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

Dimitroglou Rizell, Georgios¹ ; Golovko, Roman²

¹Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK
²Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Realtanoda u. 13–15, Budapest, H-1053, Hungary

Algebraic and Geometric Topology, Tome 15 (2015) no. 5, pp. 2885-2918

Résumé

Given a chord-generic, horizontally displaceable Legendrian submanifold Λ ⊂ P × 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 ℂn × ℝ, 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 ⊂ ℂn × 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.

Détail

DOI : 10.2140/agt.2015.15.2887

Classification : 53D12, 53D42
Keywords: Legendrian contact homology, characteristic algebra, linear representation, Arnold-type inequality

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