Obstruction
G-bundles are classified by the classifying space BG, and similarly H-bundles are classified by the classifying space BH, and the induced G-structure on an H-bundle corresponds to the induced map . Thus given a G-bundle with classifying map, the obstruction to the reduction of the structure group is the class of as a map to the cofiber ; the structure group can be reduced if and only if the class of is null-homotopic.
When is a homotopy equivalence, the cofiber is contractible, so there is no obstruction to reducing the structure group, for example for .
Conversely, the cofiber induced by the inclusion of the trivial group is again, so the obstruction to an absolute parallelism (trivialization of the bundle) is the class of the bundle.
Read more about this topic: Reduction Of The Structure Group
Famous quotes containing the word obstruction:
“The short lesson that comes out of long experience in political agitation is something like this: all the motive power in all of these movements is the instinct of religious feeling. All the obstruction comes from attempting to rely on anything else. Conciliation is the enemy.”
—John Jay Chapman (1862–1933)
“Every obstruction of the course of justice,—is a door opened to betray society, and bereave us of those blessings which it has in view.... It is a strange way of doing honour to God, to screen actions which are a disgrace to humanity.”
—Laurence Sterne (1713–1768)