nLab
Stiefel-Whitney class

Idea

A family of characteristic classes that obstruct orientation, spin structure, spin^c structure, etc.

Stiefel-Whitney classes have coefficients in ${\mathbb{Z}}_2$, but via the Bockstein homomorphism they are lifted to integral Stiefel-Whitney classes.

Definition

Definition

(axiomatic definition)

The Stiefel-Whitney classes are characteristic classes $w_i\in H^i(BO(n),{\mathbb{Z}}_2)$ on the classifying space of the orthogonal group in dimension $n$, defined by

1. $w_0=1$ and if $i>n$ then $w_i=0$;

2. for $n=1$, $w_1\ne 0$;

3. for the inclusion $\iota :BO(n)\hookrightarrow BO(n-1)$ we have ${\iota }^{*}{w}_i^{(n+1)}=w_i^{(n)}$;

4. sum rule: for all $k,l\in \mathbb{N}$ with the canonical inclusion

   $$\iota :BO(k)\times BO(l)\to BO(k+l)$$\iota : B O(k) \times B O(l) \to B O(k+l)

   we have for all $i\in \mathbb{N}$ that

   $${\iota }^{*}{w}_i=\sum _{j=0}^i{w}_j\cup w_{i-j}$$\iota^* w_i = \sum_{j = 0}^i w_j \cup w_{i-j}

   (on the right the cup product).

Definition

For $E\to X$ a real vector bundle/orthogonal group-principal bundle, the total universal Stiefel-Whitney class $w(E)$ is

$$w(E)≔\sum _i{w}_i(E)\in H^{•}(X,{\mathbb{Z}}_2)$$w(E) \coloneqq \sum_i w_i(E) \in H^\bullet(X, \mathbb{Z}_2)

as an element in the cohomology ring.

Remark

For the total SW class of def. 2, the sum rule of def. 1 says equivalently that for $E_1,E_2$ two real vector bundles, then the total SW class of their direct sum of vector bundles is the cup product of the separate classes:

$$w(E_1\oplus E_2)=w(E_1)\cup w(E_2).$$w(E_1 \oplus E_2) = w(E_1) \cup w(E_2) \,.

Constructions

(…)

Properties

Spanning the cohomology ring

Proposition

Every class in $H^{•}(BO(n),{\mathbb{Z}}_2)$ can be written uniquely as a polynomial in the Stiefel-Whitney classes. In fact the cohomology ring is the polynomial algebra over ${\mathbb{Z}}_2$ in the SW classes:

$$H^{•}(BO(n),{\mathbb{Z}}_2)\simeq {\mathbb{Z}}_2[w_1,\cdots ,w_n].$$H^\bullet(B O(n), \mathbb{Z}_2) \simeq \mathbb{Z}_2[w_1, \cdots, w_n] \,.

Whitney duality formula

For $X\hookrightarrow {ℝ}^q$ an embedding of a compact manifold, write $\tau :=TX$ for the tangent bundle and $\nu$ for the corresponding normal bundle. Then since

$$\tau \oplus \nu \simeq T{ℝ}^q$$\tau \oplus \nu \simeq T \mathbb{R}^q

and the class of the vector bundle on the right is trivial, the sum rule for the SW classes says gives the cup product duality

$$w(\tau )\cup w(\nu )=1.$$w(\tau) \cup w(\nu) = 1 \,.

Relation to Chern-classes

Proposition

If $E_{ℂ}$ is a complex vector bundle/ $U(n)$-principal bundle and $E_{ℝ}$ is the underlying real vector bundle / $O(2n)$-principal bundle then the second Stiefel-Whitney class is given by the first Chern class mod 2:

$$w_2(E_{ℝ})=c_1(E_{ℂ})\mathrm{mod}2.$$w_2(E_{\mathbb{R}}) = c_1(E_{\mathbb{C}}) \; mod 2 \,.

Corollary

An almost complex structure on the tangent bundle of a manifold induces a spin^c structure.

This is discussed at Spin^c-striucture – From almost complex structures.

Remark

More generally, the SW classes are then given by the Chern character. See for instance Milnor-Stasheff, p. 171.

Related concepts

• Chern class

• Pontryagin class

References

A classic work on the subject is

• John Milnor, James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press (1974).

A concise introduction is in chapter 23, section 3 of

• Peter May, A concise course in algebraic topology (pdf)

See also

• wikipedia: Stiefel-Whitney class