Atiyah–Bott fixed-point theorem

In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds "M" , which uses an elliptic complex on "M". This is a system of elliptic differential operators on vector bundles, generalizing the de Rham complex constructed from smooth differential forms which appears in the original Lefschetz fixed-point theorem.

Formulation

The idea is to find the correct replacement for the Lefschetz number, which in the classical result is an integer counting the correct contribution of a fixed point of a smooth mapping

:"f":"M" → "M".

Intuitively, the fixed points are the points of intersection of the graph of "f" with the diagonal (graph of the identity mapping) in "M"×"M", and the Lefschetz number thereby becomes an intersection number. The Atiyah-Bott theorem is an equation in which the LHS must be the outcome of a global topological (homological) calculation , and the RHS a sum of the local contributions at fixed points of "f".

Counting codimensions in "M"×"M", a transversality assumption for the graph of "f" and the diagonal should ensure that the fixed point set is zero-dimensional. Assuming "M" a closed manifold should ensure then that the set of intersections is finite, yielding a finite summation as the RHS of the expected formula.Further data needed relates to the elliptic complex of vector bundles "E"_"j", namely a bundle map from

:φ_"j":"f"⁻¹ "E"_"j" → "E"_"j"

for each "j", such that the resulting maps on sections give rise to an endomorphism of the elliptic complex "T". Such a "T" has its "Lefschetz number"

:"L"("T")

which by definition is the alternating sum of its traces on each graded part of the homology of the elliptic complex.

The form of the theorem is then

:"L"("T") = Σ (Σ (−1)^"j" trace φ_"j","x")/δ("x").

Here trace φ_"j","x" means the trace of φ_"j", at a fixed point "x" of "f", and δ("x") is the determinant of the endomorphism I − "Df" at "x", with "Df" the derivative of "f" (the non-vanishing of this is a consequence of transversality). The outer summation is over the fixed points "x", and the inner summation over the index "j" in the elliptic complex.

Specializing the Atiyah-Bott theorem to the de Rham complex of smooth differential forms yields the original Lefschetz fixed-point formula. A famous application of the Atiyah-Bott theorem is a simple proof of the Weyl character formula in the theory of Lie groups.

History

The early history of this result is entangled with that of the Atiyah-Singer index theorem. There was other input, as is suggested by the alternate name "Woods Hole fixed-point theorem" [http://www.whoi.edu/mpcweb/meetings/atiyah_bott_35.html] that was used in the past (referring properly to the case of isolated fixed points). A 1964 meeting at Woods Hole brought together a varied group:

"Eichler started the interaction between fixed-point theorems and automorphic forms. Shimura played an important part in this development by explaining this to Bott at the Woods Hole conference in 1964" [http://www.math.ubc.ca/~cass/macpherson/talk.pdf] .

As Atiyah puts it ["Collected Papers" III p.2.] :

" [at the conference] ...Bott and I learnt of a conjecture of Shimura concerning a generalization of the Lefschetz formula for holomorphic maps. After much effort we convinced ourselves that there should be a general formula of this type [...] "; .

and they were led to a version for elliptic complexes.

In the recollection of William Fulton, who was also present at the conference, the first to produce a proof was Jean-Louis Verdier.

External links

* [http://brauer.math.harvard.edu/history/bott/bottbio/node18.html]

References

*M. F. Atiyah; R. Bott "A Lefschetz Fixed Point Formula for Elliptic Differential Operators." Bull. Am. Math. Soc. 72 (1966), 245-50. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex.
*M. F. Atiyah; R. Bott "A Lefschetz Fixed Point Formula for Elliptic Complexes:" [http://links.jstor.org/sici?sici=0003-486X%28196709%292%3A86%3A2%3C374%3AALFPFF%3E2.0.CO%3B2-N "A Lefschetz Fixed Point Formula for Elliptic Complexes: I"] [http://links.jstor.org/sici?sici=0003-486X%28196811%292%3A88%3A3%3C451%3AALFPFF%3E2.0.CO%3B2-B "II. Applications"] The Annals of Mathematics 2nd Ser., Vol. 86, No. 2 (Sep., 1967), pp. 374-407 and Vol. 88, No. 3 (Nov., 1968), pp. 451-491. These gives the proofs and some applications of the results announced in the previous paper.

Notes