Can anybody show me an example which prove that:
A pushout of a homotopy equivalence along a arbitrary map (in Top) doesn't have to be a homotopy equivalence.
I know that if we change "arbitrary map" to cofibration then the pushout have to be a homotopy equivalence, but I try to find an example which show that assumption of cofibration is necessary.
Maybe this is a counterexample: Let $A=S^1\cup[1,2]\times\{0\}$ as a subspace of $\Bbb R^2$. For $f:A\to S^1$ take the map $$ f(a)=\begin{cases} a, &\text{if }a\in S^1\\ (1,0), &\text{if }a\in[1,2] \end{cases} $$ and for $g:A\to S^1$ take $$ g(a)=\begin{cases} a, &\text{if }a\in S^1\\ e^{2\pi i(t-1)}, &\text{if }a\in[1,2] \end{cases} $$ Now $f$ is a homotopy equivalence. But the induced map $S^1\to Y$ is a constant map since the pushout $Y=(S^1\sqcup S^1)/(f(a)\sim g(a))$ is a singleton.