h-cobordism

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

$M \hookrightarrow W \quad\mbox{and}\quad N \hookrightarrow W$

are homotopy equivalences. If $\mbox{dim}\,M = \mbox{dim}\,N = n$ and $\mbox{dim}\,W = n+1$ , it is called an $n + 1$ -dimensional h-cobordism.^[1]

The h-cobordism theorem states that if:

W is a compact h-cobordism between M and N
in the category Cat=Diff, PL, or Top
M and N are simply connected
dimension M and N > 4

then W is Cat-isomorphic^[2] to M × [0, 1] and (hence) M is Cat-isomorphic to N. Informally, "an h-cobordism is a cylinder".

The theorem was first proved by Stephen Smale and is the fundamental result in the theory of high-dimensional manifolds: for a start, it almost immediately proves the Generalized Poincaré Conjecture. The theorem is still true topologically but not smoothly for n = 4; a later result of Michael Freedman.

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-tangled 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.

1 Low dimensions
2 The s-cobordism theorem
3 Footnotes
4 References

[edit] Low dimensions

For n = 4, the h-cobordism theorem is true topologically (proved by Michael Freedman using a 4-dimensional Whitney trick) but is false PL and smoothly (as shown by Simon Donaldson).

For n = 3, the h-cobordism theorem for smooth manifolds is probably also false, but this has not been proved and, due to the Poincaré conjecture, is equivalent to the hard open question of whether the 4-sphere has non-standard smooth structures.

For n = 2, the h-cobordism theorem^[3] is equivalent to the Poincaré conjecture, which has been proved by Grigori Perelman.

For n = 1, h-cobordism theorem is vacuously true, since there is no closed simply-connected 1-dimensional manifold.

For n = 0, the h-cobordism theorem is trivially true: the interval is the only connected cobordism between connected 0-manifolds.

[edit] The s-cobordism theorem

If the assumption that M and N are simply connected is dropped, h-cobordisms need not be cylinders; the obstruction is exactly the Whitehead torsion τ (W, M) of the inclusion $M \hookrightarrow W$ .

Precisely, the s-cobordism theorem (proved independently by Barry Mazur, John Stallings, and Dennis Barden) states (assumptions as above but where M and N need not be simply connected):

an h-cobordism is a cylinder if and only if Whitehead torsion τ (W, M) vanishes

The torsion vanishes if and only if the inclusion $M \hookrightarrow W$ is not just a homotopy equivalence, but a simple homotopy equivalence.

Note that one need not assume that the other inclusion $N \hookrightarrow W$ is also a simple homotopy equivalence—that follows from the theorem.

Categorically, h-cobordisms form a groupoid, just as for categories of cobordisms.

Then a finer statement of the s-cobordism theorem is that the isomorphism classes of this category (up to Cat-isomorphism of h-cobordisms) are torsors for the respective^[4] Whitehead groups $Wh(π)$ , where $\pi \cong \pi_1(M) \cong \pi_1(W) \cong \pi_1(N)$

[edit] Footnotes

^ This notation is to clarify the dimension of all manifolds in question, otherwise it is unclear whether a "5-dimensional h-cobordism" refers to a 5-dimensional cobordism between 4-dimensional manifolds or a 6-dimensional cobordism between 5-dimensional manifolds.
^ So diffeomorphic, PL-isomorphic, homeomorphic.
^ In 3 dimensions and below, the categories are the same: Diff=PL=Top.
^ Note that identifying the Whitehead groups of the various manifolds requires that one choose base points $m\in M, n\in N$ and a path in $W$ connecting them.

[edit] References

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.
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.
S. Smale, "On the structure of manifolds" Amer. J. Math. , 84 (1962) pp. 387–399
Rudyak, Yu.B. (2001), “h-cobordism”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104