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Atiyah-Singer index theorem

In the mathematics of manifolds and differential operators, the Atiyah-Singer index theorem is a basic general result that came at the end of a long development on the theory of elliptic operators (such as Laplacians), going back to the Riemann-Roch theorem. There have been a number of subsequent developments, in particular in the work of Alain Connes.

We start with a compact smooth manifold (without boundary) and an elliptic operator E on it. Here E is a differential operator acting on smooth sections of a given vector bundle. The property of being elliptic is expressed by a symbol s that can be seen as coming from the coefficients of the highest order part of E; s is a bundle section and required to be non-zero. E.g. for a Laplacian s is a positive-definite quadratic form.

By some basic analytic theory the differential operator E gives rise to a Fredholm operator. Such a Fredholm operator has an index, defined as the difference between the dimension of the kernel of E (solutions of Ef = 0, the harmonic functions in a general sense) and the dimension of the cokernel of E (the constraints on the right-hand-side of an inhomogeneous equation like Ef = g).

The index problem is the following: compute the index of E using only the symbol s and topological data derived from the manifold and the vector bundle. This problem may have been posed in generality first in the late 1950s by Israel Gel'fand. Given the examples from Hodge theory, Cauchy-Riemann operators in several variables, and the topologists' work on the Riemann-Roch Theorem at the time, the required concepts were perhaps all 'up in the air' by 1960.

The precise statement of the Index Theorem requires K-theory, as well as the background in functional analysis and pseudo-differential operators in the manifold setting (sometimes called global analysis). In papers written or published in the period around 1962-1965 the theorem was stated and proved by Michael Atiyah, Raoul Bott and Isadore Singer; it served as a notable unification. The proof required (in effect) the rediscovery of the Dirac equation, and the use of complexes of operators.

Atiyah promoted for a while a notion of elliptic topology for which the index theorem was the central notion. Applications were found on a broad front, for example to fixed-point theory, and the representations of Lie groups.

In a further wave of development, the heat equation was introduced to re-prove the Index Theorem. This became a more standard analytical approach, removing perhaps some of the apparent depth of the theory.

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