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.
A nonzero nth Stiefel–Whitney class indicates that every section of the bundle must vanish at some point.
The Stiefel–Whitney class was named for Eduard Stiefel and Hassler Whitney and is an example of a
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.
It is an element of the cohomology ring where X is the base space of the bundle E, and
While it is in general difficult to decide whether two real vector bundles E and F are isomorphic, the Stiefel–Whitney classes
This happens for instance when E and F are trivial real vector bundles of different ranks over the same base space X.
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.
To be precise, provided X is a CW-complex, Whitney defined classes
in the i-th cellular cohomology group of X with twisted coefficients.
linearly independent vectors in the fibres of E. Whitney proved that
of a finite rank real vector bundle E on a paracompact base space X is defined as the unique class such that the following axioms are fulfilled: 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.
The pullback operation thus gives a morphism from the set of maps
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.)
arising from a standard cell decomposition, and it then turns out that these generators are in fact just given by
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.
The bijection above for line bundles implies that any functor θ satisfying the four axioms above is equal to w, by the following argument.
is injective, θ = w. Thus the Stiefel–Whitney class is the unique functor satisfying the four axioms above.
is a bijection, the corresponding map is not necessarily injective in higher dimensions.
But, provided n is even, TSn → Sn is not trivial; its Euler class
If we work on a manifold of dimension n, then any product of Stiefel–Whitney classes of total degree n can be paired with the Z/2Z-fundamental class of the manifold to give an element of Z/2Z, a Stiefel–Whitney number of the vector bundle.
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] One Stiefel–Whitney number of importance in surgery theory is the de Rham invariant of a (4k+1)-dimensional manifold,