Obstruction theory

In topology, a branch of mathematics, the theory of obstruction or obstacle theory describes the obstacles to the existence of sections in fiber bundles.

Obstruktionskozykel

Be a fibration over a simplicial complex with fiber. We assume that even a section was constructed on the skeleton of and ask if this section can be continued on the skeleton.

For each simplex is homotopy equivalent to the mapping and

Defines an element of the -th homotopy group of the fiber

Clearly, the given section can not be continued when

It can be shown that a local cocycle coefficient, it will be referred to as Obstruktionskozykel. Its cohomology class ( in the cohomology with local coefficients)

Ie -th obstruction class. Although it depends on the selected interface, but you can show that it really depends only on its restriction to the skeleton.

Sections in vector bundles

The most important application of obstruction theory is the question of the existence of linearly independent sections in a vector bundle of rank, for, or, equivalently, the existence of a cut in the frame bundle

Whose fiber is the Stiefel- manifold.

Because for one can construct such a cut on the skeleton, the obstacle to the continuation of the skeleton is then defined above obstruction class

Boots -Whitney classes

The Stiefel- Whitney classes were originally defined by Stiefel -Whitney classes as obstruction. The homotopy group is isomorphic to either (if and straight ) or otherwise infinite cyclic, so it can in any event be shown on surjective. The image of the obstruction class this illustration is the Stiefel- Whitney class

Euler class

For is, for orientable vector bundle is the cohomology with local coefficients and isomorphic to the so- defined obstruction class is the Euler class

Similarly, to define the class for any Euler spheres bundle, thus for the fiber bundle having fiber: because of there is an intersection on the skeleton of the base and the obstruction to the continuation of the skeleton is the Euler class

( In the case of the beam unit of a sphere based vector bundle Euler class of spheres beam having the Euler vector class beam match. )