Estimating the number of Reeb chords using a linear representation of the characteristic algebra

Dimitroglou Rizell, Georgios; Golovko, Roman

doi:10.2140/agt.2015.15.2887

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Estimating the number of Reeb chords using a linear representation of the characteristic algebra

Georgios Dimitroglou Rizell and Roman Golovko

Algebraic & Geometric Topology 15 (2015) 2885–2918

DOI: 10.2140/agt.2015.15.2887

Abstract

Given a chord-generic, horizontally displaceable Legendrian submanifold $Λ \subset P \times ℝ$ with the property that its characteristic algebra admits a finite-dimensional matrix representation, we prove an Arnold-type lower bound for the number of Reeb chords on $Λ$ . This result is a generalization of the results of Ekholm, Etnyre, Sabloff and Sullivan, which hold for Legendrian submanifolds whose Chekanov–Eliashberg algebras admit augmentations. We also provide examples of Legendrian submanifolds $Λ$ of $ℂ^{n} \times ℝ$ , $n \geq 1$ , whose characteristic algebras admit finite-dimensional matrix representations but whose Chekanov–Eliashberg algebras do not admit augmentations. In addition, to show the limits of the method of proof for the bound, we construct a Legendrian submanifold $Λ \subset ℂ^{n} \times ℝ$ with the property that the characteristic algebra of $Λ$ does not satisfy the rank property. Finally, in the case when a Legendrian submanifold $Λ$ has a non-acyclic Chekanov–Eliashberg algebra, using rather elementary algebraic techniques we obtain lower bounds for the number of Reeb chords of $Λ$ . These bounds are slightly better than the number of Reeb chords it is possible to achieve with a Legendrian submanifold whose Chekanov–Eliashberg algebra is acyclic.

Keywords

Legendrian contact homology, characteristic algebra, linear representation, Arnold-type inequality

Mathematical Subject Classification 2010

Primary: 53D12

Secondary: 53D42

References

Publication

Received: 29 September 2014

Revised: 6 February 2015

Accepted: 13 February 2015

Published: 10 December 2015

Authors

Georgios Dimitroglou Rizell
Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB UK
http://www.dimitroglou.name/
Roman Golovko
Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Realtanoda u. 13–15 Budapest H-1053 Hungary
http://sites.google.com/site/ragolovko/

Volume 15, issue 5 (2015)