Knowlpedia

Theorem

Isotopy extension theorem

A smooth isotopy of a compact embedded manifold extends to a smooth ambient isotopy.

Statement

Let NN be a compact , let MM be a smooth manifold without boundary, and let F:N×[0,1]→MF:N\times[0,1]\to M be a of embeddings with F0=iF_0=i. The isotopy extension theorem states that there is a smooth Φ:M×[0,1]→M\Phi:M\times[0,1]\to M such that Φ0=id⁡M\Phi_0=\operatorname{id}_M and

Φt∘i=Ft\Phi_t\circ i=F_t

for every tt. Moreover, Φt\Phi_t can be chosen to have compact support in an arbitrarily prescribed neighborhood of the track F(N×[0,1])F(N\times[0,1]), after that neighborhood is chosen suitably. Thus deforming the embedded copy of NN can be realized by deforming all of MM Hirsch, Chapter 8, §1.

Construction idea

Differentiating FtF_t in tt gives a along the moving submanifold Ft(N)F_t(N). A , extension in the normal directions, and a cutoff function extend it to a compactly supported time-dependent vector field on MM. The flow of this ambient field is the required Φt\Phi_t. Compact support ensures that the flow exists for the whole parameter interval.

Relative and boundary forms

There are relative versions that keep a region fixed when the original isotopy is stationary there. Manifolds with boundary also admit versions when the embeddings and extended vector fields respect the boundary, often after requiring the isotopy to be fixed near it. These additional hypotheses are part of the theorem being applied; the boundaryless statement in the core avoids silently imposing one convention.

Consequences and limitations

Two compact embeddings joined by a smooth isotopy are therefore ambiently isotopic, so their complements are diffeomorphic. Compactness, properness, or a suitable support condition cannot be discarded indiscriminately: for a noncompact embedded manifold, a moving velocity field may admit no complete ambient extension. A mere is also insufficient because its intermediate maps may fail to remain embeddings.

References
  1. Morris W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, 1976. DOI record. Relevant: Chapter 8, §1, isotopy extension and support control.