Be a fiber over a simplicial complex with fiber . We assume that an incision has already been made over the skeleton of and ask whether this incision can be continued onto the skeleton.
is called -th obstruction class. Although it depends on the cut chosen , it can be shown that it really only depends on its restriction to the skeleton.
Sections in vector bundles
The most important application of the obstruction theory is to the question of the existence of linearly independent cuts in a vector bundle of rank , for , or equivalently, the existence of a cut in the - frame bundle
Because of , one can construct such a cut on the -Skeleton , the obstacle to the continuation on the -Skeleton is then the obstruction class defined above
Boots Whitney Classes
The Stiefel-Whitney classes were originally defined as obstruction classes by Stiefel and Whitney. The homotopy group is either isomorphic to (if and is even) or otherwise infinitely cyclic, so it can be mapped to surjectively in each case . The image of the obstruction class below this figure is the Stiefel-Whitney class
.
Euler class
For is , for orientable vector bundles, the cohomology with local coefficients is isomorphic to and the obstruction class defined in this way is the Euler class
.
The Euler class can be defined analogously for any bundle of spheres , i.e. for fiber bundles with fibers : because of there is a cut on the base of the skeleton and the obstruction for the continuation to the skeleton is the Euler class
.
(In the case of the unit sphere bundle of an oriented vector bundle, the Euler class of the spherical bundle coincides with the Euler class of the vector bundle.)
literature
Norman Steenrod: The Topology of Fiber Bundles. (= Princeton Mathematical Series. Vol. 14). Princeton University Press, Princeton, NJ 1951 (Chapters 25, 35, 38)
John W. Milnor, James D. Stasheff: Characteristic classes. In: Annals of Mathematics Studies. No. 76. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo 1974. (Chapter 12)
George W. Whitehead: Elements of Homotopy Theory. (= Graduate Texts in Mathematics. 61). Springer Verlag, 1978, ISBN 1-4612-6320-4 .