h-cobordism

Before Smale proved this theorem, mathematicians became stuck while 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 >4. 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 entanglement.

Let n be at least 5 and let W be a compact (n + 1)-dimensional h-cobordism between M and N in the category C=Diff, PL, or Top such that W, M and N are simply connected. Then W is C-isomorphic to M × [0, 1]. The isomorphism can be chosen to be the identity on M × {0}.

This means that the homotopy equivalence between M and N (or, between M × [0, 1], W and N × [0, 1]) is homotopic to a C-isomorphism.

For n = 4, the h-cobordism theorem is false. This can be seen since Wall proved{{Cite journal |last=Wall |first=C.T.C. |date=1964 |title=On simply-connected 4-manifolds |journal=Journal of the London Mathematical Society |volume=39 |pages=141-49}} that closed oriented simply-connected topological four-manifolds with equivalent intersection forms are h-cobordant. However, if the intersection form is odd there are non-homeomorphic 4-manifolds with the same intersection form (distinguished by the Kirby-Siebenmann class). For example, CP² and a fake projective plane with the same homotopy type are not homeomorphic but both have intersection form of (1). It fails even more dramatically in the smooth category, where smooth compact simply-connected 4-manifolds often carry infinitely many distinct smooth structures that can be distinguished by their Seiberg-Witten invariants.

For n = 3, the h-cobordism theorem for smooth manifolds has not been proved and, due to the 3-dimensional 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 is equivalent to the Poincaré conjecture stated by Poincaré in 1904 (one of the Millennium Problems{{Cite web|url = http://www.claymath.org/millennium-problems|title = Millennium Problems {{!}} Clay Mathematics Institute|website = www.claymath.org|access-date = 2016-03-30}}) and was proved by Grigori Perelman in a series of three papers in 2002 and 2003,{{Cite arXiv |last = Perelman|first = Grisha|date = 2002-11-11|title = The entropy formula for the Ricci flow and its geometric applications |eprint = math/0211159}}{{Cite arXiv|last = Perelman|first = Grisha|date = 2003-03-10|title = Ricci flow with surgery on three-manifolds |eprint = math/0303109}}{{Cite arXiv|last = Perelman|first = Grisha|date = 2003-07-17|title = Finite extinction time for the solutions to the Ricci flow on certain three-manifolds |eprint = math/0307245}} where he follows Richard S. Hamilton's program using Ricci flow.

For n = 1, the 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.

A Morse function $f:W\backslash to[a,b]$ induces a handle decomposition of W, i.e., if there is a single critical point of index k in $f^\{-1\}([c,c\text{'}])$, then the ascending cobordism $W\_\{c\text{'}\}$ is obtained from $W\_c$ by attaching a k-handle. The goal of the proof is to find a handle decomposition with no handles at all so that integrating the non-zero gradient vector field of f gives the desired diffeomorphism to the trivial cobordism.

This is achieved through a series of techniques.

1) Handle rearrangement

First, we want to rearrange all handles by order so that lower order handles are attached first. The question is thus when can we slide an i-handle off of a j-handle? This can be done by a radial isotopy so long as the i attaching sphere and the j belt sphere do not intersect. We thus want $(i-1)+(n-j)\backslash leq\backslash dim\backslash partial\; W-1=n-1$ which is equivalent to $i\backslash leq\; j$.

We then define the handle chain complex $(C\_*,\backslash partial\_*)$ by letting $C\_k$ be the free abelian group on the k-handles and defining $\backslash partial\_k:C\_k\backslash to\; C\_\{k-1\}$ by sending a k-handle $h\_\{\backslash alpha\}^k$ to $\backslash sum\_\backslash beta\; \backslash langle\; h\_\backslash alpha^k\backslash mid\; h\_\backslash beta^\{k-1\}\backslash rangle\; h\_\backslash beta^\{k-1\}$, where $\backslash langle\; h\_\backslash alpha^k\backslash mid\; h\_\backslash beta^\{k-1\}\backslash rangle$ is the intersection number of the k-attaching sphere and the (k − 1)-belt sphere.

