Smooth isotopy - Imported Knowlpedia Content

Definition

Let $f_0,f_1:M\to N$ be smooth embeddings between smooth manifolds. A smooth isotopy from $f_0$ to $f_1$ is a smooth map in the manifold-with-boundary sense $F:M\times[0,1]\to N$ such that $F(x,0)=f_0(x)$ , $F(x,1)=f_1(x)$ , and every time slice $F_t(x)=F(x,t)$ is a smooth embedding. Thus both the dependence on $t$ and the combined map $F$ are smooth, while injectivity and the embedding condition are required slice by slice. An isotopy of diffeomorphisms analogously requires every $F_t$ to be a diffeomorphism.

Relation to homotopy

Every isotopy is a smooth homotopy, but the converse need not hold: intermediate maps in a homotopy may develop self-intersections, lose rank, or cease to be invertible. Isotopy therefore records deformation within a geometrically constrained class rather than merely deformation through arbitrary continuous or smooth maps.

Ambient isotopy

An ambient isotopy of $N$ is a smooth family of diffeomorphisms $\Phi_t:N\to N$ with $\Phi_0=\operatorname{id}_N$ . It carries an embedding $f_0$ to $f_t=\Phi_t\circ f_0$ . The isotopy extension theorem gives hypotheses under which an isotopy of embeddings extends to an ambient isotopy; see Hirsch, Chapter 8.

Conventions and examples

A rotation of the circle through a continuously varying angle is an isotopy of diffeomorphisms. A family of embedded knots is an isotopy of embeddings. Authors sometimes say “smooth isotopy” for any smooth homotopy; the intended slice condition must be checked from context.

References

Morris W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, 1976. DOI record. Relevant: Chapter 8, isotopy and isotopy extension.
Victor Guillemin and Alan Pollack, Differential Topology, Prentice-Hall, 1974. AMS record. Relevant: isotopy and differential-topological deformation arguments.