Persistent homology and FloerNovikov theory
Abstract
We construct "barcodes" for the chain complexes over Novikov rings that arise in Novikov's Morse theory for closed oneforms and in Floer theory on notnecessarilymonotone 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 Floertheoretic invariants such as boundary depth and torsion exponents, and also reflect information about spectral invariants. We moreover 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 C^0robustness 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 arXiv:
 arXiv:1502.07928
 Bibcode:
 2015arXiv150207928U
 Keywords:

 Mathematics  Symplectic Geometry;
 Mathematics  Algebraic Topology;
 53D40;
 55U15
 EPrint:
 71 pages. v2: several typos have been fixed. v3: the start of the proof of the stability theorem has been simplified, Appendix A has been moved to the body of the paper, and some clarifying remarks have been added