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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

Given a chord-generic, horizontally displaceable Legendrian submanifold Λ P × 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 n × , n 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 n × 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.

Legendrian contact homology, characteristic algebra, linear representation, Arnold-type inequality
Mathematical Subject Classification 2010
Primary: 53D12
Secondary: 53D42
Received: 29 September 2014
Revised: 6 February 2015
Accepted: 13 February 2015
Published: 10 December 2015
Georgios Dimitroglou Rizell
Department of Pure Mathematics and Mathematical Statistics
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge CB3 0WB
Roman Golovko
Alfréd Rényi Institute of Mathematics
Hungarian Academy of Sciences
Realtanoda u. 13–15