Parametric transversality theorem - Imported Knowlpedia Content

Statement

Let $M,P,N$ be finite-dimensional, second-countable smooth manifolds without boundary, let $S\subseteq N$ be an embedded submanifold, and let $F:M\times P\to N$ be a smooth map. Write $F_p(x)=F(x,p)$ . The parametric transversality theorem states that if $F$ is transverse to $S$ , then $F_p$ is transverse to $S$ for almost every $p\in P$ . Equivalently, the exceptional parameters form a measure-zero subset in every coordinate chart of $P$ . In particular, every nonempty open subset of $P$ contains a parameter whose slice is transverse.

Reduction to Sard's theorem

Transversality makes $Z=F^{-1}(S)$ an embedded submanifold of the product manifold $M\times P$ . For the restricted projection $\pi:Z\to P$ , a parameter $p$ is a regular value exactly when $F_p\pitchfork S$ . The conclusion therefore follows by applying Sard's theorem to $\pi$ . This reduction is presented in Guillemin–Pollack, Chapter 2.

Consequences and examples

If $S=\{y\}$ , the theorem says that almost every member of a family is transverse to the point $y$ , hence has $y$ as a regular value. Translation families in Euclidean space give a standard application: once the total evaluation map is a submersion, almost every translate meets a fixed submanifold transversely.

The theorem is a principal device for turning an adjustable finite-dimensional parameter into a generic geometric position.

Conventions and scope

References

Victor Guillemin and Alan Pollack, Differential Topology, AMS Chelsea Publishing, 2010 reprint. DOI record. Relevant: Chapter 2, the transversality theorem with parameters.
Morris W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, 1976. DOI record. Relevant: Chapter 3, parametric transversality and Sard's theorem.