Comparison of spectral invariants in Lagrangian and Hamiltonian Floer theory

Abstract

We compare spectral invariants in periodic orbits and Lagrangian Floer homology case, for a closed symplectic manifold P and its closed Lagrangian submanifolds L, when ω| π 2 (P,L) = 0, and µ| π 2 (P,L) = 0. We define a product HF * (H) ⊗ HF * (H : L) → HF * (H : L) and prove subadditivity of invariants with respect to this product.

A brief survey of the spectral numbers in floer homology

“…where ♯ denotes the concatenation of Hamiltonians with respect to the time variable. Similar comparison formulae for spectral numbers with periodic and Lagrangian boundary conditions were proved in [61] in cotangent bundles, in [25] for weakly exact Lagrangian submanifolds and in [55] in a more general case. A genuine section : → * , such that ( ) ⊂ can be understood as a section that selects a branch of understood as an image of a multi-valued section ( | ) −1 .…”

“…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].…”

Spectral invariants for monotone Lagrangians

“…An early version of this result was establised in [MVZ12] for the zero section of a cotangent bundle, in case a = [M ] and α = [L]. An inequality identical to ours in the setting of weakly exact Lagrangians was proved in [DKM14]. Triangle inequality for Floer-homological spectral invariants was first proved by Oh [Oh99], although the inequality proved there included an error term.…”

Piunikhin-Salamon-Schwarz isomorphisms and spectral invariants for conormal bundle

“…Another advantage of using our isomorphism is its naturality. Using Poźniak's type isomorphism it is not obvious whether this diagram (6) HF…”