Stiefel–Whitney class

In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. Stiefel–Whitney classes are indexed from 0 to n, where n is the rank of the vector bundle. If the Stiefel–Whitney class of index i is nonzero, then there cannot exist everywhere linearly independent sections of the vector bundle. A nonzero nth Stiefel–Whitney class indicates that every section of the bundle must vanish at some point. A nonzero first Stiefel–Whitney class indicates that the vector bundle is not orientable. For example, the first Stiefel–Whitney class of the Möbius strip, as a line bundle over the circle, is not zero, whereas the first Stiefel–Whitney class of the trivial line bundle over the circle, , is zero. The Stiefel–Whitney class was named for Eduard Stiefel and Hassler Whitney and is an example of a -characteristic class associated to real vector bundles. In algebraic geometry one can also define analogous Stiefel–Whitney classes for vector bundles with a non-degenerate quadratic form, taking values in etale cohomology groups or in Milnor K-theory. As a special case one can define Stiefel–Whitney classes for quadratic forms over fields, the first two cases being the discriminant and the Hasse–Witt invariant . For a real vector bundle E, the Stiefel–Whitney class of E is denoted by w(E). It is an element of the cohomology ring where X is the base space of the bundle E, and (often alternatively denoted by ) is the commutative ring whose only elements are 0 and 1. The component of in is denoted by and called the i-th Stiefel–Whitney class of E. Thus, where each is an element of . The Stiefel–Whitney class is an invariant of the real vector bundle E; i.e., when F is another real vector bundle which has the same base space X as E, and if F is isomorphic to E, then the Stiefel–Whitney classes and are equal.