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Tight contact structures on solid tori. (English) Zbl 0894.53036

Author’s abstract: “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 an \(\mathbb{R}/2\pi\mathbb{Z}\)-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 structures 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 (see also Y. Eliashberg [Invent. Math. 98, 623-637 (1989; Zbl 0684.57012)]. 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”.
Reviewer: D.Perrone (Lecce)

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Citations:

Zbl 0684.57012
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References:

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