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h-Cobordism Theorem


If W is a simply connected, compact manifold with a boundary that has two components, M_1 and M_2, such that inclusion of each is a homotopy equivalence, then W is diffeomorphic to the product M_1×[0,1] for dim(M_1)>=5. In other words, if M and M^' are two simply connected manifolds of dimension >=5 and there exists an h-cobordism W between them, then W is a product M×I and M is diffeomorphic to M^'.

The proof of the h-cobordism theorem can be accomplished using surgery. A particular case of the h-cobordism theorem is the Poincaré conjecture in dimension n>=5. Smale proved this theorem in 1961.


See also

Diffeomorphism, Poincaré Conjecture, Surgery

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References

Smale, S. "Generalized Poincaré's Conjecture in Dimensions Greater than Four." Ann. Math. 74, 391-406, 1961.

Referenced on Wolfram|Alpha

h-Cobordism Theorem

Cite this as:

Weisstein, Eric W. "h-Cobordism Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/h-CobordismTheorem.html

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