Approaches to critical point theory via sequential and parametrized topological complexity

TL;DR

This paper explores how advanced topological invariants like sequential and parametrized topological complexity can provide new lower bounds on the number of critical points of functions, extending classical critical point theory.

Contribution

It establishes new connections between topological complexity variants and critical point theory, including bounds and computations for specific manifolds.

Findings

Lower bounds on critical points using sequential and parametrized TC

Application to unit tangent bundles of spheres

Computation of parametrized TC for specific manifolds

Abstract

The Lusternik-Schnirelmann category of a space was introduced to obtain a lower bound on the number of critical points of a $C^1$-function on a given manifold. Related to Lusternik-Schnirelmann category and motivated by topological robotics, the topological complexity (TC) of a space is a numerical homotopy invariant whose topological properties are an active field of research. The notions of sequential and parametrized topological complexity extend the ideas of topological complexity. While the definition of TC is closely related to Lusternik-Schnirelmann category, the connections of sequential and parametrized TC to critical point theory have not been fully explored yet. In this article we apply methods from Lusternik-Schnirelmann theory to establish various lower bounds on numbers of critical points of functions in terms of sequential and parametrized TCs. We carry out several consequences and applications of these bounds, among them a computation of the parametrized TC of the unit tangent bundles of $(4m-1)$-spheres.