Let $f:X\to Y$ be a homotopy equivalence between closed smooth manifolds. Is there a closed manifold $Z$ such that $X$ and $Y$ are (strong) deformation retracts of $Z$? (There is the mapping cylinder, but it isn't a manifold.)
EDIT. As Ryan Budney pointed out, there are counterexamples in dimension 3. Probably, the right question must include $\dim X=\dim Y>4$ (or maybe even higher dimension).
EDIT. Not a very good question. The dimension is a homotopy invariant of a closed manifold, so $Z$ must have the same dimension, which can't be. Why nobody pointed it out?