the Atiyah-Singer index theorem
/ uh-TEE-uh / SING-er /
Solve a differential equation Du = f on a compact manifold and you can ask two crude counting questions: how many independent solutions of Du = 0 are there (the kernel), and how many obstructions f must satisfy for a solution to exist (the cokernel). Their difference, dim ker D - dim coker D, is the analytic index — a single integer. The astonishing claim of the Atiyah-Singer index theorem is that this analytic number, born from hard analysis, equals a purely topological number computed from characteristic classes, with no analysis visible in the answer.
Precisely, let D be an elliptic differential (or pseudodifferential) operator between sections of vector bundles E and F over a closed manifold M. Ellipticity means the symbol of D — its leading-order part, a bundle map on the cotangent sphere — is invertible away from the zero section; this is what makes ker and coker finite-dimensional, so the analytic index ind(D) = dim ker D - dim coker D is a well-defined integer. The theorem states ind(D) equals the topological index: the integral over M of a specific characteristic-class expression built from the symbol's K-theory class via the Chern character and the Todd class of the tangent bundle, ind(D) = integral over M of ch(symbol) Td(TM) (schematically). The symbol lives in K-theory, the Chern character translates it to cohomology, and integration produces the number.
Its reach is enormous: the Gauss-Bonnet theorem, the Hirzebruch signature theorem, and the Riemann-Roch theorem are all special cases, recovered by feeding in the de Rham, signature, and Dolbeault operators respectively. It launched K-theory's role in analysis and underlies index theory in geometry and physics. Two honest cautions: it is an equality of an analytic index with a topological index, NOT a curvature identity — though the heat-kernel proof passes through curvature, the statement itself is topological; and it requires ellipticity and a closed (compact, no boundary) manifold. On manifolds with boundary one needs the far subtler Atiyah-Patodi-Singer version with an eta-invariant correction.
Feed in the de Rham operator d + d^* mapping even forms to odd forms on a closed even-dimensional M. Its analytic index is the alternating sum of Betti numbers, the Euler characteristic chi(M). The topological index in this case integrates the Euler class of TM, and the theorem reduces to the Gauss-Bonnet-Chern theorem: chi(M) = integral over M of the Euler form. Same machine, classical special case.
The de Rham operator's index is the Euler characteristic; Atiyah-Singer specializes to Gauss-Bonnet-Chern.
It equates an analytic index (a kernel-minus-cokernel dimension count) with a topological index built from characteristic classes; it is NOT a curvature identity, even though heat-kernel proofs use curvature. It needs ellipticity and a closed manifold — boundaries require the Atiyah-Patodi-Singer refinement with an eta-invariant.