On the equivalence problem for toric contact structures on \(S^3\)-bundles over \(S^2\)

DOI10.2140/PJM.2014.267.277zbMATH Open1301.53080arXiv1204.2209OpenAlexW2142947808MaRDI QIDQ2510044

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Publication date: 31 July 2014

Published in: (Search for Journal in Brave)

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 $peq{p}^{\prime }$.

Full work available at URL: https://arxiv.org/abs/1204.2209

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