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

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

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. ^[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 $\pi_{[0,1]} \circ f_t$. One then applies Cerf theory.^[2]

References

1. ↑ French mathematician, born 1928
2. ↑ 2.0 2.1 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.