The diagonal fibration, $H$-spaces, and duality

Abstract:

For any pointed topological space X, there is the fibration $\Omega X \to {X^I} \to X \times X$ whose projection sends the map $f:I \to X$ to $(f(0),f(1))$. We show that if X has the based homotopy type of a CW complex, then the above fibration is equivalent to one induced from the path space fibration $\Omega X \to PX \to X$ if and only if X admits an H-space multiplication. Dually, we obtain a characterization of simply-connected co-H-spaces.