Pseudoisotopy theorem
On the connectivity of a group of diffeomorphisms of a manifold
title: "Pseudoisotopy theorem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["theorems-in-differential-topology", "singularity-theory"] description: "On the connectivity of a group of diffeomorphisms of a manifold" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Pseudoisotopy_theorem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary On the connectivity of a group of diffeomorphisms of a manifold ::
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 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. This implies that, 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 \pi_{[0,1]} \circ f_t. One then applies Cerf theory.
References
References
- Cerf, Jean. (1971). "The pseudo-isotopy theorem for simply connected differentiable manifolds". Springer.
- 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..
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