nLab Stiefel-Whitney class

Idea

A family of characteristic classes that obstruct orientation, spin structure, spin^c structure, orientation of EO(2)-theory 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 $w_i \in H^{i}(B O(n), \mathbb{Z}_2)$ on the classifying space of the orthogonal group in dimension $n$, defined by

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

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

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

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

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

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

   $$\iota^* w_i = \sum_{j = 0}^i w_j \cup w_{i-j}$$

   (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

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

as an element in the cohomology ring.

Remark

For the total SW class of def. , the sum rule of def. 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:

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

Constructions

(…)

Properties

Spanning the cohomology ring of $B SO$

Proposition

Every class in the ordinary cohomology $H^\bullet(B O(n), \mathbb{Z}_2)$ and $H^\bullet(B S O(n), \mathbb{Z}_2)$ of the classifying spaces of the (special) orthogonal group with coefficients in $\mathbb{Z}_2$ is uniquely a polynomial in the Stiefel-Whitney classes. In fact the cohomology rings are polynomial algebras over $\mathbb{Z}_2$ in the SW classes:

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

$$H^\bullet(B S O(n), \mathbb{Z}_2) \simeq \mathbb{Z}_2[w_2, \cdots, w_n] \,.$$

(Milnor & Stasheff 74, Theorem 7.1. & Theorem 12.4.)

Whitney duality formula

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

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

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

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

Relation to Chern-classes

Proposition

If $E_{\mathbb{C}}$ is a complex vector bundle/ $U(n)$-principal bundle and $E_{\mathbb{R}}$ 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:

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

Corollary

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

This is discussed at Spin^c-structure – 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

• orientation, Spin structure, w4-structure

• Chern class

• Pontryagin class, Wu class, one-loop anomaly polynomial I8

• orthogonal calculus

References

Named after Eduard Stiefel and Hassler Whitney.

Textbook accounts include

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

• Stanley Kochmann, section 2.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Peter May, chapter 23, section 3 of A concise course in algebraic topology (pdf)

Discussion with an eye towards mathematical physics

• Gerd Rudolph, Matthias Schmidt, Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields, Theoretical and Mathematical Physics series, Springer 2017 (doi:10.1007/978-94-024-0959-8)

Stiefel-Whitney classes generating the cohomology ring of $BO(n)$

• John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974) (ISBN:9780691081229, doi:10.1515/9781400881826, pdf)

See also

• Wikipedia, Stiefel-Whitney class