For an elliptic differential operator on a compact smooth manifold, the Atiyah–Singer index theorem states that its analytical index (dimension of kernel minus dimension of cokernel) equals a topological index computable from characteristic classes of the manifold and the symbol of the operator.

Principle

Analytic invariants of elliptic operators coincide with topological invariants of underlying geometric data; the index is stable under continuous deformations and can be computed via characteristic-class formulas.

Demonstration

The index of the Dirac operator on a compact spin manifold equals the Â-genus of the manifold; similarly, the signature operator's index recovers the signature of the intersection form on middle cohomology, illustrating topology-analytic equality.

Misapplication

Applying the theorem to non-elliptic operators, noncompact manifolds without growth conditions, or operators lacking Fredholm property leads to invalid conclusions; compactness and ellipticity hypotheses are essential.

Consequence

The theorem yields powerful existence and obstruction results: vanishing or nonvanishing of indices implies existence of solutions to PDEs, constraints on manifold topology, and relations used in geometry and mathematical physics.

Reversal

Viewed in reverse, topological computations of the index predict analytic phenomena such as the dimension of solution spaces; a computed vanishing index suggests no net imbalance between solutions and obstructions but does not guarantee trivial kernel and cokernel individually.

Boundary

The classical theorem requires elliptic operators on compact smooth manifolds (possibly with additional structure like spin) and uses K-theory and characteristic classes; extensions treat boundary value problems, noncompact settings, and families with extra terms.

Semantic Tension

There is tension between the analytic Fredholm index viewpoint and the algebraic-topological K-theory/formula viewpoint: the theorem identifies these a priori different invariants, but concrete computation often requires translating between analytic and topological languages.

Synthesis

Atiyah–Singer connects analysis and topology by equating the Fredholm index of elliptic operators with a computable topological index, providing a deep tool to translate differential-operator problems into characteristic-class computations and vice versa.