What are known examples of two smooth, closed, oriented Manifolds $M,N$ of the same dimension that are simple homotopy equivalent, but not homeomorphic ?
It is well-known that the homotopy type of a given such $2$-manifold is the same as its homeomorphism type, so there won't be any such easy examples. Moreover (and this is where for me, the question becomes really interesting), I've heard at a recent conference that the simple homotopy type of a closed, oriented $3$-manifold is the same as its homeomorphism type, so any known examples must be in even higher dimension. Also, as the Borel conjecture is still open, any known such example must consist of non-aspherical manifolds.
Best Answer
The manifold $*\mathbb{C}P^2$ (or the Chern manifold) is homotopy equivalent to $\mathbb{C}P^2$, but it is not homeomorphic to $\mathbb{C}P^2$, since its Kirby-Siebenmann invariant is non-trivial.