# Stiefel–Whitney class

__: Set of topological invariants__

**Short description**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 [math]\displaystyle{ (n-i+1) }[/math] 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, [math]\displaystyle{ S^1 \times\R }[/math], is zero.

The Stiefel–Whitney class was named for Eduard Stiefel and Hassler Whitney and is an example of a [math]\displaystyle{ \Z/2\Z }[/math]-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

- [math]\displaystyle{ H^\ast(X; \Z/2\Z) = \bigoplus_{i\geq0} H^i(X; \Z/2\Z) }[/math]

here *X* is the base space of the bundle *E*, and [math]\displaystyle{ \Z/2\Z }[/math] (often alternatively denoted by [math]\displaystyle{ \Z_2 }[/math]) is the commutative ring whose only elements are 0 and 1. The component of [math]\displaystyle{ w(E) }[/math] in [math]\displaystyle{ H^i(X; \Z/2\Z) }[/math] is denoted by [math]\displaystyle{ w_i(E) }[/math] and called the ** i-th Stiefel–Whitney class of E**. Thus,

- [math]\displaystyle{ w(E) = w_0(E) + w_1(E) + w_2(E) + \cdots }[/math],

where each [math]\displaystyle{ w_i(E) }[/math] is an element of [math]\displaystyle{ H^i(X; \Z/2\Z) }[/math].

The Stiefel–Whitney class [math]\displaystyle{ w(E) }[/math] 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 [math]\displaystyle{ w(E) }[/math] and [math]\displaystyle{ w(F) }[/math] are equal. (Here *isomorphic* means that there exists a vector bundle isomorphism [math]\displaystyle{ E \to F }[/math] which covers the identity [math]\displaystyle{ \mathrm{id}_X\colon X\to X }[/math].) While it is in general difficult to decide whether two real vector bundles *E* and *F* are isomorphic, the Stiefel–Whitney classes [math]\displaystyle{ w(E) }[/math] and [math]\displaystyle{ w(F) }[/math] can often be computed easily. If they are different, one knows that *E* and *F* are not isomorphic.

As an example, over the circle [math]\displaystyle{ S^1 }[/math], 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 [math]\displaystyle{ H^1(S^1; \Z/2\Z) }[/math] has just one element other than 0. This element is the first Stiefel–Whitney class [math]\displaystyle{ w_1(L) }[/math] of *L*. Since the trivial line bundle over [math]\displaystyle{ S^1 }[/math] 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 [math]\displaystyle{ S^2 }[/math] and the trivial real vector bundle of rank 2 over [math]\displaystyle{ S^2 }[/math] 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 [math]\displaystyle{ w_i(E) }[/math] get their name because Eduard Stiefel and Hassler Whitney discovered them as mod-2 reductions of the obstruction classes to constructing [math]\displaystyle{ n-i+1 }[/math] 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 [math]\displaystyle{ F\to E\to X }[/math].

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

Since [math]\displaystyle{ \pi_{i-1}V_{n-i+1}(F) }[/math] is either infinite-cyclic or isomorphic to [math]\displaystyle{ \Z/2\Z }[/math], there is a canonical reduction of the [math]\displaystyle{ W_i(E) }[/math] classes to classes [math]\displaystyle{ w_i(E) \in H^i(X; \Z/2\Z) }[/math] which are the Stiefel–Whitney classes. Moreover, whenever [math]\displaystyle{ \pi_{i-1}V_{n-i+1}(F) = \Z/2\Z }[/math], the two classes are identical. Thus, [math]\displaystyle{ w_1(E) = 0 }[/math] if and only if the bundle [math]\displaystyle{ E\to X }[/math] is orientable.

The [math]\displaystyle{ w_0(E) }[/math] 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 [math]\displaystyle{ w(E_1 \oplus E_2) = w(E_1)w(E_2) }[/math] to be true.

