If $S\times\mathbb{R}$ is homeomorphic to $T\times\mathbb{R}$, and $S$ and $T$ are compact, can we conclude that $S$ and $T$ are homeomorphic?

If $S \times \mathbb{R}$ is homeomorphic to $T \times \mathbb{R}$ and $S$ and $T$ are compact, connected manifolds (according to an earlier question if one of them is compact the other one needs to be compact) can we conclude that $S$ and $T$ are homeomorphic?

I know this is not true for non compact manifolds.

I am mainly interested in the case where $S, T$ are 3-manifolds.

Solution 1:

For closed 3-manifolds, taking the product with $\mathbb{R}$ doesn't change the fundamental group, so if the two products are homemorphic, the original spaces have the same fundamental group, and closed 3-manifolds are uniquely determined by their fundamental group, if they are irreducible and non-spherical.