"Dividing out" a cartesian product

Question

The specific case I'm wondering about, is the case where $M,N$ are smooth manifolds, and we have $$M\times\mathbb{R}\cong N\times\mathbb{R}$$meaning that they are diffeomorphic. Does this imply that $M\cong N$? If yes, at what point does this break? What if we replaced $\mathbb{R}$ with another arbitrary smooth manifold? Does this kind of relation hold in other contexts as well? Like if $M,N$ were groups or vector spaces and instead of diffeomorphism we had isomorphism?

Answer 1

This is false let M be a 2 sphere minus 3 points and N be a torus minus 1 point. Then MxR and NxR are diffeomorphic. The both are regular neighborhoods of a wedge of 2 circles in the euclidian space (a 8 drawn in a plane)