Thom transversality theorem

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.