2) Handle cancellation

Next, we want to "cancel" handles. The idea is that attaching a k-handle $h\_\backslash alpha^k$ might create a hole that can be filled in by attaching a (k + 1)-handle $h\_\backslash beta^\{k+1\}$. This would imply that $\backslash partial\_\{k+1\}h\_\backslash beta^\{k+1\}=\backslash pm\; h\_\backslash alpha^k$ and so the $(\backslash alpha,\backslash beta)$ entry in the matrix of $\backslash partial\_\{k+1\}$ would be $\backslash pm\; 1$. However, when is this condition sufficient? That is, when can we geometrically cancel handles if this condition is true? The answer lies in carefully analyzing when the manifold remains simply-connected after removing the attaching and belt spheres in question, and finding an embedded disk using the Whitney trick. This analysis leads to the requirement that n must be at least 5. Moreover, during the proof one requires that the cobordism has no 0-,1-,n-, or (n + 1)-handles which is obtained by the next technique.

3) Handle trading

The idea of handle trading is to create a cancelling pair of (k + 1)- and (k + 2)-handles so that a given k-handle cancels with the (k + 1)-handle leaving behind the (k + 2)-handle. To do this, consider the core of the k-handle which is an element in $\backslash pi\_k(W,M)$. This group is trivial since W is an h-cobordism. Thus, there is a disk $D^\{k+1\}$ which we can fatten to a cancelling pair as desired, so long as we can embed this disk into the boundary of W. This embedding exists if $\backslash dim\backslash partial\; W-1=n-1\backslash geq\; 2(k+1)$. Since we are assuming n is at least 5 this means that k is either 0 or 1. Finally, by considering the negative of the given Morse function, −f, we can turn the handle decomposition upside down and also remove the n- and (n + 1)-handles as desired.

4) Handle sliding

Finally, we want to make sure that doing row and column operations on $\backslash partial\_k$ corresponds to a geometric operation. Indeed, it isn't hard to show (best done by drawing a picture) that sliding a k-handle $h\_\backslash alpha^k$ over another k-handle $h\_\{\backslash beta\}^k$ replaces $h\_\backslash alpha^k$ by $h\_\backslash alpha^k\backslash pm\; h\_\backslash beta^k$ in the basis for $C\_k$.

The proof of the theorem now follows: the handle chain complex is exact since $H\_*(W,M;\backslash mathbb\{Z\})=0$. Thus $C\_k\backslash cong\; \backslash operatorname\{coker\}\; \backslash partial\_\{k+1\}\backslash oplus\backslash operatorname\{im\}\; \backslash partial\_\{k+1\}$ since the $C\_k$ are free. Then $\backslash partial\_k$, which is an integer matrix, restricts to an invertible morphism which can thus be diagonalized via elementary row operations (handle sliding) and must have only $\backslash pm\; 1$ on the diagonal because it is invertible. Thus, all handles are paired with a single other cancelling handle yielding a decomposition with no handles.

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 $M\; \backslash hookrightarrow\; W$.

Precisely, the s-cobordism theorem (the s stands for simple-homotopy equivalence), 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 $M\; \backslash hookrightarrow\; W$ is not just a homotopy equivalence, but a simple homotopy equivalence.

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

Categorically, h-cobordisms form a groupoid.

Then a finer statement of the s-cobordism theorem is that the isomorphism classes of this groupoid (up to C-isomorphism of h-cobordisms) are torsors for the respectiveNote that identifying the Whitehead groups of the various manifolds requires that one choose base points $m\backslash in\; M,\; n\backslash in\; N$ and a path in W connecting them. Whitehead groups Wh(π), where $\backslash pi\; \backslash cong\; \backslash pi\_1(M)\; \backslash cong\; \backslash pi\_1(W)\; \backslash cong\; \backslash pi\_1(N).$

• Semi-s-cobordism

• {{cite book|last1=Freedman|first1=Michael H|authorlink1=Michael Freedman|last2=Quinn|first2=Frank|authorlink2=Frank Quinn (mathematician)|title=Topology of 4-manifolds|series=Princeton Mathematical Series|volume=39|publisher=Princeton University Press|location=Princeton, NJ|year=1990|isbn=0-691-08577-3|url-access=registration|url=https://archive.org/details/topologyof4manif0000free}} (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
• {{springer|id=H/h046010|title=h-cobordism|first=Yu.B.|last= Rudyak}}

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