Question

Consider the following interesting theorem:(7.5.7, p.294 in Topology and Groupoids by Ronald Brown)

Gluing theorem for adjunction spaces: Suppose that we have the following commutative diagram of topological spaces and continuous maps:

where $\varphi_{A}$, $\varphi_{X}$ and $\varphi_{Y}$ are homotopy equivalences, and the inclusions $i$ and $i'$ are closed cofibrations. Then the map

$$ \varphi:X\cup_{f}Y\rightarrow X'\cup_{f\phantom{l}'}Y' $$

induced by $\varphi_{A}$, $\varphi_{X}$ and $\varphi_{Y}$ is a homotopy equivalence.

In this post I asked a question that would become a corollary if the gluing theorem were true for arbitrary cofibrations $i$ and $i'$. I would like to know if this is the case. Otherwise, does anybody know of a counterexample?

Perhaps, it is possible that although the map $\varphi$ may not in general be a homotopy equivalence when the cofibrations $i$ and $i'$ are arbitrary, the adjunction spaces $Y\cup_{f}X$ and $Y'\cup_{f'}X'$ will nonetheless be homotopy equivalent. A possible approach to this would be to try to find a third space containing both as deformation retracts.

Note on notation: That an inclusion map $j:B\rightarrow Z$ is a closed cofibration means that it is a cofibration (i.e., the pair $(B,Z)$ has the homotopy extension property) and the set $B$ is closed in $X'$.

Thank you

Answer 1

Yes, this is true even for not necessarily closed cofibrations. If you want a single source that gives a complete proof, then the only one that comes to my mind is this preprint.

Definition 1.1.1 introduces cofibration categories and then Lemma 1.4.1 says that the desired result holds in any cofibration category. Section 3.1 contains a detailed proof that the category of topological spaces equipped with Hurewicz cofibrations and homotopy equivalences is a cofibration category and thus the lemma applies. The crux of the matter is that acyclic cofibrations are closed under pushouts and this follows from a classical result of Dold (Lemma 3.1.9) that acyclic cofibrations admit deformation retractions, which doesn't depend on closedness.