I have a topological manifold whose suspension is homeomorphic to the sphere $S^{k+1}$. Is it necessarily itself homeomorphic to $S^k$?


I know that this is not true if I replace "suspension" with "double suspension", because I found the helpfully named <http://en.wikipedia.org/wiki/Double_suspension_theorem>.



If a manifold suspends to a sphere...
Suppose $M$ is a closed $n$-manifold whose suspension is homeomorphic to $S^{n+1}$.
Removing the two "singular" points from the suspension gives $M\times \mathbb R$, while
removing two points from $S^{n+1}$ gives $S^n\times\mathbb R$. Thus $M\times \mathbb R$ and $S^n\times\mathbb R$ are homeomorphic, which easily implies that
$M$ and $S^n$ are h-cobordant, and hence $M$ and $S^n$ are homeomorphic. 


