TITLE: Suspension of a topological space
QUESTION [17 upvotes]: Let $X$ be a topological space such that its suspension is a topological manifold. Can we prove that $X$ itself is a topological manifold?

REPLY [14 votes]: As Jeff Strom said, the answer is no. For references to Cannon-Edwards theorem, see https://mathoverflow.net/a/316175/121665.