H-cobordism

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A cobordism W between M and N is an h-cobordism if the inclusion maps

M → W

and

N → W

are homotopy equivalences. The h-cobordism theorem states that if W is a compact smooth h-cobordism between M and N, and if in addition M and N are simply connected and of dimension > 4, then W is diffeomorphic to M × [0, 1] and M is diffeomorphic to N.

It was first proved by Stephen Smale and is the fundamental result in the theory of high dimensional manifolds. Before he proved this theorem, mathematicians had got stuck trying to understand manifolds of dimension 3 or 4, and assumed that the higher dimensional cases were even harder. The h-cobordism theorem showed that manifolds of dimension at least 5 are much easier than those of dimension 3 or 4. An informal reason why manifolds of dimension 3 or 4 are unusually hard is that in lower dimensions there is no room for tangles to form, and in higher dimensions there is enough room to undo any tangles that do form.

Low dimensions

If the manifolds M and N have dimension 4, then the h-cobordism theorem is still true for topological manifolds (proved by Michael Freedman) but is false for PL or smooth manifolds of dimension 4 (as shown by Simon Donaldson).

If M and N have dimension 3 then the h-cobordism theorem for smooth manifolds is probably also false, but this has not been proved and (assuming the Poincare conjecture) is equivalent to the hard open question of whether the 4-sphere has non-standard smooth structures.

If M and N have dimension 2, then the h-cobordism theorems for smooth, PL, or topological manifolds are all equivalent to the Poincare conjecture, which has probably been proved by Grigori Perelman

If M and N have dimension 0 or 1 the h-cobordism theorem is true (and not very interesting).

The s-cobordism theorem

If the assumption that M and N are simply connected is dropped, the theorem becomes false. It is true, however, if (and only if) the Whitehead torsion τ(W, M) vanishes; this is the s-cobordism theorem. It was proved independently by Barry Mazur, John Stallings, and Denis Barden.

References

Milnor, John Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow Princeton University Press, Princeton, N.J. This gives the proof for smooth manifolds.

Rourke, Colin Patrick; Sanderson, Brian Joseph Introduction to piecewise-linear topology. Springer Study Edition. Springer-Verlag, Berlin-New York, 1982. ISBN 3540111026 This proves the theorem for PL manifolds.

Topology of 4-Manifolds. by Michael H. Freedman, Frank Quinn, does the theorem for topological 4-manifolds.

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