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 (, ). 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 elliptic complex on a manifold.
Let and be vector bundles of ranks and respectively over . An -linear map
is a differential operator if about every point in there is a coordinate chart and trivializations for and relative to which can be written in the form
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 symbol of a differential operator of order to be
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 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 Lefschetz number of is then defined to be
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 lifting of to . At each point , a lifting is nothing other than a linear map .
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.