Topological Gluing

Regular Interval Theorem

Let be a cobordism, and let be a Morse function without any critical points, then there is a diffeomorphism from the cylinder to that is compatible with the projection to : And similarly there is another diffeomorphism compatible with the projection.

Corollary

Let be a cobordism. Then there is a decomposition such that is diffeomorphic to the cylinder .

Proof Take a Morse function , then there is some such that has no critical points in . By the regular interval theorem, there is a diffeomorphism .

Gluing of General Cobordisms

Theorem

Suppose and are cobordisms, then there always exists a smooth manifold which is homeomorphic to and whose smooth structure agrees with both and .