# 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 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, $S^{1}\times \mathbb {R}$, is zero.
The Stiefel–Whitney class was named for Eduard Stiefel and Hassler Whitney and is an example of a $\mathbb {Z} /2\mathbb {Z}$-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
    $H^{\ast }(X;\mathbb {Z} /2\mathbb {Z} )=\bigoplus _{i\geq 0}H^{i}(X;\mathbb {Z} /2\mathbb {Z} )$
where X is the base space of the bundle E, and $\mathbb {Z} /2\mathbb {Z}$ (often alternatively denoted by $\mathbb {Z} _{2}$) is the commutative ring whose only elements are 0 and 1. The component of $w(E)$ in $H^{i}(X;\mathbb {Z} /2\mathbb {Z} )$ is denoted by $w_{i}(E)$ and called the i-th Stiefel–Whitney class of E. Thus,
    $w(E)=w_{0}(E)+w_{1}(E)+w_{2}(E)+\cdots$,
where each $w_{i}(E)$ is an element of $H^{i}(X;\mathbb {Z} /2\mathbb {Z} )$.
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 $\mathrm {id} _{X}\colon X\to X$.) 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 $H^{1}(S^{1};\mathbb {Z} /2\mathbb {Z} )$ 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 $F\to E\to X$.
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 $\pi _{i-1}V_{n-i+1}(F)$ is either infinite-cyclic or isomorphic to $\mathbb {Z} /2\mathbb {Z}$, there is a canonical reduction of the $W_{i}(E)$ classes to classes $w_{i}(E)\in H^{i}(X;\mathbb {Z} /2\mathbb {Z} )$ which are the Stiefel–Whitney classes. Moreover, whenever $\pi _{i-1}V_{n-i+1}(F)=\mathbb {Z} /2\mathbb {Z}$, the two classes are identical. Thus, $w_{1}(E)=0$ if and only if the bundle $E\to X$ 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 $w(E_{1}\oplus E_{2})=w(E_{1})w(E_{2})$ to be true.

## 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 $w(E)\in H^{*}(X;\mathbb {Z} /2\mathbb {Z} )$ 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 $\mathbf {P} ^{1}(\mathbb {R} )$ is nontrivial, i.e., $w(\gamma _{1}^{1})=1+a\in H^{*}(\mathbf {P} ^{1}(\mathbb {R} );\mathbb {Z} /2\mathbb {Z} )=(\mathbb {Z} /2\mathbb {Z} )[a]/(a^{2})$.
* Rank: $w_{0}(E)=1\in H^{0}(X),$ and for i above the rank of E, $w_{i}=0\in H^{i}(X)$, that is, $w(E)\in H^{\leqslant \mathrm {rank} (E)}(X).$
* Whitney product formula: $w(E\oplus F)=w(E)\smile w(F)$, 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 $E\to X$ and map $f\colon X'\to X$, 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
    $Gr_{n}=Gr_{n}(\mathbb {R} ^{\infty })$.
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 W.
Let $f\colon X\to Gr_{n}$, be a continuous map to the infinite Grassmannian. Then, up to isomorphism, the bundle induced by the map f on X
    $f^{*}\gamma ^{n}\in \mathrm {Vect} _{n}(X)$
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 $X\to Gr_{n}$ modulo homotopy equivalence, to the set
    $\mathrm {Vect} _{n}(X)$
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, $w_{j}(f^{*}\gamma ^{n})=f^{*}w_{j}(\gamma ^{n})$. So it suffices in principle to know the values of $w_{j}(\gamma ^{n})$ for all j. However, the cohomology ring $H^{*}(Gr_{n},\mathbb {Z} _{2})$ is free on specific generators $x_{j}\in H^{j}(Gr_{n},\mathbb {Z} _{2})$ arising from a standard cell decomposition, and it then turns out that these generators are in fact just given by $x_{j}=w_{j}(\gamma ^{n})$. 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.

