I know that some manifolds which are homotopically equivalent become homeomorphic after taking the product with $\mathbb{R}$, e.g. $\mathbb{T}^{2}$ minus a point and $\mathbb{S}^{2}$ minus three points, or Whitehead manifold and $\mathbb{R}^{3}$.
I was wondering if there are some general conditions under which this happens, i.e. when given two homotopically equivalent $n-$manifolds $X_1$ and $X_2$ we have that $X_1\times\mathbb{R}$ and $X_2\times\mathbb{R}$ are homeomorphic.