1

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.

GrasshopperAndFlies
  • 33
  • 1
    This is false at least in disconnected case. Take $X_1=X_2=Y_1=Y_2=S^1\sqcup [0,1]$. Then glueing along points in $S^1$ and glueing along points in $[0,1]$ will result in non-homotopic spaces. I'm not sure about path connected case though. – freakish Apr 13 '22 at 20:10
  • @freakish Oops. You're right. I forgot to mention that $X_\alpha$ and $Y_\alpha$ are assumed to be path connected. I'll make an edit to the question. – GrasshopperAndFlies Apr 13 '22 at 20:47
  • It's true if each space is well-pointed, which is the case for CW complexes. – Cheerful Parsnip Apr 14 '22 at 00:39

1 Answers1

2

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 $* \vee * \approx *$.

Paul Frost
  • 76,394
  • 12
  • 43
  • 125