In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps are homotopy equivalences.
Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds.
The theorem was first proved by Stephen Smale for which he received the Fields Medal and is a fundamental result in the theory of high-dimensional manifolds.
For a start, it almost immediately proves the generalized Poincaré conjecture.
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 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.
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.
This can be seen since Wall proved[1] that closed oriented simply-connected topological four-manifolds with equivalent intersection forms are h-cobordant.
For example, CP2 and a fake projective plane with the same homotopy type are not homeomorphic but both have intersection form of (1).
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[2]) and was proved by Grigori Perelman in a series of three papers in 2002 and 2003,[3][4][5] 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.
induces a handle decomposition of W, i.e., if there is a single critical point of index k in
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.
be the free abelian group on the k-handles and defining
might create a hole that can be filled in by attaching a (k + 1)-handle
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.
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.
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
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
Indeed, it isn't hard to show (best done by drawing a picture) that sliding a k-handle
The proof of the theorem now follows: the handle chain complex is exact since
, 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
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
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): The torsion vanishes if and only if the inclusion
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 respective[6] Whitehead groups Wh(π), where