Pseudoisotopy theorem

Short description: On the connectivity of a group of diffeomorphisms of a manifold

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 (with or without boundary), a pseudo-isotopy diffeomorphism of M is a diffeomorphism of M × [0, 1] which restricts to the identity on [math]\displaystyle{ M \times \{0\} \cup \partial M \times [0,1] }[/math].

Given [math]\displaystyle{ f : M \times [0,1] \to M \times [0,1] }[/math] a pseudo-isotopy diffeomorphism, its restriction to [math]\displaystyle{ M \times \{1\} }[/math] is a diffeomorphism [math]\displaystyle{ g }[/math] 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 [math]\displaystyle{ M \times \{t\} }[/math] for [math]\displaystyle{ t \in [0,1] }[/math].

Cerf's theorem states that, provided M is simply-connected and dim(M) ≥ 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 [math]\displaystyle{ \pi_{[0,1]} \circ f_t }[/math]. One then applies Cerf theory.