Characteristic Classes

Characteristic classes are cohomological invariants associated to vector bundles and principal bundles. They measure the "twisting" of a bundle and provide obstructions to the existence of certain geometric structures.

Stiefel-Whitney Classes

Definition

The Stiefel-Whitney classes of a real vector bundle $\xi : E \to B$ are cohomology classes $w_i(\xi) \in H^i(B; \mathbb{Z}/2\mathbb{Z})$ for $i \geq 0$ , satisfying:

$w_0(\xi) = 1$ and $w_i(\xi) = 0$ for $i > \operatorname{rank}(\xi)$
Naturality: $f^*(w_i(\xi)) = w_i(f^*\xi)$ for any map $f$
Whitney sum formula: $w(\xi \oplus \eta) = w(\xi) \smile w(\eta)$ , where $w = 1 + w_1 + w_2 + \cdots$
Normalization: $w_1(\gamma^1) \neq 0$ for the tautological line bundle $\gamma^1 \to \mathbb{RP}^\infty$

Definition

The Chern classes of a complex vector bundle $\xi : E \to B$ are classes $c_i(\xi) \in H^{2i}(B; \mathbb{Z})$ for $i \geq 0$ , satisfying analogous axioms with integral coefficients. The total Chern class is $c(\xi) = 1 + c_1(\xi) + c_2(\xi) + \cdots \in H^*(B; \mathbb{Z})$ .

Key Properties and Computations

ExampleStiefel-Whitney classes detect orientability and spin

For a smooth manifold $M$ with tangent bundle $TM$ :

$w_1(TM) = 0$ if and only if $M$ is orientable
$w_1(TM) = w_2(TM) = 0$ if and only if $M$ admits a spin structure

For instance, $\mathbb{RP}^n$ has $w(T\mathbb{RP}^n) = (1 + a)^{n+1}$ where $a$ generates $H^1(\mathbb{RP}^n; \mathbb{Z}/2)$ . So $\mathbb{RP}^n$ is orientable if and only if $n$ is odd.

Pontryagin Classes

Definition

The Pontryagin classes of a real vector bundle $\xi$ are $p_i(\xi) \in H^{4i}(B; \mathbb{Z})$ , defined by $p_i(\xi) = (-1)^i c_{2i}(\xi \otimes_\mathbb{R} \mathbb{C})$ . The Pontryagin numbers of a closed oriented $4k$ -manifold $M$ are obtained by evaluating products of Pontryagin classes on the fundamental class $[M]$ .

Theorem10.5Cobordism Invariance

The Stiefel-Whitney numbers and Pontryagin numbers of a closed manifold are cobordism invariants. Two closed oriented manifolds are oriented cobordant if and only if they have the same Pontryagin numbers and Stiefel-Whitney numbers.

RemarkChern-Weil theory

In the smooth category, characteristic classes can be computed using differential geometry via Chern-Weil theory: given a connection on a vector bundle, the characteristic classes are represented by differential forms built from the curvature. For instance, $c_1$ is represented by $\frac{i}{2\pi}\operatorname{tr}(\Omega)$ where $\Omega$ is the curvature $2$ -form. This connects algebraic topology with Riemannian geometry and gauge theory.