Conley pairs in geometry - Lusternik-Schnirelmann theory and more

TL;DR

This paper explores the use of Conley pairs in geometric topology, providing new proofs and insights into Lusternik-Schnirelmann theory, and demonstrating their utility in Morse theory and CW decompositions.

Contribution

It introduces novel applications of Conley pairs in organizing unstable manifolds and offers an exposition of LS theory based on Conley pairs, including new proofs and concepts.

Findings

Conley pairs facilitate Morse filtration constructions.

An alternative proof of the cell attachment theorem in Morse theory.

A new exposition of LS theory using Conley pairs.

Abstract

Firstly, we wish to motivate that Conley pairs, realized via Salamon's definition [17], are rather useful building blocks in geometry: Initially we met Conley pairs in an attempt to construct Morse filtrations of free loop spaces [21]. From this fell off quite naturally, firstly, an alternative proof [20] of the cell attachment theorem in Morse theory [13] and, secondly, some ideas [12] how to try to organize the closures of the unstable manifolds of a Morse-Smale gradient flow as a CW decomposition of the underlying manifold. Relaxing non-degeneracy of critical points to isolatedness we use these Conley pairs to implement the gradient flow proof of the Lusternik-Schnirelmann Theorem [10] proposed in Bott's survey [3]. Secondly, we shall use this opportunity to provide an exposition of Lusternik-Schnirelmann (LS) theory based on thickenings of unstable manifolds via Conley pairs. We shall cover the Lusternik-Schnirelmann Theorem [10], cuplength, subordination, the LS refined minimax principle, and a variant of the LS category called ambient category.