Pseudoisotopy theorem
Encyclopedia
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.
M (with or without boundary), a pseudo-isotopy diffeomorphism of M is a diffeomorphism
of M × [0, 1] which restricts to the identity on
.
Given
a pseudo-isotopy diffeomorphism, its restriction to 
is a diffeomorphism 
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
for 
.
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.
. One then applies Cerf theory
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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 manifoldDifferentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
M (with or without boundary), a pseudo-isotopy diffeomorphism of M is a diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
of M × [0, 1] which restricts to the identity on
.Given
a pseudo-isotopy diffeomorphism, its restriction to is a diffeomorphism of M. We say g is pseudo-isotopic to the identity. One should think of a pseudo-isotopy as something that is almost an isotopyHomotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
-- the obstruction to ƒ being an isotopy of g to the identity is whether or not ƒ preserves the level-sets
for .Cerf's theorem states that, provided M is simply-connected
Simply connected space
In topology, a topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question .If a space is not simply connected, it is convenient...
and dim(M) ≥ 5, the group of pseudo-isotopy diffeomorphisms of M is connected. Equivalently, a diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
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. One then applies Cerf theoryCerf theory
In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functionsf:M \to \mathbb R...
.