a Pontryagin class
/ pon-TRYAH-gin /
Pontryagin classes are the characteristic classes of REAL vector bundles — the real-bundle analogue of Chern classes. A real bundle has no natural complex structure, so you cannot directly write the determinant formula that gives Chern classes; the trick is to complexify the bundle (tensor with C) and read off its Chern classes, then keep only the ones that survive. The result is a sequence of invariants p_1, p_2, ... that detect the twisting of real bundles and, importantly, of the tangent bundles of manifolds.
Precisely, for a real vector bundle E over M, complexify to E (x) C and define the j-th Pontryagin class by p_j(E) = (-1)^j c_{2j}(E (x) C), living in H^{4j}(M; Z). The odd Chern classes of the complexification are 2-torsion and are discarded, which is why Pontryagin classes sit in degrees 4, 8, 12, ... and only even-indexed Chern classes appear. Via Chern-Weil, with a metric connection of curvature Omega (an so(n)-valued, i.e. antisymmetric, 2-form), the total Pontryagin class is represented by det(I + (1/2pi) Omega) keeping the degree-4j pieces; concretely p_1 is represented by -(1/8pi^2) tr(Omega ^ Omega). For an oriented bundle of even rank the Euler class is a finer invariant whose square is the top Pontryagin class.
Pontryagin classes are the workhorse invariants of the topology of smooth manifolds. They enter the signature theorem (Hirzebruch: the signature of a 4k-manifold is a specific polynomial in the Pontryagin numbers), they obstruct cobordism, and Pontryagin numbers are the integrals that distinguish exotic smooth structures — Milnor used them to detect exotic 7-spheres. Two honest cautions: like all Chern-Weil output the curvature formulas see only the real (non-torsion) part, and the relation to Chern classes via complexification means the odd-degree information is deliberately thrown away (it would be 2-torsion). Pontryagin classes are coarser than Chern classes precisely because real bundles carry less structure than complex ones.
The first Pontryagin class of the tangent bundle of a closed oriented 4-manifold M integrates to a number p_1(M), and the Hirzebruch signature theorem says the signature of M (the signature of the intersection form on H^2) equals p_1(M)/3. For example the complex projective plane CP^2 has signature 1, so its integral of p_1 must be 3 — a topological fact pinned down by a curvature integral.
Signature = p_1/3 in dimension 4: a Pontryagin number computes a purely topological invariant.
Pontryagin classes live in degrees 4j (not 2j like Chern classes), and they discard the odd information of the complexification, which is only 2-torsion. So they are coarser than Chern classes — a complex bundle has both, with the Pontryagin classes being a non-injective function of the Chern classes. As always, the Chern-Weil formulas see only the non-torsion (real) part.