TITLE: Simple homotopy equivalent, non-homeomorphic manifolds
QUESTION [17 upvotes]: 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.

REPLY [5 votes]: To expand a bit the answers given by others, the machinery of surgery theory outputs many examples of simple homotopy equivalent manifolds that are not homeomorphic, PL homeomorphic, or diffeomorphic (let's just say equivalent to avoid repeating all of these categories.) The only caveat is that what it produces are simple homotopy equivalences that are not homotopic to equivalences. The bible for this is Wall's book, Surgery on Compact Manifolds, although there are other treatments that are perhaps more approachable. Ben Wieland's answer has to do with one aspect of the theory, which has to do with the tangent bundle; the example of Cappell cited by Ian Agol has to do with the Wall surgery groups. I'll suggest some alternative examples along these same lines, although using easier to understand invariants.
If you want the manifolds to be non-equivalent, then a good collection of manifolds to look at are manifolds homotopy equivalent to higher dimensional lens spaces (quotients of $S^{2n-1}$ by a cyclic group $\mathbb{Z}_d$ acting linearly on $\mathbb{R}^{2n}$). In this setting, there are two fundamental sorts of invariants: Reidemeister torsion, and the multisignature (or Atiyah-Singer invariants). If you fix the simple homotopy type, that fixes the Reidemeister torsion, and the multisignature invariants can be varied more or less arbitrarily. This is detailed in Chapter 14E in Wall's book (which is not particularly easy reading). The executive summary of what is going on here is that you are using the action of the Wall surgery group $L^s_{2n}(\mathbb{Z}[\mathbb{Z}_d])$ on the structure set of a lens space $S^{2n-1}/\mathbb{Z}_d$, and the multisignature invariants are really invariants from that surgery group. They have the nice property that they are invariants of the manifold itself (together with an identification of the fundamental group with $\mathbb{Z}_d$) and so don't depend very strongly on a choice of homotopy equivalence.