Atiyah–Bott fixed-point theorem
In mathematics, the Atiyah-Bott fixed-point theorem is a general form of 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, which can replace the de Rham complex constructed from smooth differential forms.
The idea is to find the correct replacement for the Lefschetz number, which in the classical result is an integer counting the correct contribution of a fixed point of a mapping
- f:M → M.
Intuitively, the fixed points are the intersections 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 topological (homological) calculation , and the RHS a sum of the 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 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
for each j, such that the resulting maps on sections give rise to an endomorphism of the elliptic complex T. Such a T has its Lefschetz number
- L(T)
which by definition is the alternating sum of its traces on each graded part.
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 trace φj, at a fixed point x of f, and &delta(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.