H-COBORDISM

A cobordism ''W'' between ''M'' and ''N'' is an '''h''-cobordism' if the inclusion maps
: $M\; hookrightarrow\; W$
and
: $N\; hookrightarrow\; W$
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''.
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.

Contents

Low dimensions

The s-cobordism theorem

References

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 Poincaré 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).

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.

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.