Transversality theorem

The Transversalitätssatz is declining on René Thom theorem of differential topology, which forms the basis for numerous topological structures such as the Pontryagin - Thom construction, which Kobordismustheorie, surgery theory and the definition of average figures and Verschlingungszahlen.

Set

Let be a differentiable map between differentiable manifolds and a submanifold of. Then, for any strictly positive function (and any metric on ) an approximation of who is transversal.

Comments: A differentiable map is transverse to the submanifold if

Applies. ( In particular, even if. ) Is an illustration of a δ - approximation if

Applies. For sufficiently small δ, each approximation is homotopic to. In particular, therefore follows from the Transversalitätssatz the existence of homotopic to figure who is transversal. There is a for each, so that there will be any δ - approximation of a homotopy between and is, in each, the figure is a ε - approximation of.

Examples

• Is not transverse to the x-axis, but for the image of each transverse to the x-axis.
• If, then it follows from the Transversalitätssatz that there is any figure a δ - approximation whose image is to be disjoint.

Relative version and Homotopietransversalitätssatz

Let be a differentiable map between differentiable manifolds and a submanifold of. Be a submanifold of, and the restriction is to be transversal. Then, for any strictly positive function (and any metric on ) an approximation of which is transverse to and matches up with.

As a special case we obtain the Homotopietransversalitätssatz:

Be differentiable manifolds and a submanifold of. Let be a differentiable map, for and are transversal. Then there is a map that is transverse to and agrees to and with or.

In words: if two transverse images are homotopic, then there is also a transversal homotopy.