Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes

TL;DR

This paper establishes a relationship between topological complexity and LS-category for two-cell complexes, using Hopf invariant techniques to prove their equality in certain stable and metastable ranges.

Contribution

It demonstrates that TC of two-cell complexes equals the LS-category of their cofiber in specific stable and metastable ranges, extending previous understanding with new Hopf invariant methods.

Findings

TC(X) equals cat(C_X) in the almost stable range q ≤ 2p - 1

Equality holds in the metastable range 2p - 1 < q ≤ 3(p - 1) under mild conditions

Hopf invariant techniques are used to prove these equalities

Abstract

Let $X$ be a two-cell complex with attaching map $\alpha\colon S^q\to S^p$, and let $C_X$ be the cofiber of the diagonal inclusion $X\to X\times X$. It is shown that the topological complexity (${\rm TC}$) of $X$ agrees with the Lusternik-Schnirelmann category (${\rm cat}$) of $C_X$ in the (almost stable) range $q\leq2p-1$. In addition, the equality ${\rm TC}(X)={\rm cat}(C_X)$ is proved in the (strict) metastable range $2p-1<q\leq3(p-1)$ under fairly mild conditions by making use of the Hopf invariant techniques recently developed by the authors in their study of the sectional category of arbitrary maps.