A continuous map of a finite polyhedron,
, has a
*Lefschetz number*:

where is the induced homomorphism in cohomology and denotes the trace. According to the Lefschetz fixed point theorem, if the Lefschetz number of is not zero, then has a fixed point.

In the smooth category the Lefschetz fixed point theorem has a quantitative
refinement. A smooth map
from a compact manifold to itself is
*transversal* if its graph is transversal to the diagonal
in
. Analytically,
is transversal if and only if at each
fixed point
,

where is the differential of at .

The Lefschetz fixed point theorem states that the Lefschetz number of a transversal map is the number of fixed points counted with multiplicity depending on the sign of the determinant :

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 is computable from the de Rham complex

where is the th exterior power of the cotangent bundle. The de Rham complex is an example of an

Let and be vector bundles of ranks and respectively over . An -linear map

is a

where and is an matrix that depends on . The order of is the highest that occurs.

Given a cotangent vector , we write

and define the

In other words, the symbol of is obtained by first discarding all but the highest-order terms of and then replacing by . Because transforms like under a change of coordinates, it is not difficult to show that the symbol is well-defined, independent of the coordinate system.

Let be vector bundles over a manifold . A differential complex

is

is an exact sequence of vector spaces.

A fundamental consequence of ellipticity is that all the cohomology spaces are finite-dimensional.

An *endomorphism* of the complex (3) is a collection of linear
maps
such that

for all . Such a collection induces maps in cohomology . The

A map induces a natural map

by composition: . There is no natural way to induce a map of sections: . However, if there is a bundle map , then the composite

is an endomorphism of . Any bundle map is called a

In the case of the de Rham complex, a map induces a linear map and hence a linear map

which is the lifting that finally defines the pullback of differential forms .

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.