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Persistent homology and
Floer–Novikov theory
Michael Usher and Jun Zhang |
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Geometry & Topology 20 (2016) 3333–3430
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Abstract |
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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 –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. |
Keywords
persistence module, barcode, Floer homology, Novikov ring,
nonarchimedean singular value decomposition
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Mathematical Subject Classification 2010
Primary: 53D40
Secondary: 55U15
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References |
Publication
Received: 9 April 2015
Revised: 9 December 2015
Accepted: 3 January 2016
Published: 21 December 2016
Proposed: Leonid Polterovich
Seconded: Gang Tian, Yasha Eliashberg
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Authors |
| Michael Usher | |
| Department of Mathematics University of Georgia Athens, GA 30602 United States |
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| http://alpha.math.uga.edu/~usher/ | |
| Jun Zhang | |
| School of Mathematical
Sciences Tel Aviv University Ramat Aviv Tel Aviv 69978 Israel |
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| http://junzhangsite.wordpress.com/ | |
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