Let $X,Y,A,B$ be CW-complexes such that $A\subseteq X$ and $B\subseteq Y$ are subcomplexes. Assume that $(X,A)$ and $(Y,B)$ are good pairs if necessary. Suppose there is a relative homeomorphism $f:(X,A)\to (Y,B)$, meaning that $f$ is a map of pairs such that the induced map $X/A\to Y/B$ is a homeomorphism.
If we don't know $Y$, but we know $X,A,B$ and how $f$ is defined on $A$, can we recover $Y$ (at least up to homotopy) from the fact that $f$ is a relative homeomorphism?
Maybe a more precise way to state the question is: if we have the complexes $X,A,B$ with $A\subseteq X$ a subcomplex and a map $f|_A:A\to B$, can we find a complex $Y$ containing $B$ as a subcomplx such that there is a relative homeomorphism $f:(X,A)\to (Y,B)$ extending $f|_A$?
Since we know that $Y/B \cong X/A$, my guess is that we can take $X/A$, and in the 0-cell corresponding to the class $[A]$ we would attach $B$ in a way that is consistent with $f|_A$, so that $f$ outside $A$ is just the identity.
I'm not sure if this works or if we need further assumptions on $f|_A$. The source of this problem is a construction made by Stasheff in this paper (Construction 8, page 279-280), where he builds spaces in an inductive way via relative homeomorphisms of the form
$$(X_i,A_i)\xrightarrow{\alpha_i}(E_i,E_{i-1})$$
that are defined only on $A_i$. He later state who $E_1$ is and describes from this how $\alpha_2$ identifies $A_2$ with $E_{1}$, but never really describes $E_2$. There are some theorems specifying the homotopy type of these spaces, but the proof seem very implicit to me and I can't really understand what is going on.