Stiefel–Whitney class

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In mathematics, the Stiefel–Whitney class arises as a type of characteristic class associated to real vector bundles $E\rightarrow X$ . It is denoted by w(E), taking values in $H^*(X; \Z/2\Z)$ , the cohomology groups with mod 2 coefficients. 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 $E$ , so that $w(E) = w_0(E) + w_1(E) + w_2(E) + \cdots$ . As an example, over the circle, $S 1$ , there is a line bundle that is topologically non-trivial: that is, the line bundle associated to the Möbius band, usually thought of as having fibres $[0,1]$ . The cohomology group

$H^1(S^1;\mathbb Z/2\mathbb Z)$

has just one element other than 0, this element being the first Stiefel-Whitney class, $w 1$ , of that line bundle.

[edit] Origins

The Stiefel-Whitney classes $w i (E)$ get their name because Stiefel and 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 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 $\Bbb Z_2$ , there is a canonical reduction of the $W i (E)$ classes to classes $w_i(E) \in H^i(X;\Bbb Z_2)$ which are the Stiefel-Whitney classes. Moreover, whenever $\pi_{i-1} V_{n-i+1}(F) = \Bbb Z_2$ , 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 is exceptional and has no meaning. 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.

[edit] Axioms

Throughout, $H i (X; G)$ denotes singular cohomology of a space $X$ with coefficients in the group $G$ .

Naturality: $w (f * E) = f * w (E)$ for any bundle $E \to X$ and map $f:X' \to X$ , where $f * E$ denotes the induced bundle.
$w 0 (E) = 1$ in $H^0(X;\mathbb Z/2\mathbb Z)$ .
$w 1 (γ 1)$ is the generator of $H^1(\mathbb RP^1;\mathbb Z/2\mathbb Z)\cong\mathbb Z/2\mathbb Z$ (normalization condition). Here, $γ n$ is the canonical line bundle.
$w(E\oplus F)= w(E) \smallsmile w(F)$ (Whitney product formula).

Some work is required to show that such classes do indeed exist and are unique (at least for paracompact spaces X); see section 17.2 and 17.3 in Husemoller or section 8 in Milnor and Stasheff.

[edit] Line bundles

Let $X$ be a paracompact space, and let $V e c t n (X)$ denote the set of real vector bundles over X of dimension n for some fixed positive integer $n$ . For any vector space V, let $G r n (V)$ denote the Grassmannian $Gr_n(V) = \{W\subset V:\, \dim W = n\}$ . Set $Gr_n = Gr_n(\R^\infty)$ . Define the tautological bundle $\gamma^n \to Gr_n$ by $\gamma^n = \{(W, x):\, W\in Gr_n, x\in W\}$ ; this is a real bundle of dimension n, with projection $\gamma^n \to Gr_n$ given by $(W, x) \to W$ . For any map $f:X \to Gr_n$ , the induced bundle $f^*\gamma^n \in Vect_n(X)$ . Since any two homotopic maps $f, g: X \to Gr_n$ have $f * γ n$ and $g * γ n$ isomorphic, the map $\alpha:[X; Gr_n] \to Vect_n(X)$ given by $f \to f^* \gamma^n$ is well-defined, where $[X; G r n]$ denotes the set of homotopy equivalence classes of maps $X \to Gr_n$ . It's not difficult to prove that this map $α$ is actually an isomorphism (see Sections 3.5 and 3.6 in Husemoller, for example). As a result, $G r n$ is called the classifying space of real n-bundles.

