Стифель-Уитни сыныптары: Топологиялық инварианттар жиынтығы

Set of topological invariants
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 .

The Stiefel–Whitney classes get their name because Eduard Stiefel and Hassler Whitney discovered them as mod 2 reductions of the obstruction classes to constructing 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
To be precise, provided X is a CW complex, Whitney defined classes in the i th cellular cohomology group of X with twisted coefficients. The coefficient system being the st homotopy group of the Stiefel manifold of linearly independent vectors in the fibres of E. Whitney proved that if and only if E, when restricted to the i skeleton of X, has linearly independent sections. Since is either infinite cyclic or isomorphic to , there is a canonical reduction of the classes to classes which are the Stiefel–Whitney classes. Moreover, whenever , the two classes are identical. Thus, if and only if the bundle is orientable. The class contains no information, because it is equal to 1 by definition. Its creation by Whitney was an act of creative notation, allowing the Whitney sum Formula to be true.

Throughout, denotes singular cohomology of a space X with coefficients in the group G. The word map means always a continuous function between topological spaces.

If Vect1(X) is considered as a group under the operation of tensor product, then the Stiefel–Whitney class, w1 : Vect1(X) → H1(X; Z/2Z), is an isomorphism. That is, w1(λ ⊗ μ) = w1(λ) + w1(μ) for all line bundles λ, μ → X. For example, since H1(S1; Z/2Z) = Z/2Z, there are only two line bundles over the circle up to bundle isomorphism: the trivial one, and the open Möbius strip (i. e., the Möbius strip with its boundary deleted). The same construction for complex vector bundles shows that the Chern class defines a bijection between complex line bundles over X and H2(X; Z), because the corresponding classifying space is P∞(C), a K(Z, 2). This isomorphism is true for topological line bundles, the obstruction to injectivity of the Chern class for algebraic vector bundles is the Jacobian variety.

Although the map is a bijection, the corresponding map is not necessarily injective in higher dimensions. For example, consider the tangent bundle for n even. With the canonical embedding of in , the normal bundle to is a line bundle. Since is orientable, is trivial. The sum is just the restriction of to , which is trivial since is contractible. Hence w(TSn) = w(TSn)w(ν) = w(TSn ⊕ ν) = 1. But, provided n is even, TSn → Sn is not trivial; its Euler class , where [Sn] denotes a fundamental class of Sn and χ the Euler characteristic.

The Stiefel–Whitney classes are the Steenrod squares of the Wu classes , defined by Wu Wenjun in Most simply, the total Stiefel–Whitney class is the total Steenrod square of the total Wu class: Wu classes are most often defined implicitly in terms of Steenrod squares, as the cohomology class representing the Steenrod squares. Let the manifold X be n dimensional. Then, for any cohomology class x of degree ,

Or more narrowly, we can demand , again for cohomology classes x of degree .

Over the Steenrod algebra, the Stiefel–Whitney classes of a smooth manifold (defined as the Stiefel–Whitney classes of the tangent bundle) are generated by those of the form In particular, the Stiefel–Whitney classes satisfy the Wu formula, named for Wu Wenjun: