If a manifold suspends to a sphere... - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:18:37Z http://mathoverflow.net/feeds/question/64029 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64029/if-a-manifold-suspends-to-a-sphere If a manifold suspends to a sphere... James Cranch 2011-05-05T17:58:22Z 2011-05-05T19:06:15Z <p>I have a topological manifold whose suspension is homeomorphic to the sphere $S^{k+1}$. Is it necessarily itself homeomorphic to $S^k$?</p> <p>I know that this is not true if I replace "suspension" with "double suspension", because I found the helpfully named <a href="http://en.wikipedia.org/wiki/Double_suspension_theorem" rel="nofollow">http://en.wikipedia.org/wiki/Double_suspension_theorem</a>.</p> http://mathoverflow.net/questions/64029/if-a-manifold-suspends-to-a-sphere/64036#64036 Answer by Igor Belegradek for If a manifold suspends to a sphere... Igor Belegradek 2011-05-05T19:06:15Z 2011-05-05T19:06:15Z <p>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. </p>