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.
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 formula_1.
Given formula_2 a pseudo-isotopy diffeomorphism, its restriction to formula_3 is a diffeomorphism formula_4 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 formula_5 for formula_6.
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 formula_7. One then applies Cerf theory.