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
M x \{0\}\cup\partialM x [0,1]
Given
f:M x [0,1]\toM x [0,1]
M x \{1\}
g
M x \{t\}
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.
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]\circft