Thom transversality theorem

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.

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.

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.