In
mathematics, the Atiyah–Bott fixed-point theorem, proven by
Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded t ...
and
Raoul Bott in the 1960s, is a general form of the
Lefschetz fixed-point theorem for
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
s ''M'', which uses an
elliptic complex on ''M''. This is a system of
elliptic differential operators on
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 (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
s, generalizing the
de Rham complex
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
constructed from smooth
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s which appears in the original Lefschetz fixed-point theorem.
Formulation
The idea is to find the correct replacement for the
Lefschetz number
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. It is name ...
, which in the classical result is an integer counting the correct contribution of a
fixed point of a smooth mapping
Intuitively, the fixed points are the points of intersection of the
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
of ''f'' with the diagonal (graph of the identity mapping) in
, and the Lefschetz number thereby becomes an
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 (more than 2) curves, and accounting properly for ...
. 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
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equal ...
s in
, 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
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example ...
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
, namely a
bundle map
In mathematics, a bundle map (or bundle morphism) 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. The ...
:
for each ''j'', such that the resulting maps on
sections
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
give rise to an
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a ...
of an
elliptic complex . Such an endomorphism
has ''Lefschetz number''
:
which by definition is the
alternating sum
In mathematics, an alternating series is an infinite series of the form
\sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n
with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternati ...
of its
traces
Traces may refer to:
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* ''Traces'' (book), a 1998 short-story collection by Stephen Baxter
* ''Traces'' series, a series of novels by Malcolm Rose
Music Albums
* ''Traces'' (Classics IV album) or the title song (see below), 1969
* ''Tra ...
on each graded part of the homology of the elliptic complex.
The form of the theorem is then
:
Here trace
means the trace of
at a fixed point ''x'' of ''f'', and
is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
of the endomorphism
at ''x'', with
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 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 . There is a closely related formula for the char ...
in the theory of
Lie groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
.
History
The early history of this result is entangled with that of the
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the spac ...
. There was other input, as is suggested by the alternate name ''Woods Hole fixed-point theorem'' that was used in the past (referring properly to the case of isolated fixed points). A 1964 meeting at
Woods Hole
Woods Hole is a census-designated place in the town of Falmouth in Barnstable County, Massachusetts, United States. It lies at the extreme southwest corner of Cape Cod, near Martha's Vineyard and the Elizabeth Islands. The population was 781 a ...
brought together a varied group:
Eichler started the interaction between fixed-point theorems and automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s. Shimura played an important part in this development by explaining this to Bott at the Woods Hole conference in 1964.
As Atiyah puts it:
[''Collected Papers'' III p.2.]
t the conference
T, or t, is the twentieth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''tee'' (pronounced ), plural ''tees''. It is deri ...
..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 (; 2 February 1935 – 25 August 1989) was a French mathematician who worked, under the guidance of his doctoral advisor Alexander Grothendieck, on derived categories and Verdier duality. He was a close collaborator of Grothe ...
.
Proofs
In the context of
algebraic geometry, the statement applies for smooth and proper varieties over an algebraically closed field. This variant of the Atiyah–Bott fixed point formula was proved by by expressing both sides of the formula as appropriately chosen
categorical trace In category theory, a branch of mathematics, the categorical trace is a generalization of the trace of a matrix.
Definition
The trace is defined in the context of a symmetric monoidal category ''C'', i.e., a category equipped with a suitable noti ...
s.
See also
*
Bott residue formula In mathematics, the Bott residue formula, introduced by , describes a sum over the fixed points of a holomorphic vector field of a compact complex manifold.
Statement
If ''v'' is a holomorphic vector field on a compact complex manifold ''M'', t ...
Notes
References
*. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex.
* and . These gives the proofs and some applications of the results announced in the previous paper.
*
External links
*
*
{{DEFAULTSORT:Atiyah-Bott Fixed-Point Theorem
Fixed-point theorems
Theorems in differential topology