the Euler class
/ OY-ler /
The Euler class measures whether an oriented bundle has a nowhere-zero section — and if not, how badly it fails. Think of a vector field on a surface: the hairy-ball theorem says you cannot comb a sphere flat, every field must have a zero, and the Euler class is the invariant that counts those zeros with signs. For the tangent bundle of a closed surface that count is exactly the Euler characteristic — hence the name — which is why a torus (Euler characteristic 0) can be combed but a sphere (Euler characteristic 2) cannot.
Precisely, the Euler class e(E) of an oriented real vector bundle E of rank k over M lives in H^k(M; Z); it is defined as the obstruction to a nowhere-zero section, equivalently the primary obstruction encountered when trying to extend a section over the cells of M, equivalently (Poincare dual) the homology class of the zero set of a generic section. For an oriented even-rank bundle Chern-Weil gives a curvature representative via the Pfaffian: e(E) is represented by (1/2pi)^{k/2} Pf(Omega), the Pfaffian of the antisymmetric curvature matrix — a closed form whose integral over a closed oriented manifold is an integer. The Pfaffian is the 'square root' of the determinant for antisymmetric matrices, which is why the Euler class squares to the top Pontryagin class, e(E)^2 = p_{k/2}(E), and refines it. For a complex bundle viewed as a real oriented bundle, the Euler class equals the top Chern class c_k.
The Euler class is the bridge between local geometry and global topology. Integrating its Pfaffian representative over a closed surface recovers the Gauss-Bonnet theorem: the integral of the Gaussian curvature equals 2 pi times the Euler characteristic. In higher dimensions the generalized Gauss-Bonnet-Chern theorem does the same with the Pfaffian. Two honest cautions: the Euler class is defined only for oriented bundles — without an orientation you only get the mod-2 reduction, the top Stiefel-Whitney class; and it is defined via the Pfaffian only in even rank, since odd-rank oriented bundles always have vanishing Euler class (a generic section has a codimension-equal-to-rank zero set, which in odd rank forces cancellation). The Euler class is finer than the Pontryagin classes, not a function of them.
On a closed oriented surface, the Euler class of the tangent bundle integrates to the Euler characteristic: integral over S of e(TS) = chi(S). The Pfaffian representative is (1/2pi) K dA where K is the Gaussian curvature, so this is precisely Gauss-Bonnet: integral of K dA = 2 pi chi(S). For the 2-sphere chi = 2, so the total curvature is 4 pi; for a torus chi = 0, consistent with the existence of a nowhere-zero tangent field (you can comb a torus).
Gauss-Bonnet is the Euler class of the tangent bundle: total curvature counts the Euler characteristic.
The Euler class needs an orientation — reverse the orientation and it flips sign; without one you only have its mod-2 image, the top Stiefel-Whitney class. It is also nontrivial only for even-rank bundles (odd-rank oriented bundles have zero Euler class). And it is finer than Pontryagin: e^2 = top Pontryagin class, so e is a 'square root' carrying strictly more information.