nLab
Thom's transversality theorem

Statement

Theorem

Let $P$ be a smooth manifold and let $i:M\hookrightarrow E$ be a smooth submanifold. The subspace

$$C_{\mathrm{tr}i}^{\mathrm{\infty }}(P,E)\subset C^{\mathrm{\infty }}(P,E)$$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).

References

The result is due to

• René Thom, Quelques propriétés globales des variétés différentiables Comment. Math. Helv. 28, (1954). 17-86

A review is in

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

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