# 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 (*n*−*i*+1) everywhere linearly independent sections of the vector bundle. A nonzero *n*th 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, *S*^{1}×**R** is zero.

The Stiefel–Whitney class was named for Eduard Stiefel and Hassler Whitney and is an example of a **Z**/2**Z**-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).

## Introduction

### General presentation

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**/2**Z** (often alternatively denoted by **Z**_{2}) is the commutative ring whose only elements are 0 and 1. The component of *w*(*E*) in *H ^{i}*(

*X*;

**Z**/2

**Z**) is denoted by

*w*

_{i}(

*E*) and called the

**. Thus**

*i*-th Stiefel–Whitney class of*E**w*(

*E*) =

*w*

_{0}(

*E*) +

*w*

_{1}(

*E*) +

*w*

_{2}(

*E*) + ⋅⋅⋅, where each

*w*(

_{i}*E*) is an element of

*H*(

^{i}*X*;

**Z**/2

**Z**).

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 id_{X} : *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 *S*^{1}, 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 *H*^{1}(*S*^{1}; **Z**/2**Z**) has just one element other than 0. This element is the first Stiefel–Whitney class *w*_{1}(*L*) of *L*. Since the trivial line bundle over *S*^{1} 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 *S*^{2} and the trivial real vector bundle of rank 2 over *S*^{2} 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.

### Origins

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

The *w*_{0}(*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*(*E*_{1} ⊕ *E*_{2}) = *w*(*E*_{1})*w*(*E*_{2}) to be true. However, for generalizations of manifolds (namely certain homology manifolds), one can have *w*_{0}(*M*) ≠ 1: it only needs to equal 1 mod 8.

## Definitions

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.

### 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:

**Normalization:**The Whitney class of the tautological line bundle over the real projective space**P**^{1}(**R**) is nontrivial, i.e. .**Rank:***w*_{0}(*E*) = 1 ∈*H*^{0}(*X*), and for*i*above the rank of*E*, , that is,**Whitney product formula:**, that is, the Whitney class of a direct sum is the cup product of the summands' classes.**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

*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 *Gr _{n}*(

*V*) 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* → *Gr _{n}*, 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* → *Gr _{n}*

*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.

#### The case of line bundles

We now restrict the above construction to line bundles, *ie* we consider the space, Vect_{1}(*X*) of line bundles over *X*. The Grassmannian of lines *Gr*_{1} 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**/2**Z**, 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*, *Gr*_{1}] → Vect_{1}(*X*) is also a bijection, we obtain a bijection

this defines the Stiefel–Whitney class *w*_{1} for line bundles.

#### The group of line bundles

If Vect_{1}(*X*) is considered as a group under the operation of tensor product, then the Stiefel–Whitney class, *w*_{1} : Vect_{1}(*X*) → *H*^{1}(*X*; **Z**/2**Z**), is an isomorphism. That is, *w*_{1}(λ ⊗ μ) = *w*_{1}(λ) + *w*_{1}(μ) for all line bundles λ, μ → *X*.

For example, since *H*^{1}(*S*^{1}; **Z**/2**Z**) = **Z**/2**Z**, 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 *H*^{2}(*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.

## Properties

### Topological interpretation of vanishing

*w*(_{i}*E*) = 0 whenever*i*> rank(*E*).- If
*E*has sections which are everywhere linearly independent then the top degree Whitney classes vanish: .^{k} - 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*w*_{1}(*TM*) = 0. - The bundle admits a spin structure if and only if both the first and second Stiefel–Whitney classes are zero.
- For an orientable bundle, the second Stiefel–Whitney class is in the image of the natural map
*H*^{2}(*M*,**Z**) →*H*^{2}(*M*,**Z**/2**Z**) (equivalently, the so-called third**integral**Stiefel–Whitney class is zero) if and only if the bundle admits a spin^{c}structure. - All the Stiefel–Whitney
*numbers*(see below) of a smooth compact manifold*X*vanish if and only if the manifold is the boundary of some smooth compact (unoriented) manifold (Warning: Some Stiefel-Whitney*class*could still be non-zero, even if all the Stiefel Whitney*numbers*vanish!)

### 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(γ^{1}) = 1 + θ_{1}(γ^{1}). For the inclusion map *i* : **P**^{1}(**R**) → **P**^{∞}(**R**), the pullback bundle is equal to . Thus the first and third axiom imply

Since the map

is an isomorphism, and θ(γ^{1}) = *w*(γ^{1}) 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 *w*_{1} : Vect_{1}(*X*) → *H*^{1}(*X*; **Z**/2**Z**) is a bijection, the corresponding map is not necessarily injective in higher dimensions. For example, consider the tangent bundle *TS ^{n}* for

*n*even. With the canonical embedding of

*S*in

^{n}**R**

^{n+1}, the normal bundle ν to

*S*is a line bundle. Since

^{n}*S*is orientable, ν is trivial. The sum

^{n}*TS*⊕ ν is just the restriction of

^{n}*T*

**R**

^{n+1}to

*S*, which is trivial since

^{n}**R**

^{n+1}is contractible. Hence

*w*(

*TS*) =

^{n}*w*(

*TS*)

^{n}*w*(ν) = w(

*TS*⊕ ν) = 1. But

^{n}*TS*→

^{n}*S*is not trivial; its Euler class , where [

^{n}*S*] denotes a fundamental class of

^{n}*S*and χ the Euler characteristic.

^{n}## 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**/2**Z**-fundamental class of the manifold to give an element of **Z**/2**Z**, 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 (*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 (4*k*+1)-dimensional manifold,

### Wu classes

The Stiefel–Whitney classes *w _{k}* are the Steenrod squares of the

**Wu classes**

*v*, defined by Wu Wenjun in (Wu 1955). Most simply, the total Stiefel–Whitney class is the total Steenrod square of the total Wu class:

_{k}*Sq*(

*v*) =

*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*.[3]

## 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**/2**Z**:

For instance, the third integral Stiefel–Whitney class is the obstruction to a Spin^{c} structure.

### Relations over the Steenrod algebra

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:[4]

## See also

- Characteristic class for a general survey, in particular Chern class, the direct analogue for complex vector bundles
- Real projective space

## References

- Pontryagin, Lev S. (1947). "Characteristic cycles on differentiable manifolds".
*Mat. Sbornik N.S.*(in Russian).**21**(63): 233–284. - Milnor, John W.; Stasheff, James D. (1974).
*Characteristic Classes*. Princeton University Press. pp. 50–53. ISBN 0-691-08122-0. - Milnor, J. W.; Stasheff, J. D. (1974).
*Characteristic Classes*. Princeton University Press. pp. 131–133. ISBN 0-691-08122-0. - (May 1999, p. 197)

- Dale Husemoller,
*Fibre Bundles*, Springer-Verlag, 1994. - May, J. Peter (1999),
*A Concise Course in Algebraic Topology*(PDF), Chicago: University of Chicago Press, retrieved 2009-08-07 - Milnor, John Willard (1970), With an appendix by J. Tate, "Algebraic
*K*-theory and quadratic forms",*Inventiones Mathematicae*,**9**: 318–344, doi:10.1007/BF01425486, ISSN 0020-9910, MR 0260844, Zbl 0199.55501

## External links

- Wu class at the Manifold Atlas