Stiefel–Whitney class

For a real vector bundle {{math|E}}, the Stiefel–Whitney class of {{math|E}} is denoted by {{math|w(E)}}. It is an element of the cohomology ring

:$H^\backslash ast(X;\; \backslash Z/2\backslash Z)\; =\; \backslash bigoplus\_\{i\backslash geq0\}\; H^i(X;\; \backslash Z/2\backslash Z)$

where {{math|X}} is the base space of the bundle {{math|E}}, and $\backslash Z/2\backslash Z$ (often alternatively denoted by $\backslash Z\_2$) is the commutative ring whose only elements are 0 and 1. The component of $w(E)$ in $H^i(X;\; \backslash Z/2\backslash Z)$ is denoted by $w\_i(E)$ and called the {{math|i}}-th Stiefel–Whitney class of {{math|E}}. Thus,

:$w(E)\; =\; w\_0(E)\; +\; w\_1(E)\; +\; w\_2(E)\; +\; \backslash cdots$,

where each $w\_i(E)$ is an element of $H^i(X;\; \backslash Z/2\backslash Z)$.

The Stiefel–Whitney class $w(E)$ is an invariant of the real vector bundle {{math|E}}; i.e., when {{math|F}} is another real vector bundle which has the same base space {{math|X}} as {{math|E}}, and if {{math|F}} is isomorphic to {{math|E}}, then the Stiefel–Whitney classes $w(E)$ and $w(F)$ are equal. (Here isomorphic means that there exists a vector bundle isomorphism $E\; \backslash to\; F$ which covers the identity $\backslash mathrm\{id\}\_X\backslash colon\; X\backslash to\; X$.) While it is in general difficult to decide whether two real vector bundles {{math|E}} and {{math|F}} are isomorphic, the Stiefel–Whitney classes $w(E)$ and $w(F)$ can often be computed easily. If they are different, one knows that {{math|E}} and {{math|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 {{math|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 $H^1(S^1;\; \backslash Z/2\backslash Z)$ has just one element other than 0. This element is the first Stiefel–Whitney class $w\_1(L)$ of {{math|L}}. Since the trivial line bundle over $S^1$ has first Stiefel–Whitney class 0, it is not isomorphic to {{math|L}}.

Two real vector bundles {{math|E}} and {{math|F}} which have the same Stiefel–Whitney class are not necessarily isomorphic. This happens for instance when {{math|E}} and {{math|F}} are trivial real vector bundles of different ranks over the same base space {{math|X}}. It can also happen when {{math|E}} and {{math|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 {{math|X}} have the same Stiefel–Whitney class, then they are isomorphic.

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 {{math|E}} restricted to the i-skeleton of X. Here n denotes the dimension of the fibre of the vector bundle $F\backslash to\; E\backslash to\; X$.

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 that $W\_i(E)=0$ if and only if E, when restricted to the i-skeleton of X, has $n-i+1$ linearly-independent sections.

Since $\backslash pi\_\{i-1\}V\_\{n-i+1\}(F)$ is either infinite-cyclic or isomorphic to $\backslash Z/2\backslash Z$, there is a canonical reduction of the $W\_i(E)$ classes to classes $w\_i(E)\; \backslash in\; H^i(X;\; \backslash Z/2\backslash Z)$ which are the Stiefel–Whitney classes. Moreover, whenever $\backslash pi\_\{i-1\}V\_\{n-i+1\}(F)\; =\; \backslash Z/2\backslash Z$, the two classes are identical. Thus, $w\_1(E)\; =\; 0$ if and only if the bundle $E\backslash to\; X$ 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 $w(E\_1\; \backslash oplus\; E\_2)\; =\; w(E\_1)w(E\_2)$ to be true.

Throughout, $H^i(X;\; G)$ denotes singular cohomology of a space {{math|X}} with coefficients in the group {{math|G}}. The word map means always a continuous function between topological spaces.

