli group of $ S$. The $ J_k$-equivalence relation on $ \mathcal {IC}$ is defined in a similar way using the $ k$-th term of the Johnson filtration. In this paper, we characterize the $ Y_3$-equivalence with three classical invariants: (1) the action on the third nilpotent quotient of the fundamental group of $ \Sigma $, (2) the quadratic part of the relative Alexander polynomial, and (3) a by-product of the Casson invariant. Similarly, we show that the $ J_3$-equivalence is classified by (1) and (2). We also prove that the core of the Casson invariant (originally defined by Morita on the second term of the Johnson filtration of $ \mathcal {I}$) has a unique extension (to the corresponding submonoid of $ \mathcal {IC}$) that is preserved by $ Y_3$-equivalence and the mapping class group action.

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Gwénaël Massuyeau
Affiliation: Institut de Recherche Mathématique Avancée, Université de Strasbourg & CNRS, 7 rue René Descartes, 67084 Strasbourg, France

Jean–Baptiste Meilhan
Affiliation: Institut Fourier, Université de Grenoble 1 & CNRS, 100 rue des Maths – BP 74, 38402 Saint Martin d’Hères, France

Received by editor(s): July 28, 2011
Received by editor(s) in revised form: February 28, 2012
Published electronically: February 25, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.