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 × [0, 1] 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(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 .  One then applies Cerf theory.

References 

Category:Theorems in differential topology
Category:Singularity theory