# Homotopy equivalence of certain kinds of adjunction spaces

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.

Thanks!

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Also posted on MO: mathoverflow.net/questions/94830 – 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