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 dimension of the vector space fiber of the vector bundle. If the Stiefel–Whitney class of index i is nonzero, then there cannot exist (n−i+1) 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, S1×R is zero. 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 dimension of the vector space fiber of the vector bundle. If the Stiefel–Whitney class of index i is nonzero, then there cannot exist (n−i+1) 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, S1×R is zero. The Stiefel–Whitney class was named for Eduard Stiefel and Hassler Whitney and is an example of a Z/2Z-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 (Milnor 1970). 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 here X is the base space of the bundle E, and Z/2Z (often alternatively denoted by Z2) is the commutative ring whose only elements are 0 and 1. The component of w(E) in Hi(X; Z/2Z) is denoted by wi(E) and called the i-th Stiefel–Whitney class of E. Thus w(E) = w0(E) + w1(E) + w2(E) + ⋅⋅⋅, where each wi(E) is an element of Hi(X; Z/2Z). The Stiefel–Whitney class w(E) 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 w(E) and w(F) are equal. (Here isomorphic means that there exists a vector bundle isomorphism E → F which covers the identity idX : X → X.) While it is in general difficult to decide whether two real vector bundles E and F are isomorphic, the Stiefel–Whitney classes w(E) and w(F) can often be computed easily. If they are different, one knows that E and F are not isomorphic. As an example, over the circle S1, there is a line bundle (i.e. a real vector bundle of rank 1) that is not isomorphic to a trivial bundle. This line bundle L is the Möbius strip (which is a fiber bundle whose fibers can be equipped with vector space structures in such a way that it becomes a vector bundle). The cohomology group H1(S1; Z/2Z) has just one element other than 0. This element is the first Stiefel–Whitney class w1(L) of L. Since the trivial line bundle over S1 has first Stiefel–Whitney class 0, it is not isomorphic to L. Two real vector bundles E and F which have the same Stiefel–Whitney class are not necessarily isomorphic. This happens for instance when E and F are trivial real vector bundles of different ranks over the same base space X. It can also happen when E and F have the same rank: the tangent bundle of the 2-sphere S2 and the trivial real vector bundle of rank 2 over S2 have the same Stiefel–Whitney class, but they are not isomorphic. But if two real line bundles over X have the same Stiefel–Whitney class, then they are isomorphic. The Stiefel–Whitney classes wi(E) get their name because Eduard Stiefel and Hassler Whitney discovered them as mod-2 reductions of the obstruction classes to constructing n − i + 1 everywhere linearly independent sections of the vector bundle E restricted to the i-skeleton of X. Here n denotes the dimension of the fibre of the vector bundle F → E → X.