Fundamental group in stable Morse theory

TL;DR

This paper provides a dynamical description of the fundamental group within stable Morse theory, bridging the gap between classical Morse and Floer homology frameworks in symplectic geometry.

Contribution

It introduces a novel dynamical approach to understanding the fundamental group in the stable Morse setting, an intermediate case between Morse and Floer theories.

Findings

Dynamical description of the fundamental group in stable Morse theory

Bridges between Morse and Floer homology frameworks

Enhanced understanding of topological invariants in infinite-dimensional settings

Abstract

Morse theory relates algebraic topology invariants and the dynamics of the gradient flow of a Morse function, allowing to derive information about one out of the other. In the case of the homology, the construction extends to much more general settings, and in particular to the infinite dimensional setting of the celebrated Floer homology in symplectic geometry. The case of the fundamental group is quiet different however, and the object of this paper is to provide a dynamical description of the fundamental group in the stable Morse setting, which can be thought of as an intermediate case between the Morse and the Floer settings.