H-cobordism

From Wikipedia, the free encyclopedia

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

and

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.

The theorem was first proved by Stephen Smale and is the fundamental result in the theory of high-dimensional manifolds. Before Smale 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 (simply connected) manifolds of dimension at least 5 are much easier than those of dimension 3 or 4. The proof of the theorem depends on the "Whitney trick" of Hassler Whitney, which geometrically untangles homologically untangled spheres of complementary dimension in a manifold of dimension >5. An informal reason why manifolds of dimension 3 or 4 are unusually hard is that the trick fails to work in lower dimensions, which have no room for untanglement, and so have more tangles.

[edit] 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 using a 4-dimensional Whitney trick) 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 Poincaré 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).

[edit] 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 Dennis Barden.

[edit] References

• Milnor, John, Lectures on the h-cobordism theorem, notes by L. Siebenmann and J. Sondow, Princeton University Press, Princeton, NJ, 1965. v+116 pp. 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 3-540-11102-6. This proves the theorem for PL manifolds.

• Freedman, Michael H.; Quinn, Frank, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. viii+259 pp. ISBN 0-691-08577-3. This does the theorem for topological 4-manifolds.