# General gluing theorem for adjunction spaces

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

• The result you seek is given as propositions 5.3.2 and 5.3.3 of tom Dieck's book "Algebraic topology". – Ricardo Andrade May 6 '12 at 6:09

• I add that an advantage of the proof given in "Topology and Groupoids" is that it gives control over the homotopies involved. The result itself evolved from generalising the standard fact that a homotopy equivalence $Y \to Z$ of spaces induces an isomorphism of homotopy groups: now replace the pair $(S^n,a)$ by a pair $(X,A)$ and go through the same argument, to get a useful result on maps of pairs. – Ronnie Brown Oct 8 '12 at 15:51