Atiyah–Bott fixed-point theorem

Formulation

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 smooth mapping

${\displaystyle f:M\to M.}$

Intuitively, the fixed points are the points of intersection of the graph of f with the diagonal (graph of the identity mapping) in ${\displaystyle M\times M}$, 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 ${\displaystyle M\times M}$, 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 ${\displaystyle E_j}$, namely a bundle map

${\displaystyle {\phi }_j:f^{-1}(E_j)\to E_j}$

for each j, such that the resulting maps on sections give rise to an endomorphism of an elliptic complex ${\displaystyle T}$. Such an endomorphism ${\displaystyle T}$ has Lefschetz number

${\displaystyle L(T),}$

which by definition is the alternating sum of its traces on each graded part of the homology of the elliptic complex.

The form of the theorem is then

${\displaystyle L(T)=\sum _x\left(\sum _j(-1{)}^j\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}{\phi }_{j,x}\right)/\delta (x).}$

Here trace ${\displaystyle {\phi }_{j,x}}$ means the trace of ${\displaystyle {\phi }_j}$ at a fixed point x of f, and ${\displaystyle \delta (x)}$ is the determinant of the endomorphism ${\displaystyle I-Df}$ at x, with ${\displaystyle Df}$ 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.