# 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 !

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I am not an expert but it seems both questions are open. From the careful phrasing of the question I gather that the OP is aware of variour works on the subject by Cappell-Shaneson, Kwasik, Kwasik-Schultz, Akbulut, Chen, which can be easily found in mathscinet. – Igor Belegradek Dec 13 '12 at 21:43
Question 1 is on Kirby's low-dimensional topology problem list as an open problem of Marshall Cohen. – Igor Belegradek Dec 13 '12 at 22:30

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).

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.

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Thank you very much. So these are very hard questions indeed ! – Sylvain Courte Dec 14 '12 at 21:56

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.

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I did not know precisely what geometrization implies in this case, so thank you very much ! – Sylvain Courte Dec 14 '12 at 21:56

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.

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Welcome to mathoverflow! I never actually looked at the paper you quote, but somehow I assumed that by "geometric" you meant (as usual) that the manifold has a geometric structure modelled on one of Thurston's geometries. As I see now your definition of "geometric" means that its prime/JSJ pieces are geometric. – Igor Belegradek Jan 29 '13 at 16:30

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.

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I think, in both questions OP meant that cobordism is 4-dimensional (between 3-dimensional manifolds), not 5-dimensional cobordism between 4-dimensional manifolds. – Misha Dec 13 '12 at 21:02
Indeed I meant that the cobordism is 4-dimensional, but thanks anyway! – Sylvain Courte Dec 14 '12 at 21:58