Whitney C-infinity topology - Imported Knowlpedia Content

Definition

For smooth manifolds $M$ and $N$ , the Whitney $C^\infty$ topology on $C^\infty(M,N)$ is the topology generated by simultaneous, locally finite control of the derivatives of maps. Precisely, for each finite $r$ , one gives $C^\infty(M,N)$ the strong Whitney $C^r$ topology by requiring the $r$ -jet $j^rf:M\to J^r(M,N)$ to lie in a prescribed open subset of the jet bundle, and then takes the topology generated by all $r$ . The allowed tolerances may vary over $M$ , so the definition is stronger than uniform control on a fixed compact set.

Neighborhoods and convergence

Using charts and a locally finite family of compact sets, a basic neighborhood constrains finitely many derivatives on every member of that family, with bounds that may depend on the member. When $M$ is compact, the strong and weak $C^\infty$ topologies agree. For noncompact $M$ , convergence uniformly on each compact set generally does not imply strong Whitney convergence.

Role in differential topology

This topology is the standard setting for stability and genericity results involving global perturbations. Transversality theorems often assert that a specified class of maps is residual, dense, or open in an appropriate Whitney topology; the precise choice of strong or weak topology matters on noncompact source manifolds.

Conventions and scope

References

Morris W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, 1976. DOI record. Relevant: Chapter 2, weak and strong topologies on mapping spaces.
Martin Golubitsky and Victor Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics 14, Springer, 1973. DOI record. Relevant: Chapter II, Whitney topologies and stability.