The Stiefel-Whitney characteristic class $w(E)\backslash in\; H^*(X;\; \backslash Z/2\backslash Z)$ 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:

1. Normalization: The Whitney class of the tautological line bundle over the real projective space $\backslash mathbf\{P\}^1(\backslash R)$ is nontrivial, i.e., $w(\backslash gamma^1\_1)=\; 1\; +\; a\; \backslash in\; H^*(\backslash mathbf\{P\}^1(\backslash R);\; \backslash Z/2\backslash Z)=\; (\backslash Z/2\backslash Z)[a]/(a^2)$.
2. Rank: $w\_0(E)\; =\; 1\; \backslash in\; H^0(X),$ and for i above the rank of E, $w\_i\; =\; 0\; \backslash in\; H^i(X)$, that is, $w(E)\; \backslash in\; H^\{\backslash leqslant\; \backslash mathrm\{rank\}\; (E)\}(X).$
3. Whitney product formula: $w(E\backslash oplus\; F)=\; w(E)\; \backslash smile\; w(F)$, that is, the Whitney class of a direct sum is the cup product of the summands' classes.
4. Naturality: $w(f^*E)\; =\; f^*w(E)$ for any real vector bundle $E\; \backslash to\; X$ and map $f\backslash colon\; X\text{'}\; \backslash to\; X$, 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.

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

:$Gr\_n\; =\; Gr\_n(\backslash R^\backslash infty)$.

Recall that it is equipped with the tautological bundle $\backslash gamma^n\; \backslash to\; Gr\_n,$ a rank n vector bundle that can be defined as the subbundle of the trivial bundle of fiber V whose fiber at a point $W\; \backslash in\; Gr\_n\; (V)$ is the subspace represented by W.

Let $f\backslash colon\; X\; \backslash to\; Gr\_n$, be a continuous map to the infinite Grassmannian. Then, up to isomorphism, the bundle induced by the map f on X

:$f^*\backslash gamma^n\; \backslash in\; \backslash mathrm\{Vect\}\_n(X)$

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 $X\; \backslash to\; Gr\_n$ modulo homotopy equivalence, to the set

:$\backslash mathrm\{Vect\}\_n(X)$

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.)

Now, by the naturality axiom (4) above, $w\_j\; (f^*\backslash gamma^n)=\; f^*\; w\_j\; (\backslash gamma^n)$. So it suffices in principle to know the values of $w\_j\; (\backslash gamma^n)$ for all j. However, the cohomology ring $H^*(Gr\_n,\; \backslash Z\_2)$ is free on specific generators $x\_j\backslash in\; H^j(Gr\_n,\; \backslash Z\_2)$ arising from a standard cell decomposition, and it then turns out that these generators are in fact just given by $x\_j=w\_j\; (\backslash gamma^n)$. Thus, for any rank-n bundle, $w\_j=\; f^*x\_j$, where f is the appropriate classifying map. This in particular provides one proof of the existence of the Stiefel–Whitney classes.

We now restrict the above construction to line bundles, ie we consider the space, $\backslash mathrm\{Vect\}\_1(X)$ of line bundles over X. The Grassmannian of lines $Gr\_1$ is just the infinite projective space

:$\backslash mathbf\{P\}^\backslash infty(\backslash mathbf\{R\})\; =\; \backslash mathbf\{R\}^\backslash infty/\backslash mathbf\{R\}^*,$

which is doubly covered by the infinite sphere $S^\{\backslash infty\}$ with antipodal points as fibres. This sphere $S^\{\backslash infty\}$ is contractible, so we have

:$\backslash begin\{align\}$

\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 $K(\backslash Z/2\backslash Z,\; 1)$.

It is a property of Eilenberg-Maclane spaces, that

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

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

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

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

:$w\_1\backslash colon\; \backslash text\{Vect\}\_1(X)\; \backslash to\; H^1(X;\; \backslash mathbf\{Z\}/2\backslash mathbf\{Z\})$

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. 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.

1. w_i(E) = 0 whenever i > rank(E).
2. If E^k has $s\_1,\backslash ldots,s\_\{\backslash ell\}$ sections which are everywhere linearly independent then the $\backslash ell$ top degree Whitney classes vanish: $w\_\{k-\backslash ell+1\}=\backslash cdots=w\_k=0$.
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(TM) = 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(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.
6. 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 (Note that some Stiefel-Whitney class could still be non-zero, even if all the Stiefel- Whitney numbers vanish!)

