Abstract interlevel persistence for Morse-Novikov and Floer theory

Abstract

We develop a general algebraic framework involving "Poincaré-Novikov structures" and "filtered matched pairs" to provide an abstract approach to the barcodes associated to the homologies of interlevel sets ofor S 1 -valued Morse functions, which can then be applied to Floer-theoretic situations where no readily apparent analogue of an interlevel set is available. The resulting barcodes satisfy abstract versions of stability and duality theorems, and in the case of Morse or Novikov theory they coincide with the standard barcodes coming from interlevel persistence. In the case of Hamiltonian Floer theory, the lengths of the bars yield multiple quantities that are reminiscent of the spectral norm of a Hamiltonian diffeomorphism. CONTENTSMICHAEL USHER 8.3. Building blocks 79 8.4. Direct sum decompositions and barcodes 84 8.5. Chain-level Poincaré-Novikov structures 92 9. The chain-level Poincaré-Novikov structures of Morse and Novikov theory 94 9.1. The Morse complex, continuation maps, and Poincaré-Novikov structures with Γ = {0} 94 9.2. Orientations 99 9.3. Lifted Morse complexes on covering spaces 103 9.4. Novikov complexes 106 9.5. The associated chain-level filtered matched pairs 108 9.6. The isomorphism with interlevel persistence 114 References 121