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.

sthwhich make $SM$ a manifold ? In the same time result should be similar to suspension of $SM$ e.g. it should be 1-connected. Is such construction known ? $\endgroup$ – user21230 Oct 1 '18 at 11:35