Reeb vector fields and open book decompositions
60 citations recorded by the source corpus.
We determine parts of the contact homology of certain contact 3-manifolds in the framework of open book decompositions, due to Giroux. We study two cases: when the monodromy map of the compatible open book is periodic and when it is pseudo-Anosov. For an open book with periodic monodromy, we verify the Weinstein conjecture. In the case of an open book with pseudo-Anosov monodromy, suppose the boundary of a page of the open book is connected and the fractional Dehn twist coefficient $c={k\over n}$, where $n$ is the number of prongs along the boundary. If $k\geq 2$, then there is a well-defined linearized contact homology group. If $k\geq 3$, then the linearized contact homology is exponentially growing with respect to the action, and every Reeb vector field of the corresponding contact structure admits an infinite number of simple periodic orbits.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5f94d395c48fc99c659c65ef508e2cf518604c33
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In Geometry and Topology
- Michael Hutchings
- Clifford H. Taubes
- Basak Z. Gurel
- V. Ginzburg
- Daniel Cristofaro-Gardiner
- F. Schlenk
- Richard Siefring
- Andrew Cotton-Clay
In Mathematical Physics
- Michael Hutchings
- Clifford H. Taubes
- V. Ginzburg
- Basak Z. Gurel
- Daniel Cristofaro-Gardiner
- F. Schlenk
- Richard Siefring
- Andrew Cotton-Clay
Citations in the corpus
- The Seiberg–Witten equations and the Weinstein conjecture (2006, result): “The most recent results known to the author are those of Etnyre and Gay [8], Colin and Honda [6] and Abbas, Cielebak and Hofer [1].”
- The Conley Conjecture and Beyond (2014, background): “, Colin and Honda (2013), Hryniewicz and Macarini (2015) and McLean (2012) and the results generalize and are inspired by a theorem from Gromoll and Meyer (1969), establishing the existence of infinitely many closed geodesics for manifolds whose free loop space homology grows. (A technical but impor”
- Lecture Notes on Embedded Contact Homology (2013, methodology): “To put this in perspective, Colin-Honda [8] used linearized contact homology to show that for many contact structures, every contact form has infinitely many embedded Reeb orbits.”
- Torsion contact forms in three dimensions have two or infinitely many Reeb orbits (2017, unknown): “For more complicated three-manifolds, Colin and Honda [3] used linearized contact homology to show that for many contact three-manifolds .”