Atiyah–Singer Index Theorem

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.

A deep theorem equating the analytical index of an elliptic differential operator on a compact manifold (the difference of dimensions of kernel and cokernel) with a topologically defined index computed from characteristic classes of the manifold and the operator symbol.

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.