Manifolds cross a circle (Ex)

Suppose that $M$$<wikitex>; Suppose that $M$ and $N$ are closed smooth simply-connected $n$-manifolds, $n \geq 5$, and that there is a diffeomorphism $$ f \colon M \times S^1 \cong N \times S^1.$$ Show that $M$ and $N$ are diffeomorphic. If $M$ and $N$ are not simply-connected, what condition would you place on the fundamental group to obtain a similar theoroem? </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]M$ and $N$$N$ are closed smooth simply-connected $n$$n$-manifolds, $n \geq 5$$n \geq 5$, and that there is a diffeomorphism

$\displaystyle f \colon M \times S^1 \cong N \times S^1.$ $$\displaystyle f \colon M \times S^1 \cong N \times S^1.$$

Show that $M$$M$ and $N$$N$ are diffeomorphic.

If $M$$M$ and $N$$N$ are not simply-connected, what condition would you place on the fundamental group to obtain a similar theoroem?