## Definitions

Throughout, [math]\displaystyle{ H^i(X; G) }[/math] 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 [math]\displaystyle{ w(E)\in H^*(X; \Z/2\Z) }[/math] 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 [math]\displaystyle{ \mathbf{P}^1(\R) }[/math] is nontrivial, i.e., [math]\displaystyle{ w(\gamma^1_1)= 1 + a \in H^*(\mathbf{P}^1(\R); \Z/2\Z)= (\Z/2\Z)[a]/(a^2) }[/math].**Rank:**[math]\displaystyle{ w_0(E) = 1 \in H^0(X), }[/math] and for*i*above the rank of*E*, [math]\displaystyle{ w_i = 0 \in H^i(X) }[/math], that is, [math]\displaystyle{ w(E) \in H^{\leqslant \mathrm{rank} (E)}(X). }[/math]**Whitney product formula:**[math]\displaystyle{ w(E\oplus F)= w(E) \smallsmile w(F) }[/math], that is, the Whitney class of a direct sum is the cup product of the summands' classes.**Naturality:**[math]\displaystyle{ w(f^*E) = f^*w(E) }[/math] for any real vector bundle [math]\displaystyle{ E \to X }[/math] and map [math]\displaystyle{ f\colon X' \to X }[/math], where [math]\displaystyle{ f^*E }[/math] 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 [math]\displaystyle{ Gr_n(V) }[/math] denote the Grassmannian, the space of *n*-dimensional linear subspaces of *V*, and denote the infinite Grassmannian

- [math]\displaystyle{ Gr_n = Gr_n(\R^\infty) }[/math].

Recall that it is equipped with the tautological bundle [math]\displaystyle{ \gamma^n \to Gr_n, }[/math] a rank *n* vector bundle that can be defined as the subbundle of the trivial bundle of fiber *V* whose fiber at a point [math]\displaystyle{ W \in Gr_n (V) }[/math] is the subspace represented by *Ẃ*.

Let [math]\displaystyle{ f\colon X \to Gr_n }[/math], be a continuous map to the infinite Grassmannian. Then, up to isomorphism, the bundle induced by the map *f* on *X*

- [math]\displaystyle{ f^*\gamma^n \in \mathrm{Vect}_n(X) }[/math]

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

- [math]\displaystyle{ [X; Gr_n] }[/math]

of maps [math]\displaystyle{ X \to Gr_n }[/math] *modulo* homotopy equivalence, to the set

- [math]\displaystyle{ \mathrm{Vect}_n(X) }[/math]

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.)

Now, by the naturality axiom (4) above, [math]\displaystyle{ w_j (f^*\gamma^n)= f^* w_j (\gamma^n) }[/math]. So it suffices in principle to know the values of [math]\displaystyle{ w_j (\gamma^n) }[/math] for all *j*. However, the coholomology ring [math]\displaystyle{ H^*(Gr_n, \Z_2) }[/math] is free on specific generators [math]\displaystyle{ x_j\in H^j(Gr_n, \Z_2) }[/math] arising from a standard cell decomposition, and it then turns out that these generators are in fact just given by [math]\displaystyle{ x_j=w_j (\gamma^n) }[/math]. Thus, for any rank-n bundle, [math]\displaystyle{ w_j= f^*x_j }[/math], 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, [math]\displaystyle{ \mathrm{Vect}_1(X) }[/math] of line bundles over *X*. The Grassmannian of lines [math]\displaystyle{ Gr_1 }[/math] is just the infinite projective space

- [math]\displaystyle{ \mathbf{P}^\infty(\mathbf{R}) = \mathbf{R}^\infty/\mathbf{R}^*, }[/math]

which is doubly covered by the infinite sphere [math]\displaystyle{ S^{\infty} }[/math] by antipodal points. This sphere [math]\displaystyle{ S^{\infty} }[/math] is contractible, so we have

- [math]\displaystyle{ \begin{align} \pi_1(\mathbf{P}^\infty(\mathbf{R})) &= \mathbf{Z}/2\mathbf{Z} \\ \pi_i(\mathbf{P}^\infty(\mathbf{R})) &= \pi_i(S^\infty) = 0 && i \gt 1 \end{align} }[/math]

Hence **P**^{∞}(**R**) is the Eilenberg-Maclane space [math]\displaystyle{ K(\Z/2\Z, 1) }[/math].

