Multijet transversality theorem

Builds on Smooth manifold Jet Jet bundle Whitney C-infinity topology Transversality Submanifold Residual set Thom transversality theorem

Statement

Let $X$ and $Y$ be smooth manifolds, and let $s \ge 1$ and $k \ge 0$ be integers. Let $X^{(s)} = \set{(x_1, \dots, x_s) \in X^s : x_i \ne x_j \text{ for } i \ne j}$ . Let $J^k(X, Y)$ be the bundle of $k$ -jets of smooth maps from $X$ to $Y$ , with source map $\alpha \colon J^k(X, Y) \to X$ . The $s$ -fold multijet bundle is $J^k_s(X, Y) = \set{(\sigma_1, \dots, \sigma_s) \in J^k(X, Y)^s : (\alpha(\sigma_1), \dots, \alpha(\sigma_s)) \in X^{(s)}}.$ For a smooth map $f \colon X \to Y$ , its multijet extension is $j^k_s f \colon X^{(s)} \to J^k_s(X, Y)$ , $(x_1, \dots, x_s) \mapsto (j^k f(x_1), \dots, j^k f(x_s))$ , where $j^k f(x)$ is the $k$ -jet of $f$ at $x$ . Let $W$ be a submanifold of $J^k_s(X, Y)$ , and give $C^\infty(X, Y)$ the Whitney $C^\infty$ topology. Then the set of $f \in C^\infty(X, Y)$ such that $j^k_s f$ is transverse to $W$ is a residual set in $C^\infty(X, Y)$ .

Remarks

For $s = 1$ this is the Thom transversality theorem. The proof follows the same pattern: one perturbs $f$ near the $s$ distinct points $x_1, \dots, x_s$ independently. The perturbations are polynomial maps of degree at most $k$ , multiplied by bump functions with disjoint supports. The family of perturbed multijets is a submersion onto the fibres, so the parametric transversality theorem applies, and a countable covering argument gives a residual set. The points must be distinct, since the jets of $f$ at the same point cannot be perturbed independently.

Example: let $s = 2$ , $k = 0$ , and let $W = \set{((x_1, y_1), (x_2, y_2)) : y_1 = y_2}$ , which has codimension $\dim Y$ in $J^0_2(X, Y) = X^{(2)} \times Y \times Y$ . If $\dim Y > 2 \dim X = \dim X^{(2)}$ , transversality to $W$ means that $j^0_2 f$ misses $W$ , that is, $f$ is injective. So injective maps are generic when $\dim Y > 2 \dim X$ . Combined with the genericity of immersions, this gives the easy form of the Whitney embedding theorem for compact $X$ .

Going deeper

For a student

The theorem is the main tool for global properties of generic maps. It shows that generic immersions have only transverse self-intersections, that is, normal crossings, and it is a basic tool in the theory of stable maps.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Jet concept
Jet bundle concept
Residual set concept
Smooth manifold concept
Submanifold concept
Transversality concept
Whitney C-infinity topology concept
Thom transversality theorem statement
Multijet transversality theorem statement

Tag	`BJ0X0T4SBA`
URI	`m2://BJ0X0T4SBA`
Kind	statement, theorem
Writing	W0 stub
MSC	57R35, 58A20, 58K30
Status	This copy of the site has no build data. Run `m2 status BJ0X0T4SBA`.
About	Transversality, Jet bundle
Uses	Smooth manifold, Jet, Jet bundle, Whitney C-infinity topology, Transversality, Submanifold, Residual set, Thom transversality theorem