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

TL;DR

This paper constructs a natural isomorphism between Morse and Floer homologies for conormal bundles, establishing spectral invariants and a product structure with triangle inequality, advancing symplectic topology understanding.

Contribution

It introduces a Piunikhin--Salamon--Schwarz isomorphism for conormal bundles and proves its naturality and compatibility with spectral invariants.

Findings

Constructed a natural isomorphism between Morse and Floer homologies for conormal bundles.

Defined a product on Floer homology and established triangle inequality for spectral invariants.

Proved the isomorphism commutes with changes in Morse functions and Hamiltonians.

Abstract

We give a construction of Piunikhin--Salamon--Schwarz isomorphism between the Morse homology and the Floer homology generated by Hamiltonian orbits starting at the zero section and ending at the conormal bundle. We also prove that this isomorphism is natural in the sense that it commutes with the isomorphisms between the Morse homology for different choices of the Morse function and the Floer homology for different choices of the Hamiltonian. We define a product on the Floer homology and prove triangle inequality for conormal spectral invariants with respect to this product.