  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 \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 cat-isomorphic to the cylinder m Ã [0, 1]. here cat 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 the fundamental result in the theory of high-dimensional manifolds. for a start, it almost immediately proves the generalized poincarÃ© conjecture. ==background== 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. ==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 cat=diff, pl, or top such that w, m and n are simply connected, then w is cat-isomorphic to m Ã [0, 1]. the isomorphism can be chosen to be the identity on m Ã {0}. this means that the homotopy equivalence between m, w, and n is homotopic to a cat-isomorphism. ==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 has not been proved and, due to the poincarÃ© conjecture, is equivalent to the hard open question of whether the 4-sphere has non-standard smooth structures. for n = 2, the h-cobordism theoremin 3 dimensions and below, the categories are the same: diff = pl = top. is true â it is equivalent to t
Differential_topology Surgery_theory