Atiyah–Bott fixed-point theorem
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.
- f:M → M.
Intuitively, the fixed points are the points of intersection of the graph of f with the diagonal (graph of the identity mapping) in M×M, and the Lefschetz number thereby becomes an intersection number. 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 M×M, a transversality assumption for the graph of f and the diagonal should ensure that the fixed point set is zero-dimensional. Assuming M a closed manifold should ensure then that the set of intersections is finite, yielding a finite summation as the RHS of the expected formula. Further data needed relates to the elliptic complex of vector bundles Ej, namely a bundle map from
- φj:f−1 Ej → Ej
The form of the theorem is then
- L(T) = Σ (Σ (−1)j trace φj,x)/δ(x).
Here trace φj,x means the trace of φj, at a fixed point x of f, and δ(x) is the determinant of the endomorphism I − Df at x, with Df the derivative of f (the non-vanishing of this is a consequence of transversality). The outer summation is over the fixed points x, and the inner summation over the index j in 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.[clarification needed]
The early history of this result is entangled with that of the Atiyah–Singer index theorem. 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). 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.
As Atiyah puts it:
[at the conference]...Bott and I learnt of a conjecture of Shimura concerning a generalization of the Lefschetz formula for holomorphic maps. After much effort we convinced ourselves that there should be a general formula of this type [...]; .
and they were led to a version for elliptic complexes.
- Collected Papers III p.2.
- M. F. Atiyah; R. Bott A Lefschetz Fixed Point Formula for Elliptic Differential Operators. Bull. Am. Math. Soc. 72 (1966), 245–50. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex.
- M. F. Atiyah; R. Bott A Lefschetz Fixed Point Formula for Elliptic Complexes: A Lefschetz Fixed Point Formula for Elliptic Complexes: I II. Applications The Annals of Mathematics 2nd Ser., Vol. 86, No. 2 (Sep., 1967), pp. 374–407 and Vol. 88, No. 3 (Nov., 1968), pp. 451–491. These gives the proofs and some applications of the results announced in the previous paper.