Manifolds and cell complexes Differential topology: smooth manifolds and maps Theorem

Thom transversality theorem

Builds on Smooth manifold Submanifold Transversality Residual set

Statement

Let $X$ and $Y$ be smooth manifolds, let $k \ge 0$ , and let $J^k(X, Y)$ be the jet bundle of $k$ -jets of smooth maps from $X$ to $Y$ . For a smooth map $f \colon X \to Y$ , let $j^k f \colon X \to J^k(X, Y)$ be its $k$ -jet extension, which sends $x$ to the $k$ -jet of $f$ at $x$ . Let $W$ be an embedded submanifold of $J^k(X, Y)$ . Give $C^\infty(X, Y)$ the Whitney $C^\infty$ topology. Then the set of maps $f \in C^\infty(X, Y)$ such that $j^k f$ is transverse to $W$ is a residual set in $C^\infty(X, Y)$ , and in particular it is dense.

Remarks

For $k = 0$ one has $J^0(X, Y) = X \times Y$ , and with $W = X \times Z$ the theorem says that maps transverse to a submanifold $Z \subseteq Y$ are generic. The space $C^\infty(X, Y)$ with the Whitney topology is a Baire space, so a residual set is dense. The proof reduces to the parametric transversality theorem. If a smooth family $F \colon X \times S \to Y$ is transverse to $Z$ , then $F(\cdot, s)$ is transverse to $Z$ for almost every $s$ . This follows from Sard's theorem applied to the projection $F^{-1}(Z) \to S$ . Locally one adds polynomial terms of degree at most $k$ to $f$ , which moves $j^k f$ in all directions of the fibre. Example: let $k = 1$ and let $W$ be the set of $1$ -jets whose linear part has rank $r$ . The theorem shows that for a generic $f$ the set of points where $df$ has rank $r$ is a submanifold of the expected codimension. This is how Whitney's results on generic singularities are obtained.

Going deeper

For a student

The theorem is the source of most genericity statements in differential topology and singularity theory. With $k = 0$ it gives transversality of maps to submanifolds, which is used to define intersection numbers and the Pontryagin–Thom construction. With $k = 1$ it shows that generic maps have only mild singularities, for instance that Morse functions are dense and that generic maps $M^n \to \R^{2n}$ are immersions. With higher $k$ it is the starting point of the classification of stable singularities.

Reading path

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Tag	`KWNRS3RA09`
URI	`m2://KWNRS3RA09`
Kind	statement, theorem
Writing	W0 stub
MSC	57R35, 58A20, 57R45
Status	This copy of the site has no build data. Run `m2 status KWNRS3RA09`.
About	Transversality
Uses	Smooth manifold, Submanifold, Transversality, Residual set
Changed	2 November 2026