A criterion for a Hurewicz cofibration to be a Quillen cofibration

TL;DR

The paper proves that $hG$-cofibrations between $qG$-cofibrant spaces are $qG$-cofibrations, and extends this result to a broad equivariant framework using the $\mathcal{C}$-model structure. It develops a pushout-product property for symmetrizable cofibrations, establishes local-to-global gluing for $\mathcal{C}$-cofibrations/fibrations, and shows that a $q$-fibration between $q$-cofibrant spaces is an $h$-fibration. The work further provides an equivariant proof and generalization of Waner’s theorems via an $m\mathcal{C}$-model structure, and culminates with an array of technical appendices on point-set topology to support these results. Collectively, the results yield robust recognition criteria for $q$-cofibrations in the equivariant setting and yield new tools for analyzing diagonal maps, products, and fibre sequences in $G$-spaces.

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.

Theorems & Definitions (72)

• Example 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• proof
• Lemma 2.2