# Suspension of a topological space

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

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.