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
M ↪ W and N ↪ W {\displaystyle M\hookrightarrow W\quad {\mbox{and}}\quad N\hookrightarrow W}
are homotopy equivalences. The h-cobordism theorem gives sufficient conditions for an h-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder M × [0, 1]. 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.
Background 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.
Precise statement of the h-cobordism theorem 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.
Lower dimensional versions For n = 4, the topological h-cobordism theorem was proved by Michael Freedman. The smooth 4-dimensional h-cobordism theorem was disproved by Simon Donaldson. 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 (exotic) 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) and was proved by Grigori Perelman in a series of three papers in 2002 and 2003, 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.
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