Pseudoisotopy theorem
{{short description|On the connectivity of a group of diffeomorphisms of a manifold}}
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.{{cite journal |first=J. |last=Cerf |title=La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie |journal=Inst. Hautes Études Sci. Publ. Math. |volume=39 |year=1970 |pages=5–173 |doi=10.1007/BF02684687 |s2cid=120787287 |url=http://www.numdam.org/item/PMIHES_1970__39__5_0/ }}