Let $X$ be a topological space such that its suspension is a topological manifold. Can we prove that $X$ itself is a topological manifold?
2 Answers
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.
As Jeff Strom said, the answer is no. For references to CannonEdwards theorem, see https://mathoverflow.net/a/316175/121665.

$\begingroup$ Another related MO question is Is a map a homotopy equivalence if its supension is so?. $\endgroup$ Jun 4, 2021 at 19:44