Persistent homology and Floer-Novikov theory

TL;DR

This paper introduces barcodes for chain complexes in Novikov and Floer theories, providing a unified way to characterize their filtered homotopy types and establish stability and robustness results.

Contribution

It constructs new barcodes for Novikov and Floer chain complexes that generalize persistent homology invariants and proves their stability and robustness properties.

Findings

Barcodes characterize filtered chain homotopy types.

They unify previous Floer-theoretic invariants.

A continuity theorem analogous to bottleneck stability is proved.

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. 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^0-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 non-Archimedean singular value decomposition for the boundary operator of the chain complex.