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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2 Answers
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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.
answered Jun 3, 2021 at 0:57
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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.
answered Jun 3, 2021 at 4:52
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$\begingroup$ Another related MO question is Is a map a homotopy equivalence if its supension is so?. $\endgroup$ Commented Jun 4, 2021 at 19:44