Ambient isotopy - Imported Knowlpedia Content

Definition

Let $N$ be a smooth manifold. An ambient isotopy of $N$ is a smooth map in the manifold-with-boundary sense $\Phi:N\times[0,1]\to N$ such that every time slice $\Phi_t(x)=\Phi(x,t)$ is a diffeomorphism and $\Phi_0=\operatorname{id}_N$ . Two subsets or embedded submanifolds $A_0,A_1\subseteq N$ are ambiently isotopic if some ambient isotopy satisfies $\Phi_1(A_0)=A_1$ . For parametrized embeddings $f_0,f_1:M\to N$ , one instead requires $f_1=\Phi_1\circ f_0$ . Thus ambient isotopy deforms the whole surrounding manifold, not only the object lying inside it, and records equivalence through motions of that ambient space.

Relation to smooth isotopy

An ambient isotopy produces a smooth isotopy of every embedding by $f_t=\Phi_t\circ f_0$ . The converse is the content of an isotopy-extension theorem and requires hypotheses, typically including properness or compact support conditions. Hirsch proves standard extension results for submanifolds in Hirsch, Chapter 8.

Examples and invariants

Rotating $\mathbb R^2$ continuously about the origin is an ambient isotopy. A compactly supported time-dependent vector field integrates, on its time interval, to an ambient isotopy through its flow. Ambiently isotopic embedded submanifolds have diffeomorphic complements, so embeddings whose complements have different topological invariants cannot be ambiently isotopic.

Conventions and scope

References

Morris W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, 1976. DOI record. Relevant: Chapter 8, isotopies and isotopy extension.
Victor Guillemin and Alan Pollack, Differential Topology, AMS Chelsea Publishing, 2010 reprint. DOI record. Relevant: isotopy and ambient deformation of submanifolds.