In mathematics, the pseudoisotopy theorem is a theorem of Jean Cerf's which refers to the connectivity of a group of

diffeomorphisms In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...

of a manifold.

Statement

Given a

differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

''M'' (with or without boundary), a pseudo-isotopy diffeomorphism of ''M'' is a

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given tw ...

of ''M'' × , 1which restricts to the identity on

M \times \ \cup \partial M \times,1 /math>.

Given f : M \times,1 \to M \times,1 /math> a pseudo-isotopy diffeomorphism, its restriction to M \times \ 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 \ for t \in,1 /math>.

Cerf's theorem states that, provided ''M'' is

simply-connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space ...

and dim(''M'') ≥ 5, the group of pseudo-isotopy diffeomorphisms of ''M'' is connected. Equivalently, a

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_ \circ f_t

. One then applies Cerf theory.

References

{{Reflist Theorems in differential topology Singularity theory