I would like to know the state of the art concerning the following two questions.


1) Does there exist a smooth 4-dimensional h-cobordism (so between closed 3-manifolds) with non-vanishing Whitehead torsion ?


2) Does there exist a smooth 4-dimensional s-cobordism (that is, with vanishing Whitehead torsion) which is not diffeomorphic to a product cobordism ?


Thank you !



4-dimensional h-cobordisms
I think both questions are open. The somewhat sad state of affairs is that there are nontrivial TOP 4d s-cobordisms that are either nonsmoothable or not known to be smoothable, and there are smooth 4d s-cobordisms
that may well be products. No h-cobordisms with nontrivial torsion seems to be known.


It seems the state of the art is described in the introduction to a paper by Weimin Chen  ["Smooth s-cobordisms of elliptic 3-manifolds"](http://arxiv.org/abs/math/0403396) , JDG (2006), where references can be found.


Convention: all cobordisms below are of dimension 4 (i.e. have 3-manifold boundaries).


1. There are only finitely many orientable TOP s-cobordisms
with the boundary the same elliptic 3-manifold and in some cases there is a complete classification (Cappell-Shaneson, Kwasik-Schultz).
2. There are infinitely many non-orientable TOP 
s-cobordisms (Matsumoto-Siebenmann, Kwasik).
3. Kwasik gave (modulo now known elliptization conjecture) 
a list of finite groups such that any 4-dimensional topological h-cobordism with the fundamental group on the list must have trivial Whitehead torsion, see
"On four-dimensional h-cobordism".
Of course, the Whitehead group itself of those finite groups is often nontrivial.
4. Cappell-Shaneson constructed examples of smooth s-cobordsims with ellipltic 3-manifold boundaries, but it is unknown whether the cobordisms aren't products, and partial results of Akbulut indicate they are probably smooth products.
5. Chen proved that a symplectic s-cobordism with elliptic boundaries is a product, and conjectured that a smooth s-cobordism is a product if and only if its universal cover is a product.


