Stiefel–Whitney class

In mathematics, the Stiefel–Whitney class arises as a type of characteristic class associated to real vector bundles $\scriptstyle E\rightarrow X$. It is denoted by w(E), taking values in $\scriptstyle H^{*}(X;\mathbb {Z} /2\mathbb {Z} )$, the cohomology groups with mod 2 coefficients. The component of $\scriptstyle w(E)$ in $\scriptstyle H^{i}(X;\mathbb {Z} /2\mathbb {Z} )$ is denoted by $\scriptstyle w_{i}(E)$ and called the $\scriptstyle i$-th Stiefel-Whitney class of $\scriptstyle E$, so that $\scriptstyle w(E)=w_{0}(E)+w_{1}(E)+w_{2}(E)+\cdots$. As an example, over the circle, $\scriptstyle S^{1}$, 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 $\scriptstyle [0,1]$. 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, $\scriptstyle w_{1}$, of that line bundle.

Origins

The Stiefel-Whitney classes $\scriptstyle w_{i}(E)$ get their name because Stiefel and Whitney discovered them as mod-2 reductions of the obstruction classes to constructing $\scriptstyle n-i+1$ everywhere linearly independent sections of the vector bundle $\scriptstyle E$ restricted to the $\scriptstyle i$-skeleton of $\scriptstyle X$. Here $\scriptstyle n$ denotes the dimension of the fibre of the vector bundle $\scriptstyle F\to E\to X$.

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

Since $\scriptstyle \pi _{i-1}V_{n-i+1}(F)$ is either infinite-cyclic or isomorphic to $\scriptstyle {\mathbb {Z}}_{2}$, there is a canonical reduction of the $\scriptstyle W_{i}(E)$ classes to classes $\scriptstyle w_{i}(E)\in H^{i}(X;{\mathbb {Z}}_{2})$ which are the Stiefel-Whitney classes. Moreover, whenever $\scriptstyle \pi _{i-1}V_{n-i+1}(F)={\mathbb {Z}}_{2}$, the two classes are identical. Thus, $\scriptstyle w_{1}(E)=0$ if and only if the bundle $\scriptstyle E\to X$ is orientable.

The $\scriptstyle w_{0}(E)$ class is exceptional and has no meaning a priori. Its creation by Whitney was an act of creative notation, allowing the Whitney sum Formula $\scriptstyle w(E_{1}\oplus E_{2})=w(E_{1})w(E_{2})$ to be true. However, for generalizations of manifolds (namely certain homology manifolds), one can have $\scriptstyle w_{0}(M)\neq 1$ – it only needs to equal 1 mod 8.

Axioms

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

1. Naturality: $\scriptstyle w(f^{*}E)=f^{*}w(E)$ for any bundle $\scriptstyle E\to X$ and map $\scriptstyle f:X'\to X$, where $\scriptstyle f^{*}E$ denotes the induced bundle.
2. $\scriptstyle w_{0}(E)=1$ in $\scriptstyle H^{0}(X;\mathbb {Z} /2\mathbb {Z} )$.
3. $\scriptstyle w_{1}(\gamma ^{1})$ is the generator of $\scriptstyle H^{1}(\mathbb {R} P^{1};\mathbb {Z} /2\mathbb {Z} )\cong \mathbb {Z} /2\mathbb {Z}$ (normalization condition). Here, $\scriptstyle \gamma ^{n}$ is the canonical line bundle.
4. $\scriptstyle w(E\oplus F)=w(E)\smallsmile w(F)$ (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.