Now consider the space $V e c t 1 (X)$ of line bundles over $X$ . For $n = 1$ , the Grassmannian $G r 1$ is just $\R P^\infty = \R^\infty/\R^* = S^\infty/(\Z/2\Z)$ , where the nonzero element of $\Z/2\Z$ acts by $x \to -x$ . The quotient map $S^\infty \to S^\infty/(\Z/2\Z) = \R P^\infty$ is therefore a double cover. Since $S^\infty$ is contractible, we have $\pi_i(\R P^\infty) = \pi_i(S^\infty) = 0$ for $i > 1$ and $\#\pi_1(\R P^\infty) = 2$ ; that is, $\pi_1(\R P^\infty) = \Z/2\Z$ . Hence $\R P^\infty$ is the Eilenberg-Maclane space $K(\Z/2\Z, 1)$ . Hence $[X; Gr_1] = H^1(X; \Z/2\Z)$ for any $X$ , with the isomorphism given by $f \to f^* \eta$ , where $η$ is the generator $H^1(\R P^\infty; \Z/2\Z) = \Z/2\Z$ . Since $\alpha:[X, Gr_1] \to Vect_1(X)$ is also a bijection, we have another bijection $w_1:Vect_1 \to H^1(X; \Z/2\Z)$ . This map $w 1$ is precisely the Stiefel-Whitney class $w 1$ for a line bundle. (Since the corresponding classifying space $C P^\infty$ for complex bundles is a $K(\Z, 2)$ , the same argument shows that the Chern class defines a bijection between complex line bundles over $X$ and $H^2(X; \Z)$ .) 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). If $V e c t 1 (X)$ is considered as a group under the operation of tensor product, then $α$ is an isomorphism: $w_1(\lambda \otimes \mu) = w_1(\lambda) + w_1(\mu)$ for all line bundles $\lambda, \mu \to X$ .

[edit] Higher dimensions

The bijection above for line bundles implies that any functor $θ$ satisfying the four axioms above is equal to w. Let $\xi \to X$ be an n-bundle. Then $ξ$ admits a splitting map, a map $f:X' \to X$ for some space $X'$ such that $f^*:H^*(X; \Z/2\Z) \to H^*(X'; \Z/2\Z)$ is injective and $f^*\xi = \lambda_1 \oplus \cdots \oplus \lambda_n$ for some line bundles $\lambda_i \to X'$ . Any line bundle over X is of the form $g * γ 1$ for some map g, and $θ(g * γ 1) = g * θ(γ 1) = 1 + w 1 (g * γ 1)$ by naturality. Thus $θ = w$ on $V e c t 1 (X)$ . It follows from the fourth axiom above that

$\scriptstyle f^*\theta(\xi) = \theta(f^*\xi) = \theta(\lambda_1 \oplus \cdots \oplus \lambda_n) = \theta(\lambda_1) \cdots \theta(\lambda_n) = (1 + w_1(\lambda_1)) \cdots (1 + w_1(\lambda_n)) = w(\lambda_1) \cdots w(\lambda_n) = w(f^*\xi) = f^* w(\xi).$

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

Although the map $w_1:Vect_1(X) \to H^1(X; \Z/2\Z)$ is a bijection, the corresponding map is not necessarily injective in higher dimensions. For example, consider the tangent bundle $T S n$ for $n$ even. With the canonical embedding of $S n$ in $\R^{n+1}$ , the normal bundle $ν$ to $S n$ is a line bundle. Since $S n$ is orientable, $ν$ is trivial. The sum $TS^n \oplus \nu$ is just the restriction of $T\R^{n+1}$ to $S n$ , which is trivial since $\R^{n+1}$ is contractible. Hence $w(TS^n) = w(TS^n)w(\nu) = w(TS^n \oplus \nu) = 1$ . But $TS^n \to S^n$ 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.

[edit] 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 $\scriptstyle \mathbb{Z}_2$ -fundamental class of the manifold to give an element of $\scriptstyle \mathbb{Z}_2$ , 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 $\scriptstyle w_1^3, w_1 w_2, w_3$ . 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.

[edit] Properties

If $E k$ has $s_1,\ldots,s_{\ell}$ sections which are everywhere linearly independent then $w_{k-\ell+1}=\cdots=w_k=0$ .
$w i (E) = 0$ whenever $i > rank(E)$ .
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 $w 1 (T M) = 0$ .
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 $\scriptstyle H^2(M, \Z) \rightarrow H^2(M,\Z/2\Z)$ (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 of a smooth compact manifold X vanish if and only if the manifold is a boundary (unoriented) of a smooth compact manifold.

[edit] Integral Stiefel-Whitney classes

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