While trying to solve a problem, I came up with the following claim:
- $\textbf{Claim.}$ If $X_\alpha$ is homotopy equivalent to $Y_\alpha$ for each index $\alpha$, then the wedge sums $\bigvee_\alpha X_\alpha$ are $\bigvee_\alpha Y_\alpha$ are homotopy equivalent.
If we assume further that the homotopy equivalences $X_\alpha \simeq Y_\alpha$ are appropriately pointed, then I can work out a proof for the claim. However, I could not come up with anything for the general case.
I do suspect that the general claim is false, since there is seemingly no way to guarantee that the obvious 'stitched' map is a homotopy equivalence.
So I guess my question is, is the claim actually true, or is there a counterexample?
EDIT: I forgot to mention that $X_\alpha$ and $Y_\alpha$ are assumed to be path connected. Without this assumption the claim is false as mentioned by freakish in the comments.
Best Answer
It is false. In Fundamental group of wedge sum of two coni over Hawaiian earrings you can find a contractible space $X$ such that $X \vee X$ has a nontrivial fundamental group. We have $X \simeq *$ = one-point space and $* \vee * \approx *$.