Topological complexity and the homotopy cofibre of the diagonal map

TL;DR

This paper explores the relationship between topological complexity and the category of the homotopy cofibre of the diagonal map, establishing equalities for various classes of spaces.

Contribution

It proves the equality of topological complexity and the category of the cofibre for several key classes of spaces, advancing understanding of their homotopical properties.

Findings

Equality holds for spheres, H-spaces, and projective spaces.

The invariants are equal for most lens spaces.

Provides new insights into the relationship between topological complexity and cofibre categories.

Abstract

In this paper we analyze some relationships between the topological complexity of a space $X$ and the category of $C_{\Delta_X},$ the homotopy cofibre of the diagonal map $\Delta_X:X\rightarrow X\times X.$ We establish the equality of the two invariants for several classes of spaces including the spheres, the H-spaces, the real and complex projective spaces and almost all the (standard) lens spaces.