Topological complexity and the homotopy cofibre of the diagonal map

In this paper we analyze some relationships between the topological complexity of a space $X$ and the category of $C_{{\mathrm{\Delta }}_X},$ the homotopy cofibre of the diagonal map ${\mathrm{\Delta }}_X:X\to X\times X\mathrm{.}$ 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.