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 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).

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

H^\ast(X; \mathbf{Z}/2\mathbf{Z}) = \bigoplus_{i\geq0} H^i(X; \mathbf{Z}/2\mathbf{Z})

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 $i$ -th Stiefel–Whitney class of $E$ . Thus $w (E) = w 0 (E) + w 1 (E) + w 2 (E) + \cdot\cdot\cdot$ , 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 $E \to F$ which covers the identity $id X : X \to X$ .) 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 $L$ .

However, 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. However, 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 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−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) ∈ Hⁱ(X; Z/2Z) which are the Stiefel–Whitney classes. Moreover, whenever π_i−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.

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 $w(E)\in H^{\ast }(X;\mathbf {Z} /2\mathbf {Z} )$ 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 P¹(R) is nontrivial, i.e. $w(\gamma^1_1)= 1 + a \in H^*(\mathbf{P}^1(\mathbf{R}); \mathbf Z_2)= \mathbf Z_2[a]/(a^2)$ .
Rank: w₀(E) = 1 ∈ H⁰(X), and for i above the rank of E, $w_i = 0 \in H^i(X)$ , that is, $w(E) \in H^{\le \mathrm{rank} E}(X)$ .
Whitney product formula: $w(E\oplus F)= w(E) \smallsmile w(F)$ , that is, the Whitney class of a direct sum is the cup product of the summands' classes.
Naturality: w(f*E) = f*w(E) for any real vector bundle E → X and map $f:X'\to X$ , where f*E 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 Gr_n(V) denote the Grassmannian, the space of n-dimensional linear subspaces of V, and denote the infinite Grassmannian

Gr_n = Gr_n(\mathbf{R}^\infty)

Recall that it is equipped with the tautological bundle $\gamma^n \to Gr_n$ , a rank n vector bundle that can be defined as the subbundle of the trivial bundle of fiber V whose fiber at a point $W \in Gr_n (V)$ 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

f^*\gamma^n \in Vect_n(X)

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

[X; Gr_n]

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

Vect_n(X)

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, Vect₁(X) of line bundles over X. The Grassmannian of lines Gr₁ is just the infinite projective space

\mathbf{P}^\infty(\mathbf{R}) = \mathbf{R}^\infty/\mathbf{R}^*

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

\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 > 1 \end{align}

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

It is a property of Eilenberg-Maclane spaces, that

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

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

H^1(\mathbf{P}^\infty(\mathbf{R}); \mathbf{Z}/2\mathbf{Z}) = \mathbf{Z}/2\mathbf{Z}

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

w₁ : Vect₁(X) → H¹(X; Z/2Z);

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

The group of line bundles

If Vect₁(X) is considered as a group under the operation of tensor product, then the Stiefel–Whitney class is an isomorphism: w₁ : Vect₁(X) → H¹(X; Z/2Z). 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.

Properties

Topological interpretation of vanishing

w_i(E) = 0 whenever i > rank(E).
If E^k has $s_1,\ldots,s_{\ell}$ sections which are everywhere linearly independent then the $\ell$ top degree Whitney classes vanish: $w_{k-\ell+1}=\cdots=w_k=0$ .
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(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 H²(M, Z) → H²(M, Z/2Z) (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(γ¹) = 1 + θ₁(γ¹). For the inclusion map i : P¹(R) → P^∞(R), the pullback bundle i*γ¹ is equal to $\gamma_1^1$ . Thus the first and third axiom imply $i^* \theta_1(\gamma^1) = \theta_1(i^* \gamma^1) = \theta_1(\gamma_1^1) = w_1(\gamma_1^1) = w_1(i^* \gamma^1) = i^* w_1(\gamma^1)$ . Since the map i*: H¹(P^∞(R); Z/2Z) → H¹(P¹(R); Z/2Z) is an isomorphism, $\theta_1(\gamma^1) = w_1(\gamma^1)$ and θ(γ¹) = w(γ¹) 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 $f^*:H^*(X; \mathbf{Z}/2\mathbf{Z})) \to H^*(X'; \mathbf{Z}/2\mathbf{Z})$ is injective and $f^* E = \lambda_1 \oplus \cdots \oplus \lambda_n$ for some line bundles $\lambda_i \to X'$ . Any line bundle over X is of the form g*γ¹ for some map g, and θ(g*γ¹) = g*θ(γ¹) = g*w(γ¹) = w(g*γ₁) by naturality. Thus θ = w on $Vect_1(X)$ . It follows from the fourth axiom above that

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).

Since f* 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(X) → H¹(X; Z/2Z) 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(TSⁿ) = w(TSⁿ)w(ν) = w(TSⁿ ⊕ ν) = 1. But TSⁿ → Sⁿ is not trivial; its Euler class $e(TS^n) = \chi(TS^n)[S^n] = 2[S^n] \not =0$ , where [Sⁿ] 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 $w_1^3, w_1 w_2, w_3$ . 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 (4k+1)-dimensional manifold, $w_2w_{4k-1}.$

Wu classes

The Stiefel–Whitney classes w_k are the Steenrod squares of the Wu classes v_k, defined by Wu Wenjun in (Wu 1955). Most simply, the total Stiefel–Whitney class is the total Steenrod square of the total Wu class: 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, $v_k \cup x = Sq^k(x),$ . Or more narrowly, we can demand $\langle v_k \cup x, \mu\rangle = \langle Sq^k(x), \mu \rangle$ , again for cohomology classes x of degree n-k.^[3]

Integral Stiefel–Whitney classes

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

\beta \colon H^{i}(X;\mathbf {Z} /2\mathbf {Z} )\to H^{i+1}(X;\mathbf {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 any fiber bundle) are generated by those of the form $w_{2^i}$ . In particular, the Stiefel–Whitney classes satisfy the Wu formula, named for Wu Wenjun:^[4]

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

References

↑ Pontrjagin, L. S. (1947). "Characteristic cycles on differentiable manifolds". Math. Sbornik N. S. (in Russian). 21 (63): 233–284.
↑ Milnor, J. W.; Stasheff, J. 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)

D. Husemoller, Fibre Bundles, Springer-Verlag, 1994.
May, J. P. (1999), A Concise Course in Algebraic Topology (PDF), U. Chicago Press, Chicago, 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, ISSN 0020-9910, MR 0260844, Zbl 0199.55501, doi:10.1007/BF01425486

External links

Wu class at the Manifold Atlas

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