Line bundles

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

Now consider the space $\scriptstyle Vect_{1}(X)$ of line bundles over $\scriptstyle X$. For $\scriptstyle n=1$, the Grassmannian $\scriptstyle Gr_{1}$ is just $\scriptstyle \mathbb {R} P^{\infty }=\mathbb {R} ^{\infty }/\mathbb {R} ^{*}=S^{\infty }/(\mathbb {Z} /2\mathbb {Z} )$, where the nonzero element of $\scriptstyle \mathbb {Z} /2\mathbb {Z}$ acts by $\scriptstyle x\to -x$. The quotient map $\scriptstyle S^{\infty }\to S^{\infty }/(\mathbb {Z} /2\mathbb {Z} )=\mathbb {R} P^{\infty }$ is therefore a double cover. Since $\scriptstyle S^{\infty }$ is contractible, we have $\scriptstyle \pi _{i}(\mathbb {R} P^{\infty })=\pi _{i}(S^{\infty })=0$ for $\scriptstyle i>1$ and $\scriptstyle \#\pi _{1}(\mathbb {R} P^{\infty })=2$; that is, $\scriptstyle \pi _{1}(\mathbb {R} P^{\infty })=\mathbb {Z} /2\mathbb {Z}$. Hence $\scriptstyle \mathbb {R} P^{\infty }$ is the Eilenberg-Maclane space $\scriptstyle K(\mathbb {Z} /2\mathbb {Z} ,1)$. Hence $\scriptstyle [X;Gr_{1}]=H^{1}(X;\mathbb {Z} /2\mathbb {Z} )$ for any $\scriptstyle X$, with the isomorphism given by $\scriptstyle f\to f^{*}\eta$, where $\scriptstyle \eta$ is the generator $\scriptstyle H^{1}(\mathbb {R} P^{\infty };\mathbb {Z} /2\mathbb {Z} )=\mathbb {Z} /2\mathbb {Z}$. Since $\scriptstyle \alpha :[X,Gr_{1}]\to Vect_{1}(X)$ is also a bijection, we have another bijection $\scriptstyle w_{1}:Vect_{1}\to H^{1}(X;\mathbb {Z} /2\mathbb {Z} )$. This map $\scriptstyle w_{1}$ is precisely the Stiefel-Whitney class $\scriptstyle w_{1}$ for a line bundle. (Since the corresponding classifying space $\scriptstyle CP^{\infty }$ for complex bundles is a $\scriptstyle K(\mathbb {Z} ,2)$, the same argument shows that the Chern class defines a bijection between complex line bundles over $\scriptstyle X$ and $\scriptstyle H^{2}(X;\mathbb {Z} )$.) For example, since $\scriptstyle H^{1}(S^{1};\mathbb {Z} /2\mathbb {Z} )=\mathbb {Z} /2\mathbb {Z}$, 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 $\scriptstyle Vect_{1}(X)$ is considered as a group under the operation of tensor product, then $\scriptstyle \alpha$ is an isomorphism: $\scriptstyle w_{1}(\lambda \otimes \mu )=w_{1}(\lambda )+w_{1}(\mu )$ for all line bundles $\scriptstyle \lambda ,\mu \to X$.

Higher dimensions

The bijection above for line bundles implies that any functor $\scriptstyle \scriptstyle \theta$ satisfying the four axioms above is equal to w. Let $\scriptstyle \scriptstyle \xi \to X$ be an n-bundle. Then $\scriptstyle \scriptstyle \xi$ admits a splitting map, a map $\scriptstyle \scriptstyle f:X'\to X$ for some space $\scriptstyle \scriptstyle X'$ such that $\scriptstyle \scriptstyle f^{*}:H^{*}(X;\mathbb {Z} /2\mathbb {Z} )\to H^{*}(X';\mathbb {Z} /2\mathbb {Z} )$ is injective and $\scriptstyle \scriptstyle f^{*}\xi =\lambda _{1}\oplus \cdots \oplus \lambda _{n}$ for some line bundles $\scriptstyle \scriptstyle \lambda _{i}\to X'$. Any line bundle over X is of the form $\scriptstyle \scriptstyle g^{*}\gamma ^{1}$ for some map g, and $\scriptstyle \scriptstyle \theta (g^{*}\gamma ^{1})=g^{*}\theta (\gamma ^{1})=1+w_{1}(g^{*}\gamma _{1})$ by naturality. Thus $\scriptstyle \scriptstyle \theta =w$ on $\scriptstyle \scriptstyle Vect_{1}(X)$. 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 $\scriptstyle \scriptstyle f^{*}$ is injective, $\scriptstyle \scriptstyle \theta =w$ Thus the Stiefel-Whitney class is the unique functor satisfying the four axioms above.

