Atiyah–Bott fixed-point theorem

The Atiyah - Bott fixed point theorem was proved in 1966 by Michael Atiyah and Raoul Bott and generalizes the Lefschetz fixed point theorem for smooth manifolds.

Preliminary remarks

Let be a smooth, closed manifold, then the Lefschetz number

A continuous self-map defined. With the signified by induced map. The Lefschetz number is well defined, because the singular homology of a smooth, compact manifold is finite dimensional as vector spaces. The Atiyah - Bott fixed point theorem now generalizes this statement to a class of Kohomologien and gives a formula for calculating the Lefschetz number.

Be an elliptic complex. That is is a sequence of smooth vector bundles and a sequence ( geometric ) differential operators, so that

Because of the first property you may be of any elliptic complex gain a cohomology and due to the second property, the Kohomologien are finite-dimensional. Be a Kettenendomorphismus. This induces an endomorphism of Kohomologien In analogy to the Lefschetz number one defines

Let be a differentiable function whose graph is transverse to the diagonal in. The fixed points of are precisely the points of intersection of the graph with the diagonal. From the transversality follows for all fixed points that is true, where the derivative of the point. A lift in excess of one is a sequence of complex elliptical Bündelhomomorphismen so that for having

The identity is valid. In particular is an endomorphism of cuts in the elliptic complex.

Atiyah - Bott fixed point formula

Let be a smooth, closed manifold and a differentiable map so that its graph is transverse to the diagonal is. Be also an elliptic complex, a lift of and by defined endomorphism. Then the Lefschetz number is by

Determined, the track of said at a fixed point of and the derivative of in is.

An application of the Atiyah - Bott fixed-point theorem is a simple proof of the Weyl character formula for the representation of Lie groups.

Special case

Be the de Rham complex, here is the algebra of differential forms and the Cartan derivative. This is an elliptic complex, so you can apply the fixed point formula on this complex. Be back a differentiable map so that its graph is transverse to the diagonal, and the corresponding lift. Then for the index

Since is differentiable and only isolated fixed points has this corresponds to the fixed point formula of Lefschetz.

History

The early history is connected to the Atiyah-Singer index theorem. In a narrower sense, the first ideas at a conference in 1964 (hence the Woods Hole fixed point theorem called ) originated in Woods Hole, Massachusetts. Apparently our original motivation comes from a remark of Martin Eichler on the relationship of fixed point sets and automorphic forms, which Goro Shimura explained at the conference Raoul Bott. He suspected the existence of a Lefschetz fixed point theorem for holomorphic maps.