Spectral numbers in Floer theories

By Michael Usher. Compositio Mathematica, 2007

79 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/a8abf1846ce9c6c7d77eb4d8d45f7af78dee6e84

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In Geometry and Topology

1. Michael Usher
2. E. Shelukhin
3. L. Polterovich
4. Basak Z. Gurel
5. V. Ginzburg
6. A. Kislev
7. Yong-Geun Oh
8. Michael Entov

In Mathematical Physics

1. Michael Usher
2. E. Shelukhin
3. L. Polterovich
4. A. Kislev
5. V. Ginzburg
6. Yong-Geun Oh
7. Michael Entov
8. Basak Z. Gurel

In Neurology

1. E. Shelukhin
2. A. Kislev
3. L. Polterovich
4. Michael Entov
5. Basak Z. Gurel
6. V. Ginzburg
7. Yong-Geun Oh
8. K. Fukaya

Citations in the corpus

• Bounds on spectral norms and barcodes (2018, methodology): “invariants. Spectral invariants were introduced into symplectic topology by Viterbo [118] in the context of generating functions, and by Schwarz [97] and Oh [84] in Hamiltonian Floer theory (see also [110]). Here we use the version of spectral invariants for Lagrangian Floer homology, in the monoton”
• Persistence modules with operators in Morse and Floer theory (2017, background): “Spectral invariant can be defined in complete generality (without assumptions on π2(M)), and have been used extensively in symplectic topology in the last few decades starting with the foundational papers [42,51,57] (see [40,53] for recent developments and numerous further references), before persis”
• Action and index spectra and periodic orbits in Hamiltonian dynamics (2008, background): “z H / over a certain class of perturbations z H ; cf [54; 55].”
• Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms (2010, background): “H /;AH ; !; @ ;H / comprise the structure of a “filtered Floer–Novikov complex” in the sense of Usher [60; 61], with Floer chain complex equal to CF.”
• Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds (2009, background): “3 of [41], which in fact applies in a rather general algebraic setting that includes Hamiltonian Floer homology as a special case.”
• Lusternik–Schnirelmann theory and closed Reeb orbits (2016, background): “V. For instance, when f= 1, it would be sufficient to require V to be weakly monotone (see [HS]) and rational. The latter condition is needed to ensure that spectral invariants take values in S(H); see [Us]. Now, however, the Floer homology becomes a module over a Novikov ring and periodic orbits must”
• Persistent homology and Floer-Novikov theory (2015, background): “{−∞} defined by, for α∈ Hk(C∗), ρ(α) = inf{ℓ(c)|c∈ Ck,[c] = α} PERSISTENT HOMOLOGY AND FLOER-NOVIKOV THEORY 35 (where [c] denotes the homology class of c). In a more general context the main result of [U08] shows that the infimum in the definition of ρ(α) is always attained. The spectral invariants are”
• Symplectic quasi-states and semi-simplicity of quantum homology (2007, result): “poses into a direct sum of fields: QH2n(M) = Q1 ⊕ ...⊕ Qd. Assume the idempotent aappearing in the definitions of ζand µ(see equations (7), (8)) is the unit 5According to the recent preprint of M.Usher [43] and the previous result of Y.-G.Oh [33] on the ”non-degenerate spectrality” property of spectra”
• Lagrangian Floer theory on compact toric manifolds II: bulk deformations (2008, methodology): “onalbecause the theory of spectral invariantwas developed in [Oh3] under these conditions. The rationality assumption has been removed in [Oh4] and the semi-positivity assumption of ωremoved by Usher [Us]. Thus the spectral invariant satisfying all the properties listed in [EP1] section 5 is now est”