Cancellation in topological product
Solution 1:
The answer is no.
Let $X$ be the Whitehead manifold which is a contractible three-dimensional manifold. Despite the fact that $X$ is not homeomorphic to $\mathbb{R}^3$, $X\times\mathbb{R}$ is homeomorphic to $\mathbb{R}^4 = \mathbb{R}^3\times\mathbb{R}$.
Note, there is a similar question on MathOverflow that may be of interest.