How about the converse of the property that a product of manifolds is a manifold? We know that the Cartesian product of two manifolds is a manifold, but is the converse true? Let us assume that we have $A$ and $B$ two second countable Hausdorff topological spaces, and $M = A \times B$. If we assume that $M$ is a $n$-manifold, with $n \geq 0$ finite, do we obtain that $A$ and $B$ are $k$- and $l$-manifolds with $k+l = n$?

The dogbone space is not a manifold but its product with $\mathbb{R}$ is homeomorphic to $\mathbb{R}^4$. This was proved in this paper by Bing.
