A theorem equating the analytical index of an elliptic differential operator on a compact manifold (the Fredholm index counting kernel and cokernel) with a topological index computed from characteristic classes of the operator's symbol in K-theory, thereby connecting analysis, topology, and algebraic K-theory.

Principle

The analytical index of an elliptic operator depends only on the K-theory class of its principal symbol; that K-theory class pairs with characteristic classes of the manifold to produce a computable topological number equal to the analytic Fredholm index.

Demonstration

On a compact spin manifold, the Dirac operator is elliptic; its analytical index equals the A-hat genus of the manifold, which can be computed from Pontryagin classes. This equality computes the dimension difference of harmonic spinors from purely topological data.

Misapplication

Applying the theorem to non-elliptic operators, to operators on noncompact manifolds without specifying boundary conditions, or treating analytical invariants that depend on metrics (e.g., individual eigenvalues) as determined by the topological index.

Consequence

Provides a bridge allowing computation of analytical invariants (indices of PDE operators, existence of solutions) by topological means; it yields powerful constraints in geometry, topology and mathematical physics (for example index formulas for anomalies).

Reversal

If one inverts the statement, topological index data do not determine the full analytic spectrum of an operator; conversely, analytic spectral data can vary continuously while the topological index remains fixed.

Boundary

Holds for elliptic differential operators (or suitable elliptic pseudodifferential operators) on compact manifolds possibly with prescribed boundary conditions; does not apply to non-elliptic equations, infinite-rank operators without Fredholm property, or to analytic quantities beyond the index without extra hypotheses.

Semantic Tension

Tension arises between the analytic notion of index (kernel minus cokernel, metric-dependent analytical setup) and the topological notion (K-theory class and characteristic classes); confusion can occur between the index and finer spectral invariants such as eta invariants or individual eigenvalues.

Synthesis

The Atiyah–Singer Index Theorem identifies the Fredholm index of an elliptic operator with a topological pairing of its symbol class in K-theory against characteristic classes of the manifold, unifying analytic, topological and algebraic perspectives and enabling computation of analytic indexes by topological methods.