If we have a weak homotopy equivalence $f:X \rightarrow Y$, does there necessarily exist a weak homotopy equivalence $g:Y \rightarrow X$?

Question

Suppose $X$ and $Y$ are topological spaces. Is it true that, if we have a weak homotopy equivalence $f:X \rightarrow Y$, then there exists a weak homotopy equivalence $g:Y \rightarrow X$?

I can't quite find an answer to this online. I suspect that it is true, since I think that weak homotopy equivalence should be an equivalence relation.

Thanks!

Answer 1

No, an example is the pseudocircle. See MSE/3308705.

So indeed, the existence of a weak homotopy equivalence is not an equivalence relation but instead one says that two spaces $X,Y$ are weakly equivalent if they are isomorphic in the category $\mathbf{Top}[W^{-1}]$ where $\mathbf{Top}$ is the category of (nice) spaces and $W$ is the class of weak homotopy equivalence. In other words, there exists a zig-zag of weak homotopy equivalences between $X$ and $Y$.

Answer 2

The answer is no; see Example 3.4.27 in this thesis for a counterexample.