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 . Introduction General presentation For a real vector bundle , the Stiefel–Whitney class of is denoted by . It is an element of the cohomology ring where is the base space of the bundle , 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 -th Stiefel–Whitney class of . Thus, , where each is an element of . The Stiefel–Whitney class is an invariant of the real vector bundle ; i.e., when is another real vector bundle which has the same base space as , and if is isomorphic to , then the Stiefel–Whitney classes and are equal. (Here isomorphic means that there exists a vector bundle isomorphism which covers the identity .) While it is in general difficult to decide whether two real vector bundles and are isomorphic, the Stiefel–Whitney classes and can often be computed easily. If they are different, one knows that and are not isomorphic. As an example, over the circle , there is a line bundle (i.e.,