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Reeb dynamics of the link of the $A_n$ singularity

Leonardo Abbrescia, I. Huq-Kuruvilla, J. Nelson and N. Sultani

Vol. 10 (2017), No. 3, 417–442
Abstract

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 .

Keywords
contact geometry, contact topology, Conley–Zehnder index, $A_n$ singularity, Reeb dynamic, Maslov index
Mathematical Subject Classification 2010
Primary: 37B30, 53D35, 57R17
Secondary: 53D42
Milestones
Received: 18 September 2015
Revised: 7 June 2016
Accepted: 9 June 2016
Published: 14 December 2016

Communicated by Colin Adams
Authors
Leonardo Abbrescia
Department of Mathematics
Michigan State University
1016 Chester Rd.
Apt. B14
Lansing, MI 48912
United States
I. Huq-Kuruvilla
Department of Mathematics
Columbia University
2920 Broadway
6580 Lerner Hall
New York, NY 10027
United States
J. Nelson
Institute for Advanced Study
Columbia University
2990 Broadway
Room 509, MC 4403
New York, NY 10027
United States
N. Sultani
Department of Mathematics
Columbia University
2920 Broadway
6580 Lerner Hall
New York, NY 10027
United States