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 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 w**i(E) and called the iE. 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 w**i(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.

To be precise, provided X is a CW-complex, Whitney defined classes W**i(E) in the i-th cellular cohomology group of X with twisted coefficients. The coefficient system being the (i−1)-st homotopy group of the Stiefel manifold V**n−i+1(F) of (n−i+1) linearly independent vectors in the fibres of E. Whitney proved W**i(E) = 0 if and only if E, when restricted to the i-skeleton of X, has (n−i+1) linearly-independent sections.

Since πi−1V**n−i+1(F) is either infinite-cyclic or isomorphic to Z/2Z, there is a canonical reduction of the W**i(E) classes to classes w**i(E) ∈ H**i(X; Z/2Z) which are the Stiefel–Whitney classes. Moreover, whenever πi−1V**n−i+1(F) = Z/2Z, the two classes are identical. Thus, w1(E) = 0 if and only if the bundle E → X is orientable.

The w0(E) 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 w(E1 ⊕ E2) = w(E1)w(E2) to be true. However, for generalizations of manifolds (namely certain homology manifolds), one can have w0(M) ≠ 1: it only needs to equal 1 mod 8.

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

The uniqueness of these classes is proved for example, in section 17.2 – 17.6 in Husemoller or section 8 in Milnor and Stasheff. There are several proofs of the existence, coming from various constructions, with several different flavours, their coherence is ensured by the unicity statement.

This section describes a construction using the notion of classifying space.

For any vector space V, let Grn(V) denote the Grassmannian, the space of n-dimensional linear subspaces of V, and denote the infinite Grassmannian

Let f : X → Grn, be a continuous map to the infinite Grassmannian. Then, up to isomorphism, the bundle induced by the map f on X

depends only on the homotopy class of the map [f]. The pullback operation thus gives a morphism from the set

of maps X → Grn modulo homotopy equivalence, to the set

of isomorphism classes of vector bundles of rank n over X.

The important fact in this construction is that if X is a paracompact space, this map is a bijection. This is the reason why we call infinite Grassmannians the classifying spaces of vector bundles.

We now restrict the above construction to line bundles, ie we consider the space, Vect1(X) of line bundles over X. The Grassmannian of lines Gr1 is just the infinite projective space

which is doubly covered by the infinite sphere S∞ by antipodal points. This sphere S∞ is contractible, so we have

Hence P∞(R) is the Eilenberg-Maclane space K(Z/2Z, 1).

It is a property of Eilenberg-Maclane spaces, that

for any X, with the isomorphism given by f → *f**η, where η is the generator

Applying the former remark that α : [X, Gr1] → Vect1(X) is also a bijection, we obtain a bijection

this defines the Stiefel–Whitney class w1 for line bundles.

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.

Since the map

The Stiefel–Whitney numbers of the tangent bundle of a smooth manifold are called the Stiefel–Whitney numbers of the manifold. They are known to be cobordism invariants. It was proven by Lev Pontryagin that if B is a smooth compact (n+1)–dimensional manifold with boundary equal to M, then the Stiefel-Whitney numbers of M are all zero.^[1] Moreover, it was proved by René Thom that if all the Stiefel-Whitney numbers of M are zero then M can be realised as the boundary of some smooth compact manifold.^[2]

For instance, the third integral Stiefel–Whitney class is the obstruction to a Spinc structure.