It is a property of Eilenberg-Maclane spaces, that

- [math]\displaystyle{ \left [X; \mathbf{P}^\infty(\mathbf{R}) \right ] = H^1(X; \Z/2\Z) }[/math]

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

- [math]\displaystyle{ H^1(\mathbf{P}^\infty(\mathbf{R}); \mathbf{Z}/2\mathbf{Z}) = \Z/2\Z }[/math].

Applying the former remark that α : [*X*, *Gr*_{1}] → Vect_{1}(*X*) is also a bijection, we obtain a bijection

- [math]\displaystyle{ w_1\colon \text{Vect}_1(X) \to H^1(X; \mathbf{Z}/2\mathbf{Z}) }[/math]

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 [math]\displaystyle{ s_1,\ldots,s_{\ell} }[/math] sections which are everywhere linearly independent then the [math]\displaystyle{ \ell }[/math] top degree Whitney classes vanish: [math]\displaystyle{ w_{k-\ell+1}=\cdots=w_k=0 }[/math].^{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 [math]\displaystyle{ i^*\gamma^1 }[/math] is equal to [math]\displaystyle{ \gamma_1^1 }[/math]. Thus the first and third axiom imply

- [math]\displaystyle{ i^* \theta_1 \left (\gamma^1 \right ) = \theta_1 \left (i^* \gamma^1 \right ) = \theta_1 \left (\gamma_1^1 \right ) = w_1 \left (\gamma_1^1 \right ) = w_1 \left (i^* \gamma^1 \right ) = i^* w_1 \left (\gamma^1 \right ). }[/math]

Since the map

- [math]\displaystyle{ i^*: H^1 \left (\mathbf{P}^\infty(\mathbf{R} \right ); \mathbf{Z}/2\mathbf{Z}) \to H^1 \left (\mathbf{P}^1(\mathbf{R}); \mathbf{Z}/2\mathbf{Z} \right ) }[/math]

is an isomorphism, [math]\displaystyle{ \theta_1(\gamma^1) = w_1(\gamma^1) }[/math] 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 [math]\displaystyle{ f^*:H^*(X; \mathbf{Z}/2\mathbf{Z})) \to H^*(X'; \mathbf{Z}/2\mathbf{Z}) }[/math] is injective and [math]\displaystyle{ f^* E = \lambda_1 \oplus \cdots \oplus \lambda_n }[/math] for some line bundles [math]\displaystyle{ \lambda_i \to X' }[/math]. Any line bundle over *X* is of the form [math]\displaystyle{ g^*\gamma^1 }[/math] for some map *g*, and

- [math]\displaystyle{ \theta \left (g^*\gamma^1 \right ) = g^*\theta \left ( \gamma^1 \right ) = g^* w \left ( \gamma^1 \right ) = w \left ( g^*\gamma^1 \right ), }[/math]

by naturality. Thus θ = *w* on [math]\displaystyle{ \text{Vect}_1(X) }[/math]. It follows from the fourth axiom above that

- [math]\displaystyle{ f^*\theta(E) = \theta(f^*E) = \theta(\lambda_1 \oplus \cdots \oplus \lambda_n) = \theta(\lambda_1) \cdots \theta(\lambda_n) = w(\lambda_1) \cdots w(\lambda_n) = w(f^*E) = f^* w(E). }[/math]

