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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Tight contact structures on solid tori


Author: Sergei Makar-Limanov
Journal: Trans. Amer. Math. Soc. 350 (1998), 1013-1044
MSC (1991): Primary 53C15; Secondary 57R15, 58F05
MathSciNet review: 1401526
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Abstract: In this paper we study properties of tight contact structures on solid tori. In particular we discuss ways of distinguishing two solid tori with tight contact structures. We also give examples of unusual tight contact structures on solid tori.

We prove the existence of a $\mathbb{Z}$-valued and a $\mathbb{R}/2\pi\mathbb{Z}$-valued invariant of a closed solid torus. We call them the self-linking number and the rotation number respectively. We then extend these definitions to the case of an open solid torus. We show that these invariants exhibit certain monotonicity properties with respect to inclusion. We also prove a number of results which give sufficient conditions for two solid tori to be contactomorphic.

At the same time we discuss various ways of constructing a tight contact structure on a solid torus. We then produce examples of solid tori with tight contact structures and calculate self-linking and rotation numbers for these tori. These examples show that the invariants we defined do not give a complete classification of tight contact structure on open solid tori.

At the end, we construct a family of tight contact structure on a solid torus such that the induced contact structure on a finite-sheeted cover of that solid torus is no longer tight. This answers negatively a question asked by Eliashberg in 1990. We also give an example of tight contact structure on an open solid torus which cannot be contactly embedded into a sphere with the standard contact structure, another example of unexpected behavior.


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Additional Information

Sergei Makar-Limanov
Affiliation: Département de Mathématiques, Université du Québec à Montréal, Montréal, Canada
Email: sergei@math.uqam.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01822-4
PII: S 0002-9947(98)01822-4
Received by editor(s): January 18, 1995
Article copyright: © Copyright 1998 American Mathematical Society