If $X$ and $Y$ are two CW complexes such that $\Sigma X$ and $\Sigma Y$ are homotopy equivalent, are the spaces $X$ and $Y$ homotopy equivalent? I have seen a counterexample when $\pi_1(X)$ is not zero, so I am really looking for counterexamples where $X$ and $Y$ are $1$ connected.
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1Can you include the counter example you mention in your question? Or post it as an answer to your own question? – Joshua Ruiter Feb 18 '17 at 04:56
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@JoshuaRuiter Here is the link to the counterexample I have not looked at it closely but I think he gives a counterexample to this. – happymath Feb 18 '17 at 05:53
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The spaces $\Sigma(S^n\times S^m)$ and $\Sigma\left(S^n\vee S^m\vee S^{m+n}\right) \cong S^{n+1}\vee S^{m+1}\vee S^{m+n+1}$ are homotopy eqiuvalent, while $S^n\times S^m$ and $S^n\vee S^m\vee S^{m+n}$ are not.
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For the uninitiated, this question gives an expression for the suspension of a product. – Michael Albanese Dec 15 '20 at 16:51