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
- French mathematician, born 1928
- 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.