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 $C_{\Delta_X},$ 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.