#### The case of line bundles

We now restrict the above construction to line bundles, ie we consider the space, $\mathrm {Vect} _{1}(X)$ of line bundles over X. The Grassmannian of lines $Gr_{1}$ is just the infinite projective space
    $\mathbf {P} ^{\infty }(\mathbf {R} )=\mathbf {R} ^{\infty }/\mathbf {R} ^{*},$
which is doubly covered by the infinite sphere $S^{\infty }$ with antipodal points as fibres. This sphere $S^{\infty }$ is contractible, so we have
    ${\begin{aligned}\pi _{1}(\mathbf {P} ^{\infty }(\mathbf {R} ))&=\mathbf {Z} /2\mathbf {Z} \\\pi _{i}(\mathbf {P} ^{\infty }(\mathbf {R} ))&=\pi _{i}(S^{\infty })=0&&i>1\end{aligned}}$
Hence P$^∞$(R) is the Eilenberg-Maclane space $K(\mathbb {Z} /2\mathbb {Z} ,1)$.
It is a property of Eilenberg-Maclane spaces, that
    $\left[X;\mathbf {P} ^{\infty }(\mathbf {R} )\right]=H^{1}(X;\mathbb {Z} /2\mathbb {Z} )$
for any X, with the isomorphism given by f → f*η, where η is the generator
    $H^{1}(\mathbf {P} ^{\infty }(\mathbf {R} );\mathbf {Z} /2\mathbf {Z} )=\mathbb {Z} /2\mathbb {Z}$.
Applying the former remark that α : [X, Gr$_1$] → Vect$_1$(X) is also a bijection, we obtain a bijection
    $w_{1}\colon {\text{Vect}}_{1}(X)\to H^{1}(X;\mathbf {Z} /2\mathbf {Z} )$
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/2Z), 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/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 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$^k$ 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 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/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 (Note that 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 $i^{*}\gamma ^{1}$ is equal to $\gamma _{1}^{1}$. Thus the first and third axiom imply
    $i^{*}\theta _{1}\left(\gamma ^{1}\right)=\theta _{1}\left(i^{*}\gamma ^{1}\right)=\theta _{1}\left(\gamma _{1}^{1}\right)=w_{1}\left(\gamma _{1}^{1}\right)=w_{1}\left(i^{*}\gamma ^{1}\right)=i^{*}w_{1}\left(\gamma ^{1}\right).$
Since the map
    $i^{*}:H^{1}\left(\mathbf {P} ^{\infty }(\mathbf {R} \right);\mathbf {Z} /2\mathbf {Z} )\to H^{1}\left(\mathbf {P} ^{1}(\mathbf {R} );\mathbf {Z} /2\mathbf {Z} \right)$
is an isomorphism, $\theta _{1}(\gamma ^{1})=w_{1}(\gamma ^{1})$ 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 $f^{*}:H^{*}(X;\mathbf {Z} /2\mathbf {Z} ))\to H^{*}(X';\mathbf {Z} /2\mathbf {Z} )$ 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 $g^{*}\gamma ^{1}$ for some map g, and
    $\theta \left(g^{*}\gamma ^{1}\right)=g^{*}\theta \left(\gamma ^{1}\right)=g^{*}w\left(\gamma ^{1}\right)=w\left(g^{*}\gamma ^{1}\right),$
by naturality. Thus θ = w on ${\text{Vect}}_{1}(X)$. 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 $w_{1}\colon \mathrm {Vect} _{1}(X)\to H^{1}(X;\mathbb {Z} /2\mathbb {Z} )$ 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 $\mathbb {R} ^{n+1}$, the normal bundle $\nu$ to $S^{n}$ is a line bundle. Since $S^{n}$ is orientable, $\nu$ is trivial. The sum $TS^{n}\oplus \nu$ is just the restriction of $T\mathbb {R} ^{n+1}$ to $S^{n}$, which is trivial since $\mathbb {R} ^{n+1}$ is contractible. Hence w(TS$^n$) = w(TS$^n$)w(ν) = w(TS$^n$ ⊕ ν) = 1. But, provided n is even, TS$^n$ → S$^n$ is not trivial; its Euler class $e(TS^{n})=\chi (TS^{n})[S^{n}]=2[S^{n}]\not =0$, where [S$^n$] denotes a fundamental class of S$^n$ 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. 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 (4k+1)-dimensional manifold, $w_{2}w_{4k-1}.$

### Wu classes

The Stiefel–Whitney classes $w_{k}$ are the Steenrod squares of the Wu classes $v_{k}$, defined by Wu Wenjun in 1947. Most simply, the total Stiefel–Whitney class is the total Steenrod square of the total Wu class: $\operatorname {Sq} (v)=w$. 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 $n-k$,
    $v_{k}\cup x=\operatorname {Sq} ^{k}(x)$.
Or more narrowly, we can demand $\langle v_{k}\cup x,\mu \rangle =\langle \operatorname {Sq} ^{k}(x),\mu \rangle$, again for cohomology classes x of degree $n-k$.

## 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 the tangent 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:
    $Sq^{i}(w_{j})=\sum _{t=0}^{i}{j+t-i-1 \choose t}w_{i-t}w_{j+t}.$