Lusternik-Schnirelmann theory to Topological Complexity from $A_{\infty}$-view point

TL;DR

This paper explores the connection between Lusternik-Schnirelmann theory and Topological Complexity, emphasizing the influence of Fadell and Husseini's concepts like category weight and relative category from an $A_{ infty}$-algebra perspective.

Contribution

It introduces a novel viewpoint linking L-S theory and TC through $A_{ infty}$-structures and highlights the role of category weight and relative category in this relationship.

Findings

Establishes a conceptual bridge between L-S theory and TC.

Highlights the impact of Fadell and Husseini's ideas on both theories.

Provides a new $A_{ infty}$-viewpoint for understanding topological invariants.

Abstract

We are trying to look over the Lusternik-Schnirelmann theory (L-S theory, for short) and the Topological Complexity (TC, for short) as a natural extension of the L-S theory. In particular, we focus on the impact of the ideas originated from E. Fadell and S. Husseini on both theories. More precisely, we see how their ideas on a category weight and a relative category drive the L-S theory and the TC.