The persistence of a relative Rabinowitz–Floer complex

Dimitroglou Rizell, Georgios¹ ; Sullivan, Michael G²

¹ Department of Mathematics, Uppsala University, Uppsala, Sweden
² Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, United States

Geometry & topology, Tome 28 (2024) no. 5, pp. 2145-2206.

Résumé

We give a quantitative refinement of the invariance of the Legendrian contact homology algebra in general contact manifolds. We show that in this general case, the Lagrangian cobordism trace of a Legendrian isotopy defines a DGA stable tame isomorphism, which is similar to a bifurcation invariance proof for a contactization contact manifold. We use this result to construct a relative version of the Rabinowitz–Floer complex defined for Legendrians that also satisfies a quantitative invariance, and study its persistent homology barcodes. We apply these barcodes to prove several results, including: displacement energy bounds for Legendrian submanifolds in terms of the oscillatory norms of the contact Hamiltonians; a proof of Rosen and Zhang’s nondegeneracy conjecture for the Shelukhin–Chekanov–Hofer metric on Legendrian submanifolds; and the nondisplaceability of the standard Legendrian real-projective space inside the contact real-projective space.

Détail

DOI : 10.2140/gt.2024.28.2145

Keywords: Legendrian submanifolds, Rabinowitz Floer homology

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