The Atiyah-Bott fixed point theorem

A continuous map of a finite polyhedron, $f:P \to P$ , has a Lefschetz number:

$$L(f) = \sum (-1)^i \operatorname{tr}f^* \vert H^i (P),$$

where $f^*$ is the induced homomorphism in cohomology and $\operatorname{tr}$ denotes the trace. According to the Lefschetz fixed point theorem, if the Lefschetz number of $f$ is not zero, then $f$ has a fixed point.

In the smooth category the Lefschetz fixed point theorem has a quantitative refinement. A smooth map $f: M\to M$ from a compact manifold to itself is transversal if its graph is transversal to the diagonal $\Delta$ in $M \times M$ . Analytically, $f$ is transversal if and only if at each fixed point $p$ ,

$$\det (1- f_{*,p}) \ne 0,$$

where $f_{*,p}: T_pM \to T_pM$ is the differential of $f$ at $p$ .

Figure 7: A transversal map $f$

The $C^{\infty}$ Lefschetz fixed point theorem states that the Lefschetz number of a transversal map $f$ is the number of fixed points $f$ counted with multiplicity $\pm 1$ depending on the sign of the determinant $\det (1- f_{*,p})$ :

$$L(f) = \sum_{f(p)=p} \pm 1.$$

In the Sixties Atiyah and Bott proved a far-reaching generalization of the Lefschetz fixed point theorem ([42], [44]). This type of result, relating a global invariant to a sum of local contributions, is a recurring theme in some of Bott's best work.

To explain it, recall that the real singular cohomology of $M$ is computable from the de Rham complex

$$\Gamma (\Lambda^0) \overset{d}{\to} \Gamma (\Lambda^1) \overset{d}{\to} \Gamma (\Lambda^2) \overset{d}{\to} \to \dots,$$

where $\Lambda^q = \Lambda^q T^*M$ is the $q$ th exterior power of the cotangent bundle. The de Rham complex is an example of an elliptic complex on a manifold.

Let $E$ and $F$ be vector bundles of ranks $r_E$ and $r_F$ respectively over $M$ . An $\mathbb{R}$ -linear map

$$D: \Gamma (E) \to \Gamma (F)$$

is a differential operator if about every point in $M$ there is a coordinate chart $(U, x_1, \dots, x_n)$ and trivializations for $E$ and $F$ relative to which $D$ can be written in the form

$$D=\sum_{\vert\alpha\vert \le m} A^{\alpha}(x) \frac{\partial}{\pa... ...)^{\alpha _1} \dots \left(\frac{\partial}{\partial{x_n}}\right)^{\alpha _n} ,$$

where $\vert\alpha\vert=\sum \alpha_i$ and $A^{\alpha }(x)$ is an $r_F \times r_E$ matrix that depends on $x$ . The order of $D$ is the highest $\vert \alpha \vert$ that occurs.

Given a cotangent vector $\xi = \sum \xi_i dx_i \in T_x^* M$ , we write

$$\xi_{\alpha } = \xi_1^{\alpha _1} \dots \xi_n^{\alpha _n}$$

and define the symbol of a differential operator $D$ of order $m$ to be

$$\sigma (D, \xi)_x = \sum_{\vert\alpha\vert = m} A^{\alpha } (x) \xi _{\alpha } \in \operatorname{Hom} (E_x, F_x).$$

In other words, the symbol of $D$ is obtained by first discarding all but the highest-order terms of $D$ and then replacing $\partial / \partial x^{\alpha }$ by $\xi_{\alpha }$ . Because $\xi_i$ transforms like $\partial / \partial x^i$ under a change of coordinates, it is not difficult to show that the symbol is well-defined, independent of the coordinate system.

Let $E_i$ be vector bundles over a manifold $M$ . A differential complex

$$\mathcal{E}: 0 \to \Gamma(E_0) \overset{D}{\to} \Gamma(E_1) \overset{D}{\to} \Gamma(E_0) \overset{D}{\to} \dots, \quad D^2=0,$$ (3)

is elliptic if for each nonzero cotangent vector $\xi \in T_x^*M$ , the associated symbol sequence

$$0 \to E_{0,x} \xrightarrow{\sigma(D,\xi)} E_{1,x} \xrightarrow{\sigma(D,\xi)} E_{2,x} \xrightarrow{\sigma(D,\xi)} \dots$$

is an exact sequence of vector spaces.

A fundamental consequence of ellipticity is that all the cohomology spaces $H^i = H^i(\Gamma (E_{*}))$ are finite-dimensional.

An endomorphism of the complex (3) is a collection of linear maps $T_i: \Gamma (E_i) \to \Gamma (E_i)$ such that

$$T_{i+1} \mathrel{\scriptstyle\circ}D = D \mathrel{\scriptstyle\circ}T_i$$

for all $i$ . Such a collection $T=\{T_i\}$ induces maps in cohomology $T_i^*:H^i \to H^i$ . The Lefschetz number of $T$ is then defined to be

$$L(T) = \sum (-1)^i \operatorname{tr}T_i^*.$$

A map $f: M\to M$ induces a natural map

$$\Gamma_f : \Gamma (E) \to \Gamma(f^{-1}E)$$

by composition: $\Gamma_f (s) = s \mathrel{\scriptstyle\circ}f$ . There is no natural way to induce a map of sections: $\Gamma (E) \to \Gamma (E)$ . However, if there is a bundle map $\phi: f^{-1}E \to E$ , then the composite

$$\Gamma (E) \overset{\Gamma_f}{\to} \Gamma(f^{-1}E) \overset{\tilde{\phi}}{\to} \Gamma (E)$$

is an endomorphism of $\Gamma (E)$ . Any bundle map $\phi: f^{-1}E \to E$ is called a lifting of $f$ to $E$ . At each point $x\in M$ , a lifting $\phi$ is nothing other than a linear map $\phi_x: E_{f(x)} \to E_x$ .

In the case of the de Rham complex, a map $f: M\to M$ induces a linear map $f_x^*: T_{f(x)}^*M \to T_x^*M$ and hence a linear map

$$\Lambda^q f_x^* : \Lambda^q T_{f(x)}^*M \to \Lambda T_x^* M,$$

which is the lifting that finally defines the pullback of differential forms $f^* : \Gamma(\Lambda^q T^* M) \to \Gamma (\Lambda^q T^* M)$ .

Theorem 1 (Atiyah-Bott fixed point theorem)   Given an elliptic complex (3) on a compact manifold $M$ , suppose $f: M\to M$ has a lifting $\phi_i : f^{-1}E_i \to E_i$ for each $i$ such that the induced maps $T_i: \Gamma (E_i) \to \Gamma (E_i)$ give an endomorphism of the elliptic complex. Then the Lefschetz number of $T$ is given by

$$L(T)= \sum_{f(x)=x} \dfrac{\sum (-1)^i \operatorname{tr}\phi_{i,x}} {\vert\det(1- f_{*,x})\vert}.$$

As evidence of its centrality, the Atiyah-Bott fixed point theorem has an astonishing range of applicability.

Here is an easily stated corollary in algebraic geometry: any holomorphic map of a rational algebraic manifold to itself has a fixed point.

Specializing the Atiyah-Bott fixed point theorem to the de Rham complex, one recovers the classical Lefschetz fixed point theorem. When applied to other geometrically interesting elliptic complexes, Atiyah and Bott obtained new fixed point theorems, such as a holomorphic Lefschetz fixed point theorem in the complex analytic case and a signature formula in the Riemannian case. In the homogeneous case, the fixed point theorem implies the Weyl character formula.