In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M, which uses an elliptic complex on M. This is a system of elliptic differential operators on vector bundles, generalizing the de Rham complex constructed from smooth differential forms which appears in the original Lefschetz fixed-point theorem.
The Atiyah–Bott theorem is an equation in which the LHS must be the outcome of a global topological (homological) calculation, and the RHS a sum of the local contributions at fixed points of f. Counting codimensions in
, a transversality assumption for the graph of f and the diagonal should ensure that the fixed point set is zero-dimensional.
has Lefschetz number which by definition is the alternating sum of its traces on each graded part of the homology of the elliptic complex.
Specializing the Atiyah–Bott theorem to the de Rham complex of smooth differential forms yields the original Lefschetz fixed-point formula.
A famous application of the Atiyah–Bott theorem is a simple proof of the Weyl character formula in the theory of Lie groups.
There was other input, as is suggested by the alternate name Woods Hole fixed-point theorem that was used in the past (referring properly to the case of isolated fixed points).
[1] A 1964 meeting at Woods Hole brought together a varied group: Eichler started the interaction between fixed-point theorems and automorphic forms.
Shimura played an important part in this development by explaining this to Bott at the Woods Hole conference in 1964.
[2]As Atiyah puts it:[3] [at the conference]...Bott and I learnt of a conjecture of Shimura concerning a generalization of the Lefschetz formula for holomorphic maps.
In the recollection of William Fulton, who was also present at the conference, the first to produce a proof was Jean-Louis Verdier.