A criterion for a Hurewicz cofibration to be a Quillen cofibration

Abstract

In this paper, we prove that $h$-cofibrations between $q$-cofibrant spaces are $q$-cofibrations. We also present a number of applications, including a pushout-product property for symmetrizable cofibrations, a local-to-global gluing lemma for $q$-cofibrations, a proof that $q$-fibrations between $q$-cofibrant spaces are $h$-fibrations, and an alternative proof of Waner's theorem on $G$-spaces with the $G$-homotopy type of a $G$-CW complex in fibre sequences. Moreover, all of the above generalises readily to the equivariant context, and so we work in the more general equivariant setting throughout.