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Let $X$ be a topological space such that its suspension is a topological manifold. Can we prove that $X$ itself is a topological manifold?

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It’s not true. The Poincare sphere $P$ is a manifold, and its suspension is not. But its double suspension is homeomorphic to $S^5$ by Cannon’s “Double Suspension Theorem”. I learned about this from Mark Grant in an answer to a different question of mine on MO.

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As Jeff Strom said, the answer is no. For references to Cannon-Edwards theorem, see https://mathoverflow.net/a/316175/121665.

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