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.

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 ${\displaystyle M\times \{0\}\cup \partial M\times [0,1]}$.

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

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.^[1]

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 ${\displaystyle \pi _{[0,1]}\circ f_{t}}$. One then applies Cerf theory.^[1]