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I'm little bit lost with the following question:

I have four connected closed orientable manifolds $M,N,S,S'$ such that $S$ and $S'$ are homotopy equivalent and $M\times S$ is homotopy equivalent to $N\times S'$.

Under these conditions, it seems to me that $M$ has to be homotopy equivalent to $N$. Am I wrong ?

EDIT: Skupers has answered my question by giving a counterexample, thanks!

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    $\begingroup$ I certainly can't imagine what your argument would be; I doubt that the result is true. I would try to take M, N inequivalent lens spaces and see if I can find an example with S = S' in the literature. $\endgroup$
    – mme
    Commented Jul 31, 2021 at 19:10
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    $\begingroup$ Theorem 2 of sciencedirect.com/science/article/pii/0040938378900149 provides a counterexample with $S = S'$ a 3-sphere and $M$, $N$ lens spaces. $\endgroup$
    – skupers
    Commented Jul 31, 2021 at 19:40
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    $\begingroup$ You can even take $M\times S$ diffeomorphic to $N\times S$ (i.e. $S' = S$), and the conclusion is false. In particular, you need not even have $\pi_1(M) \cong \pi_1(N)$, see here for example. $\endgroup$
    – Michael Albanese
    Commented Jul 31, 2021 at 20:36

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