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

TL;DR

This paper investigates the classification of toric contact structures on S^3-bundles over S^2, providing criteria for contact equivalence and inequivalence, especially among Sasaki-Einstein structures, using advanced geometric and homological tools.

Contribution

It offers a complete solution to the contact equivalence problem for a subclass of toric contact structures, including the Y^{p,q} structures, based on conjugacy classes and contact homology.

Findings

Toric contact structures with different 3-tori conjugacy classes are contact inequivalent.

Structures with the same first Chern class can be contact inequivalent, distinguished via Morse-Bott contact homology.

Y^{p,q} and Y^{p'q'} are inequivalent if and only if p ≠ p'.

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 $Y^{p'q'}$ are inequivalent as contact structures if and only if $p\neq p'$.