On topological complexity and LS-category

TL;DR

This paper investigates the relationship between topological complexity and LS-category, providing bounds, counterexamples, and applications to classical theorems in topology.

Contribution

It offers new bounds for topological complexity, presents a counterexample to a conjecture, and explores the connection between topological complexity and LS-category.

Findings

Bounds for $TC(X\vee Y)$ established

Counterexample to $TC(X')\le TC(X)$ conjecture provided

Short proof of Arnold-Kuiper theorem using LS-category

Abstract

We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity $TC(X)$ and monoidal topological complexity $TC^M(X)$. Using these results we provide lower and upper bounds for the topological complexity of the wedge $X\vee Y$. We use these bounds to give a counterexample to the conjecture asserting that $TC(X')\le TC(X)$ for any covering map $p:X'\to X$. We discuss a possible reduction of the monoidal topological complexity to the LS-category. Also we apply the LS-category to give a short proof of the Arnold-Kuiper theorem.