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Suppose that $X$, $Y$ and $Z$ are topological spaces, with $A\subset X$, a map $f:A\rightarrow Y$, and a homotopy equivalence $\phi:Y\rightarrow Z$. It seems fair to think that the adjunction spaces $Y\cup_{f}X$ and $Z\cup_{\phi\circ f}X$ will be homotopy equivalent. But how?

A reasonable candidate for a homotopy equivalence seems to arise from the map $(Id,\phi):X+Y\rightarrow X+Z$ after passing to the quotient ($X+Y$ denotes disjoint union, and $(Id,\phi)$ is the map defined to be the identity on $X$ and $\phi$ on Y).

Any suggestions will be appreciated.


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Also posted on MO: – t.b. Apr 22 '12 at 11:01
Thanks to t.b for pointing out that I also posted the problem on Mathoverflow. There I posted a fairly detailed answer. – VCF Apr 25 '12 at 8:56

I have posted a bit more detailed answer also on See also this discussion.

The answer is that there is a general gluing theorem for adjunction spaces which is useful for constructing some basic homotopy equivalences; indeed in algebraic topology one wants to know not only when spaces are not homotopy equivalent, but also when they are so. Some assumptions are needed, namely that certain inclusions are cofibrations. The result is 7.5.7 of my book "Topology and groupoids", and various applications are given there.

The statement of the gluing theorem is best set up as a cubical diagram, which I do not know how to do in this system!

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