On the Equivalence Problem for Toric Contact Structures on S^3-bundles over S^2$

Abstract

We study the contact equivalence problem for toric contact structures on $S^3$-bundles over $S^2$. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To find inequivalent toric contact structures that are contact equivalent, we show that the corresponding 3-tori belong to distinct conjugacy classes in the contactomorphism group. To show that two toric contact structures with the same first Chern class are contact inequivalent, we use Morse-Bott contact homology. We treat a subclass of contact structures which include the Sasaki-Einstein contact structures $Y^{p,q}$ studied by physicists. In this subcase we give a complete solution to the contact equivalence problem by showing that $Y^{p,q}$ and $Y^{p^{\prime }{q}^{\prime }}$ are inequivalent as contact structures if and only if $p\ne p^{\prime }$.

61 pages

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• Note on a Cohomological Theory of Contact-Instanton and Invariants of Contact Structures
• The Sasaki Join, Hamiltonian 2-forms, and Sasaki-Einstein Metrics
• The $S^3_\bfw$ Sasaki Join Construction
• On Positivity in Sasaki Geometry
• On the Topology of some Sasaki-Einstein Manifolds