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Persistent homology and Floer–Novikov theory

Michael Usher and Jun Zhang

Geometry & Topology 20 (2016) 3333–3430

We construct “barcodes” for the chain complexes over Novikov rings that arise in Novikov’s Morse theory for closed one-forms and in Floer theory on not-necessarily-monotone symplectic manifolds. In the case of classical Morse theory these coincide with the barcodes familiar from persistent homology. Our barcodes completely characterize the filtered chain homotopy type of the chain complex; in particular they subsume in a natural way previous filtered Floer-theoretic invariants such as boundary depth and torsion exponents, and also reflect information about spectral invariants. Moreover, we prove a continuity result which is a natural analogue both of the classical bottleneck stability theorem in persistent homology and of standard continuity results for spectral invariants, and we use this to prove a C0–robustness result for the fixed points of Hamiltonian diffeomorphisms. Our approach, which is rather different from the standard methods of persistent homology, is based on a nonarchimedean singular value decomposition for the boundary operator of the chain complex.

persistence module, barcode, Floer homology, Novikov ring, nonarchimedean singular value decomposition
Mathematical Subject Classification 2010
Primary: 53D40
Secondary: 55U15
Received: 9 April 2015
Revised: 9 December 2015
Accepted: 3 January 2016
Published: 21 December 2016
Proposed: Leonid Polterovich
Seconded: Gang Tian, Yasha Eliashberg
Michael Usher
Department of Mathematics
University of Georgia
Athens, GA 30602
United States
Jun Zhang
School of Mathematical Sciences
Tel Aviv University
Ramat Aviv
Tel Aviv 69978