4-dimensional h-cobordisms 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 !
 A: The answer to question 1 is no in the orientable case .Every topological 4-dim.h-cobordism is an s-cobordism.
I think that the argument in the paper S.Kwasik,R.Schultz:Vanishing of Whitehead torsion in dimension 4,Topology,vol.31(1992),pp.735-756 should also give the non-orientable case.
The orientable case is the main theorem on p.736.It was proved for geometric 3-manifolds,but now we know that all3-manifods are geometric.
A: 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" , JDG (2006), where references can be found.
Convention: all cobordisms below are of dimension 4 (i.e. have 3-manifold boundaries).


*

*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).

*There are infinitely many non-orientable TOP 
s-cobordisms (Matsumoto-Siebenmann, Kwasik).

*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.

*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.

*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.
A: Just a remark: it follows from geometrization that  two closed 3-manifolds are simple-homotopy equivalent, then they are diffeomorphic. So for an $s$-cobordism, you know that at least the two ends are the same. 
I'm actually not sure what's known about h-cobordant 3-manifolds. Aspherical 3-manifolds are homotopy rigid, and so are most spherical space forms, so I think it boils down to analyzing h-cobordant connect sums of lens spaces. Atiyah and Bott have shown that $h$-cobordant lens spaces are diffeomorphic, but I'm not sure what's known about connect sums.  
A: As for the second question the answer is yes. S. Donaldson, using gauge theory  showed  that there exist non-product $h$-cobordisms between  $K3$ surfaces. 
These are simply connected complex surfaces (real dimension $4$) with trivial canonical line bundle. They are all diffeomorphic and have intersection form $2E_8+3H$.   The automorphism $-1$ of the intersection form of $K3$  determines an $h$-cobordism which is not a product.   
