Morse cohomology in a Hilbert space via the Conley index

Maciej Starostka

doi:10.1007/s11784-015-0216-5

Abstract

The main theorem of this paper states thatMorse cohomology groups in a Hilbert space are isomorphic to the cohomological Conley index. It is also shown that calculating the cohomological Conley index does not require finite-dimensional approximations of the vector field. Further directions are discussed.

Highlights

The aim of this paper is to show that Morse cohomology groups defined for a certain functional in a Hilbert space can be recovered via the Conley index
This was motivated by the growing number of different Floer cohomology theories in three- and four-dimensional topology
Results presented below are obtained by the facts that those cohomology groups satisfy axioms of the generalized cohomology theory and that they are invariant under the flow deformations

Summary

Introduction

The aim of this paper is to show that Morse cohomology groups defined for a certain functional in a Hilbert space can be recovered via the Conley index. This was motivated by the growing number of different Floer cohomology theories in three- and four-dimensional topology. Some of the Floer theories are still conjectured to be equivalent, e.g. Seiberg–Witten–Floer (HSW) cohomology and Monopole–Floer cohomology (HM) The former is defined by the Conley index while the latter one by counting connecting orbits. Results presented below are obtained by the facts that those cohomology groups satisfy axioms of the generalized cohomology theory (see below for a precise statement) and that they are invariant under the flow deformations.

E-cohomology

Conley index

Main theorem

Further directions