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 ${\displaystyle M\times \{0\}\cup \partial M\times [0,1]}$.

Given ${\displaystyle f:M\times [0,1]\to M\times [0,1]}$ a pseudo-isotopy diffeomorphism, its restriction to ${\displaystyle M\times \{1\}}$ is a diffeomorphism ${\displaystyle g}$ 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 ${\displaystyle M\times \{t\}}$ for ${\displaystyle t\in [0,1]}$.

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.[1]

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 ${\displaystyle \pi _{[0,1]}\circ f_{t}}$. One then applies Cerf theory.[1]

References

1. Cerf, J. (1970). "La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie". Inst. Hautes Études Sci. Publ. Math. 39: 5–173.