The persistence of the Chekanov–Eliashberg algebra

Abstract

We apply the barcodes of persistent homology theory to the c Chekanov–Eliashberg algebra of a Legendrian submanifold to deduce displacement energy bounds for arbitrary Legendrians. We do not require the full Chekanov–Eliashberg algebra to admit an augmentation as we linearize the algebra only below a certain action level. As an application we show that it is not possible to $$C^0$$ C 0 -approximate a stabilized Legendrian by a Legendrian that admits an augmentation.

Legendrian persistence modules and dynamics

“…. The equality between the two compositions yields (17). This finishes the proof of part B and of the proposition.…”

“…For other results on the existence of Reeb chords between different Legendrian submanifolds (or equivalently, chords of positive contact Hamiltonians) see the papers of G.Dimitroglou-Rizell and M.Sullivan [16,17] (for a comparison of their results with the results in [27] and here, see [17,Sec. 1.3]).…”

“…In the simplest setting (say, where a contact manifold is the contactization of an exact symplectic manifold) the enhancement of the RSFT using the action filtration gives rise to the filtered relative symplectic field theory (FRSFT). Namely, the Legendrian contact homology can be viewed as a persistence module formed by vector spaces (over Z 2 ), while an exact Lagrangian cobordism between Legendrian submanifolds defines a morphism between the corresponding persistence modules, see papers [17,15] by Dimitroglou Rizell and M. Sullivan. In the present paper we rely on some special instances of this theory only leaving more general results for a forthcoming manuscript [29].…”

Viterbo conjecture for Zoll symmetric spaces

“…This map is Lipschitz with respect to the L 1,∞ -distance on H = H M , and the bottleneck distance on the space barcodes of barcodes. This observation was used in [76], in [3,40,78,92,99,105] and more recently in [18,31,60,66,93,95] to produce various quantitative results in symplectic topology. Set barcodes ′ for the quotient space of barcodes with respect to the isometric R-action by shifts.…”

Small energy isotopies of loose Legendrian submanifolds

“…Can we give an upper bound for d(L 0 , L 1 ) depending on L 0 and L 1 ? As a first step, it was proven in [DRS20] that the displacement energy of the standard Legendrian 2-sphere in R 5 can be made arbitrarily small by adding a stabilization contained in a sufficiently small neighbourhood of a point x ∈ L. The authors of [DRS20] conjectured that the same should hold for any closed Legendrian in a contact manifold. We will use their techniques to prove this conjecture, and, in fact, give an explicit bound of the displacement energy in terms of the size of the stabilization if dim L ≥ 2 (Corollary 1.11).…”