nLab Thom's transversality theorem

Statement

Theorem

Let $P$ be a smooth manifold and let $i \colon M \hookrightarrow E$ be a smooth submanifold. Then the topological subspace

$$C^{\infty}_{tr i}(P,E) \subset C^\infty(P,E)$$

of smooth functions that are transverse maps to $i$ is a dense subspace.

This is due to Thom 54, see e.g. Kosinski 93, IV (2.5).

Relate entries

• Thom's theorem

• Sard's theorem

References

The result is due to (Theorem I.5?)

• René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, (1954). 17-86 (doi:10.1007/BF02566923, dml:139072, digiz:GDZPPN002056259, pdf)

Textbook account:

• Antoni Kosinski, chapter IV.2 of Differential manifolds, Academic Press 1993 (pdf)

Lecture notes:

• John Francis, Topology of manifolds course notes (2010) (web)

  Lecture 4 Transversality (notes by I. Bobkova) (pdf)