Stable normal bundle

The stable normal bundle of a manifold is an important tool in differential topology, a branch of mathematics.

Idea

By the theorem of Whitney each manifold has an embedding into a Euclidean space, for which you can then consider the normal bundle. This embedding is not unique in low Kodimensionen, in sufficiently high Kodimensionen but unique up to isotopy, so that we can define a unique up to isomorphism normal bundle for embedding in high-dimensional Euclidean spaces.

Definition

It should be a differentiable n- manifold with tangent bundle. It should be

The classifying map of the tangent bundle. Herein, the Grassmann manifold, the classifying space for n-dimensional vector bundle.

For embedding the normal bundle has a classifying map

So that the Whitney sum

Is homotopic to a constant map.

It is the infinite-dimensional Grassmann manifold, the classifying space for stable vector bundles. It can be shown that the homotopy the composition is independent of the selected embedding. The classifying map defined by this stable vector bundles is called the stable normal bundle of.