Geometry & Topology

Persistent homology and Floer–Novikov theory

Michael Usher and Jun Zhang

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Abstract

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.

Article information

Source
Geom. Topol., Volume 20, Number 6 (2016), 3333-3430.

Dates
Received: 9 April 2015
Revised: 9 December 2015
Accepted: 3 January 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859087

Digital Object Identifier
doi:10.2140/gt.2016.20.3333

Mathematical Reviews number (MathSciNet)
MR3590354

Zentralblatt MATH identifier
1359.53070

Subjects
Primary: 53D40: Floer homology and cohomology, symplectic aspects
Secondary: 55U15: Chain complexes

Keywords
persistence module barcode Floer homology Novikov ring nonarchimedean singular value decomposition

Citation

Usher, Michael; Zhang, Jun. Persistent homology and Floer–Novikov theory. Geom. Topol. 20 (2016), no. 6, 3333--3430. doi:10.2140/gt.2016.20.3333. https://projecteuclid.org/euclid.gt/1510859087


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