Spectral numbers in Floer theories

TL;DR

This paper proves that spectral numbers in Floer theories are always finite and attained, extending known properties from Hamiltonian Floer homology to a broader algebraic setting.

Contribution

It establishes the finiteness and attainability of spectral numbers in general Floer-type theories using algebraic methods, applicable beyond Hamiltonian Floer homology.

Findings

Spectral numbers are finite for all nonzero Floer homology classes.

The infimum defining spectral numbers is always achieved.

Results apply to various Floer-type theories including Novikov homology.

Abstract

The chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz, and others. We prove that the spectral number associated to any nonzero Floer homology class is always finite, and that the infimum in the definition of the spectral number is always attained. In the Hamiltonian case, this implies that what is known as the ``nondegenerate spectrality'' axiom holds on all closed symplectic manifolds. Our proofs are entirely algebraic and apply to any Floer-type theory (including Novikov homology) satisfying certain standard formal properties. The key ingredient is a theorem about the existence of best approximations of arbitrary elements of finitely generated free modules over Novikov rings by elements of prescribed submodules with respect to a certain family of non-Archimedean metrics.