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The persistence of a relative Rabinowitz–Floer complex

Georgios Dimitroglou Rizell and Michael G Sullivan
Geometry & Topology 28 (2024) 2145–2206
DOI: 10.2140/gt.2024.28.2145
Abstract

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.

Keywords
Legendrian submanifolds, Rabinowitz Floer homology
Mathematical Subject Classification
Primary: 53D10, 53D42
References
Publication
Received: 4 February 2022
Revised: 28 December 2022
Accepted: 28 January 2023
Published: 24 August 2024
Proposed: Leonid Polterovich
Seconded: Yakov Eliashberg, András I Stipsicz
Authors
Georgios Dimitroglou Rizell
Department of Mathematics
Uppsala University
Uppsala
Sweden
Michael G Sullivan
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA
United States
© 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY).

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