Pseudoisotopy theorem

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 M × { 0 } ∪ ∂ M × [ 0 , 1 ] .

Given f : M × [ 0 , 1 ] → M × [ 0 , 1 ] a pseudo-isotopy diffeomorphism, its restriction to M × { 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 ƒ being an isotopy of g to the identity is whether or not ƒ preserves the level-sets M × { t } for t ∈ [ 0 , 1 ] .

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 π [ 0 , 1 ] ∘ f t . One then applies Cerf theory.