Although the map $\scriptstyle \scriptstyle w_{1}:Vect_{1}(X)\to H^{1}(X;\mathbb {Z} /2\mathbb {Z} )$ is a bijection, the corresponding map is not necessarily injective in higher dimensions. For example, consider the tangent bundle $\scriptstyle \scriptstyle TS^{n}$ for $\scriptstyle \scriptstyle n$ even. With the canonical embedding of $\scriptstyle \scriptstyle S^{n}$ in $\scriptstyle \scriptstyle \mathbb {R} ^{n+1}$, the normal bundle $\scriptstyle \scriptstyle \nu$ to $\scriptstyle \scriptstyle S^{n}$ is a line bundle. Since $\scriptstyle \scriptstyle S^{n}$ is orientable, $\scriptstyle \scriptstyle \nu$ is trivial. The sum $\scriptstyle \scriptstyle TS^{n}\oplus \nu$ is just the restriction of $\scriptstyle \scriptstyle T\mathbb {R} ^{n+1}$ to $\scriptstyle \scriptstyle S^{n}$, which is trivial since $\scriptstyle \scriptstyle \mathbb {R} ^{n+1}$ is contractible. Hence $\scriptstyle \scriptstyle w(TS^{n})=w(TS^{n})w(\nu )=w(TS^{n}\oplus \nu )=1$. But $\scriptstyle \scriptstyle TS^{n}\to S^{n}$ is not trivial; its Euler class $\scriptstyle \scriptstyle e(TS^{n})=\chi (TS^{n})[S^{n}]=2[S^{n}]\not =0$, where $\scriptstyle \scriptstyle [S^{n}]$ denotes a fundamental class of $\scriptstyle \scriptstyle S^{n}$ and $\scriptstyle \scriptstyle \chi$ the Euler characteristic.

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 $\scriptstyle \scriptstyle \scriptstyle \mathbb {Z} _{2}$-fundamental class of the manifold to give an element of $\scriptstyle \scriptstyle \scriptstyle \mathbb {Z} _{2}$, 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 $\scriptstyle \scriptstyle \scriptstyle w_{1}^{3},w_{1}w_{2},w_{3}$. 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.

One Stiefel–Whitney number of important in surgery theory is the de Rham invariant of a (4k+1)-dimensional manifold, $\scriptstyle w_{2}w_{4k-1}.$

Wu classes

The Stiefel–Whitney classes $\scriptstyle w_{k}$ are the Steenrod squares of the Wu classes $\scriptstyle v_{k},$ defined by Wu Wenjun in (Wu 1955). Most simply, the total Stiefel–Whitney class is the total Steenrod square of the total Wu class: $\scriptstyle Sq(v)=w.$ Wu classes are most often defined implicitly in terms of Steenrod squares, as the cohomology class representing the Steenrod squares: $\scriptstyle v_{k}\cup x=Sq^{k}(x),$ or more narrowly $\scriptstyle \langle v_{k}\cup x,\mu \rangle =\langle Sq^{k}(x),\mu \rangle ;$ (Milnor & Stasheff 1974, pp. 131-133).

Properties

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

Integral Stiefel-Whitney classes

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

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

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 $\scriptstyle \{w_{2^{i}}\}$ generate all the Stiefel-Whitney classes. In particular, the Stiefel-Whitney classes satisfy the Wu formula, named for Wu Wenjun:

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

References

• D. Husemoller, Fibre Bundles, Springer-Verlag, 1994.
• J. Milnor & J. Stasheff, Characteristic Classes, Princeton, 1974.
• May, J. P. (1999), A Concise Course in Algebraic Topology (PDF), U. Chicago Press, Chicago, retrieved 2009-08-07