Equivalence relations for homology cylinders and the core of the Casson invariant
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- by Gwénaël Massuyeau and Jean–Baptiste Meilhan PDF
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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.References
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Additional Information
- Gwénaël Massuyeau
- Affiliation: Institut de Recherche Mathématique Avancée, Université de Strasbourg & CNRS, 7 rue René Descartes, 67084 Strasbourg, France
- Email: massuyeau@math.unistra.fr
- Jean–Baptiste Meilhan
- Affiliation: Institut Fourier, Université de Grenoble 1 & CNRS, 100 rue des Maths – BP 74, 38402 Saint Martin d’Hères, France
- Email: jean-baptiste.meilhan@ujf-grenoble.fr
- Received by editor(s): July 28, 2011
- Received by editor(s) in revised form: February 28, 2012
- Published electronically: February 25, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 5431-5502
- MSC (2010): Primary 57M27, 57N10, 20F38
- DOI: https://doi.org/10.1090/S0002-9947-2013-05818-7
- MathSciNet review: 3074379