I’d like to take a break from the previous homotopy-theoretic series of posts and do something a bit more geometric here. I’ll describe the classical Atiyah-Bott fixed point formula for an elliptic complex and one of the applications in the paper. The ultimate goal is for me to  understand some of the more recent rigidity results for genera.

1. The Atiyah-Bott fixed point formula

Let ${M}$ be a compact manifold, and suppose given an endomorphism ${f: M \rightarrow M}$ with finitely many fixed points. The classical Lefschetz fixed-point formula counts the number of fixed points via the supertrace of the action of ${f}$ on cohomology ${H^*(M; \mathbb{R})}$. In other words, if ${F}$ is the fixed point set, we have

$\displaystyle \mathrm{Tr} (f)|_{H^*(M; \mathbb{R})} = \sum_{p \in F} (-1)^{\sigma(p)},$

where ${\sigma}$ is a sign related to the determinant of ${1 - df}$ at ${p}$.

Using the de Rham isomorphism, the groups ${H^*(M; \mathbb{R})}$ are identified with the cohomology of a complex of sections of bundles

$\displaystyle 0 \rightarrow \Gamma(1) \stackrel{d}{\rightarrow} \Gamma(T^* M) \stackrel{d}{\rightarrow} \dots.$

This is an example of an elliptic complex of differential operators: in other words, when one takes the symbol sequence at a nonzero cotangent vector, the induced map of vector spaces is exact. It is a consequence of this that the cohomology groups are finite-dimensional.

The Atiyah-Bott fixed point formula is a striking generalization of the previous fact. Consider an elliptic complex of differential operators on ${M}$,

$\displaystyle 0 \rightarrow \Gamma(E_0 ) \rightarrow \Gamma(E_1) \rightarrow \dots \rightarrow \Gamma(E_r) \rightarrow 0,$

where the ${E_i}$ are vector bundles over ${M}$. The cohomology groups of this complex are finite-dimensional and provide a generalization (not much of a generalization, actually) of the index of an elliptic operator; they thus often hold significant geometric information about ${M}$. (more…)