Spectral invariants for monotone Lagrangians

Abstract

Abstract. Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper [Vit92], they have been defined in various contexts, mainly via Floer homology theories, and then used in a great variety of applications. In this paper we extend their definition to monotone Lagrangians, which is so far the most general case for which a "classical" Floer theory has been developed. Then, we gather and prove the properties satisfied by these invariants, and which are crucial for their applications. Finally, as a demonstration, we apply these new invariants to symplectic rigidity of some specific monotone Lagrangians.

Disjoint superheavy subsets and fragmentation norms

“…We assume that QH * (L; R) = 0. Following [LZ,Section 3], one can define the Lagrangian spectral invariant c (L;R) : Ham(M ) → R associated with the fundamental class [L] ∈ QH * (L; R). Moreover, Leclercq and Zapolsky proved that c (L;R) is a subadditive invariant [LZ,Theorem 41].…”

“…To prove Corollaries 2.4, 2.6, 2.8 and Theorem 2.9, we first prove the following result. (see [Za,Section 7.4], [LZ,Section 2.5.3]). [LZ,Proposition 5] then yields the following inequality as a corollary.…”

Mather theory and symplectic rigidity

“…In particular (52) follows with = and = . Using (51) and (55) as well as the Lipschitz property of Lagrangian spectral invariants [31], we can estimate…”

Persistence Modules with Operators in Morse and Floer Theory

“…The number of the infinite intervals in the barcode is equal to the dimension of the ambient (quantum) homology in a given degree, while the left ends of these intervals correspond to the well know spectral invariants. Spectral invariant can be defined in complete generality (without assumptions on π 2 (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 persistence modules entered the field. In terms of barcodes, in complete generality one can only expect bars with endpoints in R/P(ω), where P(ω) = im ( ω : π 2 (M ) → R) is the period group of ω.…”