Indextheorie

IndexTheorie, or index theory, is a branch of differential geometry and global analysis that studies integers known as the index associated with elliptic differential operators on manifolds. For a Fredholm operator D between sections of vector bundles over a compact manifold, the analytical index is defined as ind(D) = dim ker D − dim coker D. This numerical invariant links analysis, topology, and geometry.

A central result is the Atiyah-Singer index theorem. It states that the analytical index of an elliptic operator on a compact manifold equals a topological index computed from the symbol of the operator and characteristic classes of the involved bundles, expressed via K-theory and the Chern character. The theorem encompasses and explains several classical results, including the Gauss-Bonnet, Riemann-Roch, and Hirzebruch signature theorems, by interpreting them as special cases of a unified index computation.

Historically, index theory emerged from early 20th-century developments in topology and analysis and was rigorously formulated in its modern form by Michael Atiyah and Isadore Singer in the 1960s. Since then, it has been extended to families of operators, equivariant settings, and noncommutative geometry, as well as to contexts beyond smooth manifolds.

Applications of index theory span geometry, topology, and mathematical physics. In geometry and topology, it provides tools for computing invariants and proving existence results; in physics, indices appear in the study of anomalies and topological field theories. Core ideas include elliptic complexes, the analytical and topological indices, the symbol map, K-theory, and characteristic classes, all linked by the index theorem.