Spectral invariants in Lagrangian Floer homology of open subset

Abstract

We define and investigate spectral invariants for Floer homology HF (H, U : M ) of an open subset U ⊂ M in T * M , defined by Kasturirangan and Oh as a direct limit of Floer homologies of approximations. We define a module structure product on HF (H, U : M ) and prove the triangle inequality for invariants with respect to this product. We also prove the continuity of these invariants and compare them with spectral invariants for periodic orbits case in T * M .

A note on partial quasi-morphisms and products in Lagrangian Floer homology in cotangent bundles

“…In the proof of the previous theorems, we have shown that it is possible to construct quasi-morphisms using Lagrangian spectral numbers even in the case when we do not have a product in homology that ensures that they are subadditive. With the help of constructions in [8,9,10], we can apply these conclusions to the case when N is a manifold with a boundary.…”

“…It turns out that the upper limit of this sequence exists and it satisfies well known properties of partial quasimorphism. More precisely, if we define a map (9) σ…”

“…The two choices above give rise to two Lagrangian Floer homologies, isomorphic to singular homologies H * (N ) and H * (N, ∂N ). For a homology class α ∈ H * (N ) (or a relative homology class α ∈ H * (N, ∂N ), there are well defined spectral numbers N + (α, •) defined as a minimax values in corresponding Lagrangian Floer homology (see [9,10] for details).…”

A brief survey of the spectral numbers in floer homology

“…In [45,46] the previous construction is generalized to the case where is a submanifold with boundary in . The assumption that is closed can also be somewhat weakened [57].…”

“…For periodic boundary conditions on closed symplectically aspherical symplectic manifolds, this construction is due to Schwarz [84], and for Lagrangian boundary conditions on a pair (zero-section, conormal bundle of a closed submanifold) in cotangent bundles, the construction is due to Oh [63,64]. There are several generalizations of these constructions, such as [25,45,46,54,55,57,61,63,64].…”