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My question arose when I was working on a research problem. It is as following:

Let $M$ be a homology manifold whose homology groups are the same as a sphere. Assume further that the double suspension of $M$ is homeomorphic to a sphere. Is $M$ homeomorphic to the suspension of a homology sphere?

Edition: Is $M$ homeomorphic to either suspension of a homology sphere or homeomorphic to a homology sphere.

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No. The suspension of a connected space is simply connected, and M might not be.

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  • $\begingroup$ @ Dan Petersen. You are right. I edited my question. $\endgroup$
    – Jayq
    Commented Jun 2, 2017 at 5:36

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