Involve: A Journal of Mathematics

  • Involve
  • Volume 10, Number 3 (2017), 417-442.

Reeb dynamics of the link of the $A_n$ singularity

Leonardo Abbrescia, Irit Huq-Kuruvilla, Jo Nelson, and Nawaz Sultani

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The link of the An singularity, LAn 3 admits a natural contact structure ξ0 coming from the set of complex tangencies. The canonical contact form α0 associated to ξ0 is degenerate and thus has no isolated Reeb orbits. We show that there is a nondegenerate contact form for a contact structure equivalent to ξ0 that has two isolated simple periodic Reeb orbits. We compute the Conley–Zehnder index of these simple orbits and their iterates. From these calculations we compute the positive S1-equivariant symplectic homology groups for (LAn,ξ0). In addition, we prove that (LAn,ξ0) is contactomorphic to the lens space L(n + 1,n), equipped with its canonical contact structure ξstd.

Article information

Involve, Volume 10, Number 3 (2017), 417-442.

Received: 18 September 2015
Revised: 7 June 2016
Accepted: 9 June 2016
First available in Project Euclid: 12 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37B30: Index theory, Morse-Conley indices 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 57R17: Symplectic and contact topology
Secondary: 53D42: Symplectic field theory; contact homology

contact geometry contact topology Conley–Zehnder index $A_n$ singularity Reeb dynamic Maslov index


Abbrescia, Leonardo; Huq-Kuruvilla, Irit; Nelson, Jo; Sultani, Nawaz. Reeb dynamics of the link of the $A_n$ singularity. Involve 10 (2017), no. 3, 417--442. doi:10.2140/involve.2017.10.417.

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