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

Given $$ f : M \times [0,1] \to M \times [0,1]$$ a pseudo-isotopy diffeomorphism, its restriction to $$M \times \{1\}$$ is a diffeomorphism $$g$$ 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 &fnof; being an isotopy of g to the identity is whether or not &fnof; preserves the level-sets $$M \times \{t\}$$ for $$ t \in [0,1]$$.

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