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 $i^*\backslash gamma^1$ is equal to $\backslash gamma\_1^1$. Thus the first and third axiom imply

:$i^*\; \backslash theta\_1\; \backslash left\; (\backslash gamma^1\; \backslash right\; )\; =\; \backslash theta\_1\; \backslash left\; (i^*\; \backslash gamma^1\; \backslash right\; )\; =\; \backslash theta\_1\; \backslash left\; (\backslash gamma\_1^1\; \backslash right\; )\; =\; w\_1\; \backslash left\; (\backslash gamma\_1^1\; \backslash right\; )\; =\; w\_1\; \backslash left\; (i^*\; \backslash gamma^1\; \backslash right\; )\; =\; i^*\; w\_1\; \backslash left\; (\backslash gamma^1\; \backslash right\; ).$

Since the map

:$i^*:\; H^1\; \backslash left\; (\backslash mathbf\{P\}^\backslash infty(\backslash mathbf\{R\}\; \backslash right\; );\; \backslash mathbf\{Z\}/2\backslash mathbf\{Z\})\; \backslash to\; H^1\; \backslash left\; (\backslash mathbf\{P\}^1(\backslash mathbf\{R\});\; \backslash mathbf\{Z\}/2\backslash mathbf\{Z\}\; \backslash right\; )$

is an isomorphism, $\backslash theta\_1(\backslash gamma^1)\; =\; w\_1(\backslash gamma^1)$ 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 $f^*:H^*(X;\; \backslash mathbf\{Z\}/2\backslash mathbf\{Z\}))\; \backslash to\; H^*(X\text{'};\; \backslash mathbf\{Z\}/2\backslash mathbf\{Z\})$ is injective and $f^*\; E\; =\; \backslash lambda\_1\; \backslash oplus\; \backslash cdots\; \backslash oplus\; \backslash lambda\_n$ for some line bundles $\backslash lambda\_i\; \backslash to\; X\text{'}$. Any line bundle over X is of the form $g^*\backslash gamma^1$ for some map g, and

:$\backslash theta\; \backslash left\; (g^*\backslash gamma^1\; \backslash right\; )\; =\; g^*\backslash theta\; \backslash left\; (\; \backslash gamma^1\; \backslash right\; )\; =\; g^*\; w\; \backslash left\; (\; \backslash gamma^1\; \backslash right\; )\; =\; w\; \backslash left\; (\; g^*\backslash gamma^1\; \backslash right\; ),$

by naturality. Thus θ = w on $\backslash text\{Vect\}\_1(X)$. It follows from the fourth axiom above that

:$f^*\backslash theta(E)\; =\; \backslash theta(f^*E)\; =\; \backslash theta(\backslash lambda\_1\; \backslash oplus\; \backslash cdots\; \backslash oplus\; \backslash lambda\_n)\; =\; \backslash theta(\backslash lambda\_1)\; \backslash cdots\; \backslash theta(\backslash lambda\_n)\; =\; w(\backslash lambda\_1)\; \backslash cdots\; w(\backslash lambda\_n)\; =\; w(f^*E)\; =\; f^*\; w(E).$

Since $f^*$ is injective, θ = w. Thus the Stiefel–Whitney class is the unique functor satisfying the four axioms above.

Although the map $w\_1\; \backslash colon\; \backslash mathrm\{Vect\}\_1(X)\; \backslash to\; H^1(X;\; \backslash Z/2\backslash Z)$ 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 $S^n$ in $\backslash R^\{n+1\}$, the normal bundle $\backslash nu$ to $S^n$ is a line bundle. Since $S^n$ is orientable, $\backslash nu$ is trivial. The sum $TS^n\; \backslash oplus\; \backslash nu$ is just the restriction of $T\backslash R^\{n+1\}$ to $S^n$, which is trivial since $\backslash R^\{n+1\}$ is contractible. Hence w(TSⁿ) = w(TSⁿ)w(ν) = w(TSⁿ ⊕ ν) = 1. But, provided n is even, TSⁿ → Sⁿ is not trivial; its Euler class $e(TS^n)\; =\; \backslash chi(TS^n)[S^n]\; =\; 2[S^n]\; \backslash not\; =0$, where [Sⁿ] denotes a fundamental class of Sⁿ and χ the Euler characteristic.

