Stiefel–Whitney class

Builds on Characteristic class Whitney sum Singular cohomology Cup product

Definition

Consider real vector bundles $E \to B$ over paracompact spaces $B$ . The Stiefel–Whitney classes are the characteristic classes $w_i(E) \in H^i(B; \Z/2\Z)$ , $i \ge 0$ , in singular cohomology, that satisfy the following axioms. Write $w(E) = w_0(E) + w_1(E) + w_2(E) + \cdots$ for the total class.

$w_0(E) = 1$ , and $w_i(E) = 0$ for $i$ greater than the rank of $E$ .
Naturality: $w_i(f^* E) = f^* w_i(E)$ for every continuous map $f \colon B' \to B$ , and isomorphic bundles have equal classes.
Whitney product formula: $w(E \oplus E') = w(E) \smile w(E')$ for the Whitney sum of two bundles over the same base, where $\smile$ is the cup product.
Normalization: for the tautological line bundle $\gamma^1_1$ over $\R P^1$ , $w_1(\gamma^1_1)$ is the generator of $H^1(\R P^1; \Z/2\Z) \cong \Z/2\Z$ .

There is exactly one family of classes with these properties.

Remarks

Example: a trivial bundle $B \times \R^n$ is pulled back from a point, so by axiom 2 its total class is $w = 1$ . For the tangent bundle of $\R P^n$ , one has $w(T\R P^n) = (1 + a)^{n+1}$ , where $a$ generates $H^1(\R P^n; \Z/2\Z)$ ; for $n = 2$ this gives $w = 1 + a + a^2$ .

Non-example: the rule $w(E) = 1$ for every bundle satisfies axioms 1 to 3 but fails axiom 4.

A bundle $E$ is orientable exactly when $w_1(E) = 0$ . An orientable bundle admits a spin structure exactly when also $w_2(E) = 0$ . For a smooth manifold $M$ , one writes $w_i(M) = w_i(TM)$ .

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Tag	`GKWC4EASX9`
URI	`m2://GKWC4EASX9`
Kind	definition
Writing	W0 stub
MSC	57R20, 55R40
Defines	Stiefel–Whitney class
Uses	Characteristic class, Whitney sum, Singular cohomology, Cup product
Changed	2 November 2026

Stiefel–Whitney class

Definition

Remarks

Reading path

Further reading

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