Since [math]\displaystyle{ f^* }[/math] 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 [math]\displaystyle{ w_1 \colon \mathrm{Vect}_1(X) \to H^1(X; \Z/2\Z) }[/math] is a bijection, the corresponding map is not necessarily injective in higher dimensions. For example, consider the tangent bundle [math]\displaystyle{ TS^n }[/math] for *n* even. With the canonical embedding of [math]\displaystyle{ S^n }[/math] in [math]\displaystyle{ \R^{n+1} }[/math], the normal bundle [math]\displaystyle{ \nu }[/math] to [math]\displaystyle{ S^n }[/math] is a line bundle. Since [math]\displaystyle{ S^n }[/math] is orientable, [math]\displaystyle{ \nu }[/math] is trivial. The sum [math]\displaystyle{ TS^n \oplus \nu }[/math] is just the restriction of [math]\displaystyle{ T\R^{n+1} }[/math] to [math]\displaystyle{ S^n }[/math], which is trivial since [math]\displaystyle{ \R^{n+1} }[/math] is contractible. Hence *w*(*TS ^{n}*) =

*w*(

*TS*)

^{n}*w*(ν) = w(

*TS*⊕ ν) = 1. But, provided n is even,

^{n}*TS*→

^{n}*S*is not trivial; its Euler class [math]\displaystyle{ e(TS^n) = \chi(TS^n)[S^n] = 2[S^n] \not =0 }[/math], 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 [math]\displaystyle{ w_1^3, w_1 w_2, w_3 }[/math]. 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, [math]\displaystyle{ w_2w_{4k-1}. }[/math]

### Wu classes

The Stiefel–Whitney classes [math]\displaystyle{ w_k }[/math] are the Steenrod squares of the **Wu classes** [math]\displaystyle{ w_k }[/math], defined by Wu Wenjun in (Wu 1955). Most simply, the total Stiefel–Whitney class is the total Steenrod square of the total Wu class: [math]\displaystyle{ \operatorname{Sq}(v) = w }[/math]. 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 [math]\displaystyle{ n-k }[/math],

- [math]\displaystyle{ v_k \cup x = \operatorname{Sq}^k(x) }[/math].

Or more narrowly, we can demand [math]\displaystyle{ \langle v_k \cup x, \mu\rangle = \langle \operatorname{Sq}^k(x), \mu \rangle }[/math], again for cohomology classes *x* of degree [math]\displaystyle{ n-k }[/math].^{[3]}

## Integral Stiefel–Whitney classes

The element [math]\displaystyle{ \beta w_i \in H^{i+1}(X;\mathbf{Z}) }[/math] is called the *i* + 1 *integral* Stiefel–Whitney class, where β is the Bockstein homomorphism, corresponding to reduction modulo 2, **Z** → **Z**/2**Z**:

- [math]\displaystyle{ \beta\colon H^i(X;\mathbf{Z}/2\mathbf{Z}) \to H^{i+1}(X;\mathbf{Z}). }[/math]

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 [math]\displaystyle{ w_{2^i} }[/math]. In particular, the Stiefel–Whitney classes satisfy the **Wu formula**, named for Wu Wenjun:^{[4]}

- [math]\displaystyle{ Sq^i(w_j)=\sum_{t=0}^i {j+t-i-1 \choose t} w_{i-t}w_{j+t}. }[/math]

## 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" (in Russian).
*Mat. Sbornik*. New Series**21**(63): 233–284. - ↑ Milnor, John W.; Stasheff, James D. (1974).
*Characteristic Classes*. Princeton University Press. pp. 50–53. ISBN 0-691-08122-0. https://archive.org/details/characteristiccl76miln. - ↑ Milnor, John W.; Stasheff, James D. (1974).
*Characteristic Classes*. Princeton University Press. pp. 131–133. ISBN 0-691-08122-0. https://archive.org/details/characteristiccl76miln. - ↑ (May 1999)

- Dale Husemoller,
*Fibre Bundles*, Springer-Verlag, 1994. - May, J. Peter (1999),
*A Concise Course in Algebraic Topology*, Chicago: University of Chicago Press, http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf, 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

## External links

- Wu class at the Manifold Atlas

Original source: https://en.wikipedia.org/wiki/Stiefel–Whitney class.
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