Extensions of maps as fibrations and cofibrations

Abstract:

Suppose $f:X \to Y$ is a map of 1-connected spaces. In the “stable” range, roughly where the connectivity of Y exceeds the homology, or homotopy, dimension of X, it is well known that f can be extended as a cofibration $C \to X \to Y$, or respectively a fibration $X \to Y \to B$. A criterion is given for the existence of such extensions in a less restrictive “metastable” range. A main result is that if f is at least 2-connected and 2 con $Y \geq \dim Y - 1,\dim X$, then f extends as a cofibration if and only if the map $(1 \times f)\Delta :X \to (X \times Y)/X$ factors through f.