Atiyah–Bott fixed-point theorem
Encyclopedia
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 bundle
s, generalizing the de Rham complex constructed from smooth differential form
s which appears in the original Lefschetz fixed-point theorem.
of a smooth mapping
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 codimension
s 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 Ej, namely a bundle map
from
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
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
Here trace φj,x means the trace of φj, 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.
. 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:
As Atiyah puts it:
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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
and Raoul Bott
Raoul Bott
Raoul Bott, FRS was a Hungarian mathematician known for numerous basic contributions to geometry in its broad sense...
in the 1960s, is a general form of the Lefschetz fixed-point theorem
Lefschetz fixed-point theorem
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X...
for smooth manifolds M , which uses an elliptic complex
Elliptic complex
In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for...
on M. This is a system of elliptic differential operators on vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s, generalizing the de Rham complex constructed from smooth differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s 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 pointFixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
of a smooth mapping
- f:M → M.
Intuitively, the fixed points are the points of intersection of the graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
of f with the diagonal (graph of the identity mapping) in M×M, and the Lefschetz number thereby becomes an intersection number
Intersection number
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple curves, and accounting properly for tangency...
. The Atiyah–Bott theorem is an equation in which the LHS
Sides of an equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric...
must be the outcome of a global topological (homological) calculation, and the RHS
Sides of an equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric...
a sum of the local contributions at fixed points of f.
Counting codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
s in M×M, a transversality
Transversality
In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology...
assumption for the graph of f and the diagonal should ensure that the fixed point set is zero-dimensional. Assuming M a closed manifold
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is 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 Ej, namely a bundle map
Bundle map
In mathematics, a bundle map is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely...
from
- φj:f−1 Ej → Ej
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 φj,x)/δ(x).
Here trace φj,x means the trace of φj, at a fixed point x of f, and δ(x) is the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
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
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by ....
in the theory of Lie groups.
History
The early history of this result is entangled with that of the Atiyah–Singer index theoremAtiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
. 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:
EichlerEichlerSeveral people are named Eichler:* August W. Eichler was a German botanist.* Fritz Eichler* Dr. Fritz Eichler, German industrial designer and former chief designer at Braun AG* Glenn Eichler...
started the interaction between fixed-point theorems and automorphic formAutomorphic formIn mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...
s. ShimuraGoro Shimurais a Japanese mathematician, and currently a professor emeritus of mathematics at Princeton University.Shimura was a colleague and a friend of Yutaka Taniyama...
played an important part in this development by explaining this to Bott at the Woods Hole conference in 1964.
As Atiyah puts it:
[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
Jean-Louis Verdier
Jean-Louis Verdier was a French mathematician who worked, under the guidance of Alexander Grothendieck, on derived categories and Verdier duality...
.