The H-Cobordism Theorem gives sufficient conditions under which a cobordism between two manifolds whose inclusion maps are both homotopy equivalences, called an h-cobordism, must in fact be trivial, meaning isomorphic to the simple product of one of the manifolds with an interval. Proved by Stephen Smale, who received the Fields Medal in part for this result, it is a fundamental theorem of the theory of high-dimensional manifolds, and it almost immediately implies the generalized Poincare Conjecture in dimensions five and above.
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Proof YearMathWorld gives 1961 for Smale's proof; search-result summaries also report 1962 for the theorem, so the year is not uniform across sources. StatementLet n be at least 5 and W a compact (n + 1)-dimensional h-cobordism between M and N in the category Diff, PL, or Top such that W, M and N are simply connected. Then W is isomorphic in that category to M x [0, 1]. 1 Classification
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Source H-cobordism (Wikipedia)
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1. H-cobordism (Wikipedia)
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
Proved By: Stephen Smale, Lead
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.
In Group: Geometric and Differential Topology, Lead section
In geometric topology and differential topology, an (n + 1)-dimensional cobordism
View the Source2. h-Cobordism Theorem (Wolfram MathWorld)
Prover and year lineQuote, Prover and year line
Smale proved this theorem in 1961
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