Thom transversality theorem - Imported Knowlpedia Content

Statement

Let $M,N$ be smooth manifolds, let $r\geq0$ , and let $W$ be a smooth submanifold of the $r$ -jet bundle $J^r(M,N)$ . The Thom transversality theorem states that

\{f\in C^\infty(M,N):j^rf\pitchfork W\}

is residual, hence dense, in the Whitney $C^\infty$ topology. Here $j^rf$ is the $r$ -jet extension of $f$ , and transversality is understood as transversality of a map to a submanifold. If $W$ is closed, this set is also open Hirsch, Chapter 2, Theorem 2.8.

Meaning of genericity

A residual set is a countable intersection of open dense sets. Since the Whitney mapping space is a Baire space, residual conditions are dense, so every smooth map can be approximated by maps whose $r$ -jets are transverse to $W$ . The theorem packages many general-position arguments into a single statement about jet extensions.

Important special cases

For $r=0$ , the jet bundle is $M\times N$ , and choosing $W=M\times Z$ recovers the ordinary transversality theorem for maps $M\to N$ transverse to $Z\subseteq N$ . For $r=1$ , choosing rank-defect strata in the first-jet bundle makes the theorem a starting point for the study of generic singularities of smooth maps Golubitsky–Guillemin, Chapter II, §4.

Scope and cautions

The residual conclusion does not say that every transverse map lies in one prescribed finite-dimensional perturbation family. Parametric transversality supplies that separate statement under its own hypotheses. Openness also needs the stated closedness condition on $W$ ; transversality to a nonclosed submanifold can be destroyed by behavior accumulating outside compact subsets.

References

Morris W. Hirsch, Differential Topology, Springer, 1976. Springer DOI record. Relevant: Chapter 2, §4, especially the Jet Transversality Theorem 2.8.
Martin Golubitsky and Victor Guillemin, Stable Mappings and Their Singularities, Springer, 1973. Springer DOI record. Relevant: Chapter II, jet transversality and singularity strata.