Pseudoisotopy theorem


In mathematics, the pseudoisotopy theorem is a theorem of Jean Cerf's which refers to the connectivity of a group of diffeomorphisms of a manifold.

Statement

Given a differentiable manifold M, a pseudo-isotopy diffeomorphism of M is a diffeomorphism of M × which restricts to the identity on.
Given a pseudo-isotopy diffeomorphism, its restriction to is a diffeomorphism of M. We say g is pseudo-isotopic to the identity. One should think of a pseudo-isotopy as something that is almost an isotopy—the obstruction to ƒ being an isotopy of g to the identity is whether or not ƒ preserves the level-sets for.
Cerf's theorem states that, provided M is simply-connected and dim ≥ 5, the group of pseudo-isotopy diffeomorphisms of M is connected. Equivalently, a diffeomorphism of M is isotopic to the identity if and only if it is pseudo-isotopic to the identity.

Relation to Cerf theory

The starting point of the proof is to think of the height function as a 1-parameter family of smooth functions on M by considering the function. One then applies Cerf theory.