Characteristic classes attach cohomology classes to vector or principal bundles. They detect global twisting that no single local trivialization can see and obstruct the existence of everywhere independent sections.
Naturality and classification
A characteristic class \(c\) assigns \(c(E)\in H^\bullet(M)\) to each bundle \(E\to M\) so that \[ c(f^*E)=f^*c(E). \] Universal bundles over classifying spaces show that such classes arise by pulling back universal cohomology classes.
Chern-Weil construction
Theorem (Chern-Weil). Let \(P\to M\) be a principal \(G\)-bundle, \(\Omega\) the curvature of a connection, and \(P_k:\mathfrak g^k\to\mathbb R\) an \(\operatorname{Ad}\)-invariant symmetric polynomial. Then the \(2k\)-form \[ P_k(\Omega,\ldots,\Omega) \] is closed, and its de Rham cohomology class is independent of the connection.
Proof. Invariance of \(P_k\) and the Bianchi identity imply \[ \mathrm{d}\,P_k(\Omega^k) =kP_k(\mathrm{d}^\omega\Omega,\Omega^{k-1})=0. \] For a path of connections \(\omega_t\), differentiate \(P_k(\Omega_t^k)\). The variation formula \(\dot\Omega_t=\mathrm{d}^{\omega_t}\dot\omega_t\), invariance, and Bianchi show that the derivative is exact: \[ \frac{\mathrm{d}}{\mathrm{d}t}P_k(\Omega_t^k) =k\,\mathrm{d}\,P_k(\dot\omega_t,\Omega_t^{k-1}). \] Integrating in \(t\) proves connection independence.
Chern, Pontryagin, and Euler classes
For a complex rank-\(r\) bundle with curvature \(F\), total Chern class is represented by \[ c(E)=\det\!\left(I+\frac{i}{2\pi}F\right) =1+c_1(E)+\cdots+c_r(E). \] For a real bundle, Pontryagin classes can be obtained from the complexification: \[ p_k(E)=(-1)^kc_{2k}(E\otimes\mathbb C). \] For an oriented real rank-\(2m\) bundle, the Pfaffian represents the Euler class: \[ e(E)=\operatorname{Pf}\!\left(\frac{F}{2\pi}\right). \]
Example (line bundles). A complex line bundle has only \(c_1\). It is topologically trivial only if its first Chern class vanishes under suitable hypotheses. For the Hopf bundle over \(S^2\), \[ \int_{S^2}c_1=1, \] so no global nowhere-zero section can exist.
Euler class and zeros of vector fields
Theorem (Poincar?-Hopf). If \(M\) is a compact oriented manifold and \(X\) has isolated zeros, then \[ \sum_{p:X(p)=0}\operatorname{ind}_p(X) =\langle e(TM),[M]\rangle =\chi(M). \]
Thus the impossibility of combing a sphere is cohomological: \(\chi(S^{2m})=2\), so every vector field has a zero. On odd-dimensional closed orientable manifolds, \(\chi=0\), consistent with the existence of a nowhere-zero field.
Gauss-Bonnet
Theorem (Gauss-Bonnet for surfaces). For a compact oriented Riemannian surface, \[ \frac1{2\pi}\int_MK\,\mathrm{d}A=\chi(M). \]
The left side is geometric and depends pointwise on the metric; the right side is purely topological. Chern-Weil theory explains their equality: \((2\pi)^{-1}K\,\mathrm{d}A\) represents the Euler class of \(TM\).