Non-homeomorphic manifolds that are homotopy equivalent

Question

Is there an example for the following?

Two simply-connected smooth $n$-manifolds which are homotopy equivalent, but not homeomorphic.

I know the following related results:

• If I remove "simply-connected", then there are lens space examples.
• If I remove "smooth", then there's a non-smoothable 4-manifold homotopy equivalent to $\mathbb{CP}^2$.
• If I change "homeomorphic" to "diffeomorphic", then there are exotic spheres.

But I haven't been able to find any actual example after searching for a while.

Answer 1

Take the 0 knot traces of two knots (4 manifolds with a 0 framed 2 handle glued along the knot). Then the resulting manifolds $X_0(K)$ and $X_0(K')$ are simply connected (since there is no 1 handle), homotopy equivalent (since all knots are homotopic), but not necessarily homeomorphic (their boundaries are the 0 surgeries of the knots and they are not the same in most cases).