A compact smooth cobordism is an h-cobordism when both boundary inclusions are homotopy equivalences. The name does not itself impose simple connectivity. A trivial cobordism is an h-cobordism because a product cylinder retracts onto either end.
A compact smooth h-cobordism of dimension at least six between connected simply connected closed manifolds is diffeomorphic to a product, relative to its incoming boundary. The boundary manifolds consequently are diffeomorphic. The dimension and simple-connectivity hypotheses are sufficient conditions, not necessary properties of a product cylinder. The two boundary inclusions being homotopy equivalences are necessary for a trivial cobordism.
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In topology, particularly in the field of differential topology, H-cobordism is a concept that arises in the study of smooth manifolds. It is a specific type of cobordism that deals with the structures of manifolds and the mappings between them. To provide a more precise definition, let \( M \) and \( N \) be smooth manifolds of the same dimension.