TITLE: Why is it true that if two 4-manifolds are homeomorphic then their squares are diffeomorphic?
QUESTION [23 upvotes]: Near the top of the second page of this paper, it is claimed that if two 4-manifolds $X$ and $Y$ are homeomorphic, then their squares $X \times X$ and $Y \times Y$ are diffeomorphic. Why is this true? It is trivially correct for 3-manifolds and lower dimensions. What about higher dimensions?
References would be greatly appreciated as well.

REPLY [6 votes]: The original paper of Ivan Smith assumes that the 4-manifolds are simply-connected.  So this controls $H^3(X; \mathbb{Z}/2) \cong H_1(X; \mathbb{Z}/2) = 0$.  
Possibly this gives an interesting invariant for non-simply-connected 4-manifolds in general?