Stiefel–Whitney class

In mathematics, the Stiefel–Whitney class arises as a type of characteristic class associated to real vector bundles $E\; ightarrow\; X$. It is denoted by "w"("E"), taking values in , the cohomology groups with mod 2 coefficients. The component of $w(E)$ in 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 :$H^1(S^1;mathbb\; Z/2mathbb\; Z)$has just one element other than 0, this element being the first Stiefel-Whitney class, $w\_1$, of that line bundle.

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 $F\; o\; E\; o\; X$.

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 $pi\_\{i-1\}\; V\_\{n-i+1\}(F)$ is either infinite-cyclic or isomorphic to $Bbb\; Z\_2$, their 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 $E\; o\; X$ 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.

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 $E\; o\; X$ and map $f:X\text{'}\; o\; X$, where $f^*E$ denotes the induced bundle.
# $w\_0(E)=1$ in $H^0(X;mathbb\; Z/2mathbb\; Z)$.
# $w\_1(gamma^1)$ is the generator of $H^1(mathbb\; RP^1;mathbbZ/2mathbb\; Z)congmathbb\; Z/2mathbb\; Z$ (normalization condition). Here, $gamma^n$ is the canonical line bundle.
# $w(Eoplus\; F)=\; w(E)\; smallsmile\; w(F)$ (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 3.5 and 3.6 in Husemoller or section 8 in Milnor and Stasheff.

Line bundles

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

Now consider the space $Vect\_1(X)$ of line bundles over $X$. For $n\; =\; 1$, the Grassmannian $Gr\_1$ is just , where the nonzero element of acts by $x\; o\; -x$. The quotient map 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, . Hence $R\; P^infty$ is the Eilenberg-Maclane space . Hence for any $X$, with the isomorphism given by $f\; o\; f^*\; eta$, where $eta$ is the generator . Since $alpha:\; [X,\; Gr\_1]\; o\; Vect\_1(X)$ is also a bijection, we have another bijection . 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 , the same argument shows that the Chern class defines a bijection between complex line bundles over $X$ and .) For example, since , 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 $Vect\_1(X)$ is considered as a group under the operation of tensor product, then $alpha$ is an isomorphism: $w\_1(lambda\; otimes\; mu)\; =\; w\_1(lambda)\; +\; w\_1(mu)$ for all line bundles $lambda,\; mu\; o\; X$.

Higher dimensions

The bijection above for line bundles implies that any functor $heta$ satisfying the four axioms above is equal to "w". Let $xi\; o\; X$ be an n-bundle. Then $xi$ admits a splitting map, a map $f:X\text{'}\; o\; X$ for some space $X\text{'}$ such that is injective and $f^*xi\; =\; lambda\_1\; oplus\; cdots\; oplus\; lambda\_n$ for some line bundles $lambda\_i\; o\; X\text{'}$. Any line bundle over "X" is of the form $g^*\; gamma^1$ for some map "g", and $heta(g^*gamma^1)\; =\; g^*\; heta(gamma^1)\; =\; 1\; +\; w\_1(g^*gamma\_1)$ by naturality. Thus $heta\; =\; w$ on $Vect\_1(X)$. It follows from the fourth axiom above that:$f^*\; heta(xi)\; =\; heta(f^*xi)\; =\; heta(lambda\_1\; oplus\; cdots\; oplus\; lambda\_n)\; =\; heta(lambda\_1)\; cdots\; heta(lambda\_n)\; =\; (1\; +\; w\_1(lambda\_1))\; cdots\; (1\; +\; w\_1(lambda\_n))\; =\; w(lambda\_1)\; cdots\; w(lambda\_n)\; =\; w(f^*xi)\; =\; f^*\; w(xi).$Since $f^*$ is injective, $heta\; =\; w$ Thus the Stiefel-Whitney class is the unique functor satisfying the four axioms above.

Although the map 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 $R^\{n+1\}$, the normal bundle $u$ to $S^n$ is a line bundle. Since $S^n$ is orientable, $u$ is trivial. The sum $TS^n\; oplus\; u$ is just the restriction of $TR^\{n+1\}$ to $S^n$, which is trivial since $R^\{n+1\}$ is contractible. Hence $w(TS^n)\; =\; w(TS^n)w(\; u)\; =\; w(TS^n\; oplus\; u)\; =\; 1$. But $TS^n\; o\; S^n$ is not trivial; its Euler class $e(TS^n)\; =\; chi(TS^n)\; [S^n]\; =\; 2\; [S^n]\; ot\; =0$, where $[S^n]$ denotes a fundamental class of $S^n$ and $chi$ the Euler characteristic.

tiefel–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 $mathbf\{Z\}/2$-fundamental class of the manifold to give an element of $mathbf\{Z\}/2$, 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 $w\_1^3,\; w\_1\; w\_2,\; w\_3$.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.

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 $i>mathrm\{rank\}(E)$.
#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 $w\_1(TM)\; =\; 0$.
#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 (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.

Integral Stiefel-Whitney classes

The element is called the $i+1$ "integral" Stiefel-Whitney class, where β is the Bockstein homomorphism, corresponding to reduction modulo 2, $mathbf\{Z\}\; o\; mathbf\{Z\}/2$::

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

References

*D. Husemoller, "Fibre Bundles", Springer-Verlag, 1994.
*J. Milnor & J. Stasheff, "Characteristic Classes", Princeton, 1974.