Index theorem of Atiyah–Singer

Index theorem of Atiyah–Singer
Name	Index theorem of Atiyah–Singer
Field	Differential geometry, Algebraic topology, Functional analysis
Introduced	1963
Authors	Michael Atiyah, Isadore Singer
Notable for	Relation between analytical index and topological index

Index theorem of Atiyah–Singer The Index theorem of Atiyah–Singer asserts an equality between an analytical invariant of an elliptic differential operator and a topological invariant computed from characteristic classes. It connects ideas developed in Bernhard Riemann's analysis, Henri Poincaré's topology, Élie Cartan's differential geometry and later work by John von Neumann and Israel Gelfand. The theorem catalyzed interactions among researchers at institutions like University of Cambridge, Massachusetts Institute of Technology, Princeton University, and Institute for Advanced Study.

Statement of the theorem

The theorem gives that for an elliptic differential operator on a compact manifold the analytical index (dimension of kernel minus dimension of cokernel) equals a topological index evaluated via characteristic classes. It formalizes a relationship between the Fredholm index studied by Atle Selberg and Frigyes Riesz and the topological invariants used by L.E.J. Brouwer and Henri Lebesgue. In the oriented case the formula involves the Todd class from work of Jules Henri Poincaré and Salomon Bochner, the Â-genus related to Hermann Weyl's representation theory, and Chern characters introduced by Shiing-Shen Chern. For complex elliptic operators on complex manifolds the statement reduces to classical results of Fritz Noether and Ivar Fredholm as extended by Atiyah and Singer.

Historical development and motivation

Motivations trace through attempts to generalize the Riemann–Roch theorem of Georg Riemann and Ferdinand von Riemann's successors, including Bernhard Riemann's early work and later algebraic formulations by Gustav Roch and Alexander Grothendieck. Precursors include the Hirzebruch–Riemann–Roch theorem by Friedrich Hirzebruch and index computations in spectral theory by Marcel Riesz and Erwin Schrödinger's spectral problems. The joint work of Michael Atiyah and Isadore Singer synthesized inputs from Raoul Bott's fixed point theorem, Jean-Pierre Serre's sheaf cohomology, and André Weil's characteristic classes. Seminal conferences at International Congress of Mathematicians and collaborations with researchers from University of Oxford, Stanford University, and University of Chicago accelerated adoption across topology, analysis, and mathematical physics.

Key examples and special cases

Classic instances include the Gauss–Bonnet theorem for the Euler characteristic proved earlier by Carl Friedrich Gauss and Pierre Ossian Bonnet and the Hirzebruch signature theorem studied by Hirzebruch and René Thom. The Dolbeault operator on complex manifolds yields the Hirzebruch–Riemann–Roch formula used by Kunihiko Kodaira and Shing-Tung Yau; the de Rham complex relates to Hodge theory of W.V.D. Hodge. The Dirac operator studied by Paul Dirac and extended by Friedrich Hirzebruch and Raoul Bott produces the Â-genus interpretation crucial to work by Edward Witten in quantum field theory and by Mikhail Gromov in geometry. Elliptic operators on orbifolds tie to contributions by William Thurston and later applications by Maxim Kontsevich.

Proof ideas and methods

Proofs combine analytic techniques for Fredholm operators from Iosif Gelfand and Mark Grigorievich Krein with topological K-theory introduced by Atiyah and G. B. Segal and cohomological characteristic class machinery by Chern. Core methods exploit heat equation proofs developed by Daniel Quillen and Patodi-type analyses connected to Klaus P. Friedrich and M. F. Atiyah's collaborators. Other approaches use cobordism techniques from René Thom and equivariant localization associated to Berline, Nicolas Berline, and Michèle Vergne as well as fixed point formulas from Raoul Bott. Index computations employ spectral flow concepts related to M. F. Atiyah's prior studies and analytic torsion examined by Ray and Singer.

Applications in mathematics and physics

The theorem underpins developments in Algebraic geometry via Grothendieck's Riemann–Roch, influences classification results in Differential topology by Milnor and Kervaire, and informs moduli problems studied by Donaldson and Simon Donaldson's collaborators. In physics it provides the mathematical basis for anomalies in quantum field theory explored by Alvarez-Gaumé and Edward Witten, and underlies index computations in string theory research by Cumrun Vafa and Ashoke Sen. It impacted the study of topological phases by Frank Wilczek, influenced noncommutative geometry pioneered by Alain Connes, and guided developments in geometric quantization associated with Kirillov and Julius Wess.

Generalizations and related index theorems

Generalizations include the equivariant index theorem by Berline and Vergne, the families index theorem by Atiyah and Singer themselves, and index results in noncompact settings developed by M. F. Atiyah's school and Bunke and Schick. Extensions to foliations were advanced by Alain Connes and Paul Baum while index theory on manifolds with boundary draws on work of Vladimir G. Maz'ya and Richard Melrose. Analogs in Noncommutative geometry and cyclic cohomology were developed by Connes and applied by Matilde Marcolli; arithmetic versions relate to Arakelov geometry studied by Gunnlaugur and Serre-inspired arithmetic geometers. Current research connects index theory to categorical frameworks by Jacob Lurie and homotopy-theoretic refinements influenced by Mike Hopkins and Ando.

Category:Differential geometry