# Pseudoisotopy theorem

In mathematics, the **pseudoisotopy theorem** is a theorem of Jean Cerf's^{[1]} 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.
^{[2]}

## 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.^{[3]}

## References

- ↑ French mathematician, born 1928
- ↑ J.Cerf, La stratification naturelle des espaces de fonctions deff\'erentiables r\'eelles et le th\'eor\`eme de la pseudo-isotopie, Inst. Hautes \'Etudes Sci. Publ. Math. No {\bf 39} (1970) 5–173.
- ↑ J.Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. No. 39 (1970) 5–173.