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 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, 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.

Introduction

General presentation

For a real vector bundle, the Stiefel–Whitney class of is denoted by. It is an element of the cohomology ring
here is the base space of the bundle, and cyclic group#Definition| 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. 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 that is not isomorphic to a trivial bundle. This line bundle is the Möbius strip. The cohomology group has just one element other than 0. This element is the first Stiefel–Whitney class of. Since the trivial line bundle over has first Stiefel–Whitney class 0, it is not isomorphic to.
Two real vector bundles and which have the same Stiefel–Whitney class are not necessarily isomorphic. This happens for instance when and are trivial real vector bundles of different ranks over the same base space. It can also happen when and have the same rank: the tangent bundle of the 2-sphere and the trivial real vector bundle of rank 2 over have the same Stiefel–Whitney class, but they are not isomorphic. But if two real line bundles over have the same Stiefel–Whitney class, then they are isomorphic.

Origins

The Stiefel–Whitney classes wi 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 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 Wi in the i-th cellular cohomology group of X with twisted coefficients. The coefficient system being the -st homotopy group of the Stiefel manifold Vn−i+1 of linearly independent vectors in the fibres of E. Whitney proved Wi = 0 if and only if E, when restricted to the i-skeleton of X, has linearly-independent sections.
Since πi−1Vn−i+1 is either infinite-cyclic or isomorphic to Z/2Z, there is a canonical reduction of the Wi classes to classes which are the Stiefel–Whitney classes. Moreover, whenever, the two classes are identical. Thus, w1 = 0 if and only if the bundle E → X is orientable.
The w0 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.

Definitions

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

Axiomatic definition

The Stiefel-Whitney characteristic class 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:
  1. Normalization: The Whitney class of the tautological line bundle over the real projective space P1 is nontrivial, i.e..
  2. Rank: w0 = 1 ∈ H0, and for i above the rank of E,, that is,
  3. Whitney product formula:, that is, the Whitney class of a direct sum is the cup product of the summands' classes.
  4. Naturality: for any real vector bundle E → X and map, where denotes the pullback vector bundle.
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.

Definition ''via'' infinite Grassmannians

The infinite Grassmannians and vector bundles

This section describes a construction using the notion of classifying space.
For any vector space V, let Grn denote the Grassmannian, the space of n-dimensional linear subspaces of V, and denote the infinite Grassmannian
Recall that it is equipped with the tautological bundle a rank n vector bundle that can be defined as the subbundle of the trivial bundle of fiber V whose fiber at a point is the subspace represented by Ẃ.
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 . 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.
Now, by the naturality axiom above,. So it suffices in principle to know the values of for all j. However, the coholomology ring is free on specific generators arising from a standard cell decomposition, and it then turns out that these generators are in fact just given by. Thus, for any rank-n bundle,, where f is the appropriate classifying map. This in particular provides one proof of the existence of the Stiefel-Whitney classes.

The case of line bundles

We now restrict the above construction to line bundles, ie we consider the space, Vect1 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∞ is the Eilenberg-Maclane space K.
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 : → Vect1 is also a bijection, we obtain a bijection
this defines the Stiefel–Whitney class w1 for line bundles.

The group of line bundles

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

Properties

Topological interpretation of vanishing

  1. wi = 0 whenever i > rank.
  2. If Ek has sections which are everywhere linearly independent then the top degree Whitney classes vanish:.
  3. The first Stiefel–Whitney class is zero if and only if the bundle is orientable. In particular, a manifold M is orientable if and only if w1 = 0.
  4. The bundle admits a spin structure if and only if both the first and second Stiefel–Whitney classes are zero.
  5. For an orientable bundle, the second Stiefel–Whitney class is in the image of the natural map H2 → H2 if and only if the bundle admits a spinc structure.
  6. All the Stiefel–Whitney numbers of a smooth compact manifold X vanish if and only if the manifold is the boundary of some smooth compact manifold

    Uniqueness of the Stiefel–Whitney classes

The bijection above for line bundles implies that any functor θ satisfying the four axioms above is equal to w, by the following argument. The second axiom yields θ = 1 + θ1. For the inclusion map i : P1 → P∞, the pullback bundle is equal to. Thus the first and third axiom imply
Since the map
is an isomorphism, and θ = w follow. Let E be a real vector bundle of rank n over a space X. Then E admits a splitting map, i.e. a map f : X′ → X for some space X′ such that is injective and for some line bundles. Any line bundle over X is of the form for some map g, and
by naturality. Thus θ = w on. It follows from the fourth axiom above that
Since is injective, θ = w. Thus the Stiefel–Whitney class is the unique functor satisfying the four axioms above.

Non-isomorphic bundles with the same Stiefel–Whitney classes

Although the map w1 : Vect1 → H1 is a bijection, the corresponding map is not necessarily injective in higher dimensions. For example, consider the tangent bundle TSn for n even. With the canonical embedding of Sn in Rn+1, the normal bundle ν to Sn is a line bundle. Since Sn is orientable, ν is trivial. The sum TSn ⊕ ν is just the restriction of TRn+1 to Sn, which is trivial since Rn+1 is contractible. Hence w = ww = w = 1. But, provided n is even, TSn → Sn is not trivial; its Euler class, where denotes a fundamental class of Sn and χ the Euler characteristic.

Related invariants

Stiefel–Whitney numbers

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. For example, if the manifold has dimension 3, there are three linearly independent Stiefel–Whitney numbers, given by. In general, if the manifold has dimension n, the number of possible independent Stiefel–Whitney numbers is the number of partitions of n.
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 –dimensional manifold with boundary equal to M, then the Stiefel-Whitney numbers of M are all zero. 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.
One Stiefel–Whitney number of importance in surgery theory is the de Rham invariant of a -dimensional manifold,

Wu classes

The Stiefel–Whitney classes wk are the Steenrod squares of the Wu classes vk, defined by Wu Wenjun in. Most simply, the total Stiefel–Whitney class is the total Steenrod square of the total Wu class: Sq = w. 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 n-k, . Or more narrowly, we can demand, again for cohomology classes x of degree n-k.

Integral Stiefel–Whitney classes

The element is called the i + 1 integral Stiefel–Whitney class, where β is the Bockstein homomorphism, corresponding to reduction modulo 2, Z → Z/2Z:
For instance, the third integral Stiefel–Whitney class is the obstruction to a Spinc structure.

Relations over the Steenrod algebra

Over the Steenrod algebra, the Stiefel–Whitney classes of a smooth manifold are generated by those of the form. In particular, the Stiefel–Whitney classes satisfy the , named for Wu Wenjun: