Integral Chern classes

Let $E\to X$ be a finite-rank complex vector bundle over a paracompact Hausdorff space $X$ . Its integral Chern classes are the canonical classes

c_k(E)\in H^{2k}(X;\mathbb Z),\qquad k\geq 0,

depending only on the bundle isomorphism class, and characterized as a family over all such bases and bundles by all of the following axioms:

Naturality. For every continuous map $f:Y\to X$ with $Y$ paracompact Hausdorff, $c_k(f^*E)=f^*c_k(E).$
Normalization. $c_0(E)=1$ , and $c_k(E)=0$ for $k>\operatorname{rank}_{\mathbb C}E$ .
Whitney sum. For direct sums, $c(E\oplus F)=c(E)\smile c(F),\qquad c(E)=1+c_1(E)+c_2(E)+\cdots,$ with the cup product in integral cohomology.
Line-bundle normalization. If $L=\mathcal O(-1)$ is the tautological complex line bundle over $\mathbb{CP}^{1}$ , then $c_1(L)=-u,$ where $u\in H^2(\mathbb{CP}^{1};\mathbb Z)$ is the positive generator determined by the complex orientation.

These axioms define a stable family: the same class $c_k(E)$ is used after adding trivial summands, and the Whitney identity determines the components of a direct sum from those of its factors.

Universal construction and uniqueness

For each rank $n$ , the universal rank- $n$ bundle over the classifying space $BU(n)$ has universal classes

c_k\in H^{2k}(BU(n);\mathbb Z).

For a classifying map $f:X\to BU(n)$ , one has $c_k(E)=f^*c_k$ . The classifying-space construction proves existence and uniqueness for paracompact bases because classifying maps are unique up to homotopy. Equivalently, after pulling $E$ back to a complete flag bundle, the splitting principle writes it as a sum of line bundles; the normalization and the direct-sum axiom then give the elementary symmetric expressions in the line-bundle first classes, and injectivity of the flag-bundle pullback proves uniqueness downstairs.

Relation to Chern–Weil forms

If $X$ is a smooth manifold and $E$ has a Hermitian connection with curvature $F$ , the real image of $c_k(E)$ is represented by the degree- $2k$ component of

\det\!\left(I+\frac{i}{2\pi}F\right).

The differential form fixes the image in real cohomology; it does not determine a possible torsion component of the integral class. The sign in the tautological-line normalization agrees with this convention: the tautological bundle has $c_1(\mathcal O(-1))=-u$ .

Examples

For a trivial rank- $n$ bundle, $c(E)=1$ . For a sum of line bundles $L_1\oplus\cdots\oplus L_n$ ,

c(E)=\prod_{j=1}^{n}\bigl(1+c_1(L_j)\bigr).

Thus $c_k(E)$ is the $k$ th elementary symmetric polynomial in the first Chern classes of the summands.

References

Allen Hatcher, Vector Bundles and K-Theory, version 2.2 (2017), Chapter 3, §3.1, “Stiefel-Whitney and Chern Classes,” pp. 77–84. Author's PDF.
John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974, Chapter 14. DOI record.