Conley pairs in geometry - Lusternik-Schnirelmann theory and more

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.