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.{{cite journal|title=Characteristic cycles on differentiable manifolds|journal=Mat. Sbornik |series=New Series|volume=21|issue=63|year=1947|pages=233–284|language=Russian|first=Lev S.| last=Pontryagin| author-link= Lev Pontryagin}} 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.{{cite book|first1=John W.|last1=Milnor|authorlink1=John Milnor|first2=James D.|last2=Stasheff|authorlink2=Jim Stasheff|title=Characteristic Classes|url=https://archive.org/details/characteristiccl76miln|url-access=limited|publisher=Princeton University Press|year=1974|ISBN=0-691-08122-0|pages=[https://archive.org/details/characteristiccl76miln/page/n28 50]–53}}

One Stiefel–Whitney number of importance in surgery theory is the de Rham invariant of a (4k+1)-dimensional manifold, $w\_2w\_\{4k-1\}.$

The Stiefel–Whitney classes $w\_k$ are the Steenrod squares of the Wu classes $v\_k$, defined by Wu Wenjun in 1947.{{cite journal

| last = Wu | first = Wen-Tsün | author-link = Wu Wenjun

| journal = Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences

| mr = 19914

| pages = 1139–1141

| title = Note sur les produits essentiels symétriques des espaces topologiques

| volume = 224

| year = 1947}} Most simply, the total Stiefel–Whitney class is the total Steenrod square of the total Wu class: $\backslash operatorname\{Sq\}(v)\; =\; w$. 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 $n-k$,

:$v\_k\; \backslash cup\; x\; =\; \backslash operatorname\{Sq\}^k(x)$.

Or more narrowly, we can demand $\backslash langle\; v\_k\; \backslash cup\; x,\; \backslash mu\backslash rangle\; =\; \backslash langle\; \backslash operatorname\{Sq\}^k(x),\; \backslash mu\; \backslash rangle$, again for cohomology classes x of degree $n-k$.{{cite book|first1=John W.|last1=Milnor|author1-link=John Milnor|first2=James D.|last2=Stasheff|author2-link=Jim Stasheff|title=Characteristic Classes|url=https://archive.org/details/characteristiccl76miln|url-access=limited|publisher=Princeton University Press |year=1974|ISBN=0-691-08122-0|pages=[https://archive.org/details/characteristiccl76miln/page/n69 131]–133}}

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

:$\backslash beta\backslash colon\; H^i(X;\backslash mathbf\{Z\}/2\backslash mathbf\{Z\})\; \backslash to\; H^\{i+1\}(X;\backslash mathbf\{Z\}).$

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

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 $w\_\{2^i\}$. In particular, the Stiefel–Whitney classes satisfy the {{visible anchor|Wu formula}}, named for Wu Wenjun:{{Harv|May|1999|p=197}}

:$Sq^i(w\_j)=\backslash sum\_\{t=0\}^i\; \{j+t-i-1\; \backslash choose\; t\}\; w\_\{i-t\}w\_\{j+t\}.$

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

{{Reflist}}

{{Refbegin}}

• Dale Husemoller, Fibre Bundles, Springer-Verlag, 1994.
• {{citation |last= May |first=J. Peter |author-link=J. Peter May|title=A Concise Course in Algebraic Topology |year=1999 |publisher=University of Chicago Press|location= Chicago |url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |accessdate=2009-08-07}}
• {{Citation | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Algebraic K-theory and quadratic forms | others=With an appendix by J. Tate | mr=0260844 | zbl=0199.55501 | year=1970 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=9 | pages=318–344 | doi=10.1007/BF01425486}}

{{Refend}}

• [http://www.map.mpim-bonn.mpg.de/Wu_class#External_links Wu class] at the Manifold Atlas

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Category:Characteristic classes