Lefschetz fixed-point theorem
Encyclopedia
In mathematics
, the Lefschetz fixed-point theorem is a formula that counts the fixed point
s of a continuous mapping from a compact
topological space
X to itself by means of trace
s of the induced mappings on the homology groups of X. It is named after Solomon Lefschetz
, who first stated it in 1926.
The counting is subject to an imputed multiplicity at a fixed point called the fixed point index
. A weak version of the theorem is enough to show that a mapping without any fixed point must have rather special topological properties (like a rotation of a circle).
be a continuous map from a compact triangulable space X to itself. Define the Lefschetz number Λf of f by
the alternating (finite) sum of the matrix traces of the linear maps induced by f on the Hk(X,Q), the singular homology
of X with rational
coefficients.
A simple version of the Lefschetz fixed-point theorem states: if
then f has at least one fixed point, i.e. there exists at least one x in X such that f(x) = x. In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map homotopic to f has a fixed point as well.
Note however that the converse is not true in general: Λf may be zero even if f has fixed points.
A stronger form of the theorem, also known as the Lefschetz-Hopf theorem, states that, if f has only finitely many fixed points, then
where Fix(f) is the set of fixed points of f, and i(f,x) denotes the index
of the fixed point x.
on a finite CW complex
can be easily computed by realizing that each
can be thought of as an identity matrix, and so each trace term is simply the dimension of the appropriate homology group. Thus the Lefschetz number of the identity map is equal to the alternating sum of the Betti number
s of the space, which in turn is equal to the Euler characteristic
χ(X). Thus we have
, which states that every continuous map from the n-dimensional closed unit disk Dn to Dn must have at least one fixed point.
This can be seen as follows: Dn is compact and triangulable, all its homology groups except H0 are 0, and every continuous map f : Dn → Dn induces a non-zero homomorphism f* : H0(Dn, Q) → H0(Dn, Q); all this together implies that Λf is non-zero for any continuous map f : Dn → Dn.
s of mappings.
Given two maps f and g from an orientable manifold
X to an orientable manifold Y of the same dimension, the Lefschetz coincidence number of f and g is defined as
where f∗ is as above, g∗ is the mapping induced by g on the cohomology
groups with rational coefficients, and DX and DY are the Poincaré duality
isomorphisms for X and Y, respectively.
Lefschetz proves that if the coincidence number is nonzero, then f and g have a coincidence point. He notes in his paper that letting X = Y and letting g be the identity map gives a simpler result, which we now know as the fixed point theorem.
be a variety defined over the finite field 
with 
elements and let 
be the lift of 
to the algebraic closure of 
. The Frobenius endomorphism
(often just the Frobenius), notation
, of 
maps a point with coordinates 
to the point with coordinates 
(i.e. 
is the geometric Frobenius). Thus the fixed points of 
are exactly the points of 
with coordinates in 
, notation for the set of these points: 
. The Lefschetz trace formula holds in this context and reads:
This formula involves the trace of the Frobenius on the étale cohomology, with compact supports, of
with values in the field of 
-adic numbers, where 
is a prime coprime to 
.
If
is smooth and equidimensional, this formula can be rewritten in terms of the arithmetic Frobenius 
, which acts as the inverse of 
on cohomology:
This formula involves usual cohomology, rather than cohomology with compact supports.
The Lefschetz trace formula can also be generalized to algebraic stacks over finite fields.
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 Lefschetz fixed-point theorem is a formula that counts the fixed point
Fixed 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...
s of a continuous mapping from a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X to itself by means of trace
Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
s of the induced mappings on the homology groups of X. It is named after Solomon Lefschetz
Solomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...
, who first stated it in 1926.
The counting is subject to an imputed multiplicity at a fixed point called the fixed point index
Fixed point index
In mathematics, the fixed-point index is a concept in topological fixed-point theory, and in particular Nielsen theory. The fixed-point index can be thought of as a multiplicity measurement for fixed points....
. A weak version of the theorem is enough to show that a mapping without any fixed point must have rather special topological properties (like a rotation of a circle).
Formal statement
For a formal statement of the theorem, letbe a continuous map from a compact triangulable space X to itself. Define the Lefschetz number Λf of f by
the alternating (finite) sum of the matrix traces of the linear maps induced by f on the Hk(X,Q), the singular homology
Singular homology
In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....
of X with rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
coefficients.
A simple version of the Lefschetz fixed-point theorem states: if
then f has at least one fixed point, i.e. there exists at least one x in X such that f(x) = x. In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map homotopic to f has a fixed point as well.
Note however that the converse is not true in general: Λf may be zero even if f has fixed points.
A stronger form of the theorem, also known as the Lefschetz-Hopf theorem, states that, if f has only finitely many fixed points, then
where Fix(f) is the set of fixed points of f, and i(f,x) denotes the index
Fixed point index
In mathematics, the fixed-point index is a concept in topological fixed-point theory, and in particular Nielsen theory. The fixed-point index can be thought of as a multiplicity measurement for fixed points....
of the fixed point x.
Relation to the Euler characteristic
The Lefschetz number of the identity mapIdentity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
on a finite CW complex
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
can be easily computed by realizing that each
can be thought of as an identity matrix, and so each trace term is simply the dimension of the appropriate homology group. Thus the Lefschetz number of the identity map is equal to the alternating sum of the Betti numberBetti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
s of the space, which in turn is equal to the Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
χ(X). Thus we have
Relation to the Brouwer fixed point theorem
The Lefschetz fixed point theorem generalizes the Brouwer fixed point theoremBrouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
, which states that every continuous map from the n-dimensional closed unit disk Dn to Dn must have at least one fixed point.
This can be seen as follows: Dn is compact and triangulable, all its homology groups except H0 are 0, and every continuous map f : Dn → Dn induces a non-zero homomorphism f* : H0(Dn, Q) → H0(Dn, Q); all this together implies that Λf is non-zero for any continuous map f : Dn → Dn.
Historical context
Lefschetz presented his fixed point theorem in [Lefschetz 1926]. Lefschetz's focus was not on fixed points of mappings, but rather on what are now called coincidence pointCoincidence point
In mathematics, a coincidence point of two mappings is a point in their domain having the same image point under both mappings.Formally, given two mappingsf,g \colon X \rightarrow Y...
s of mappings.
Given two maps f and g from an orientable manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
X to an orientable manifold Y of the same dimension, the Lefschetz coincidence number of f and g is defined as
where f∗ is as above, g∗ is the mapping induced by g on the cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
groups with rational coefficients, and DX and DY are the Poincaré duality
Poincaré duality
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
isomorphisms for X and Y, respectively.
Lefschetz proves that if the coincidence number is nonzero, then f and g have a coincidence point. He notes in his paper that letting X = Y and letting g be the identity map gives a simpler result, which we now know as the fixed point theorem.
Frobenius
Let be a variety defined over the finite field with elements and let be the lift of to the algebraic closure of . The Frobenius endomorphismFrobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its pth power...
(often just the Frobenius), notation
, of maps a point with coordinates to the point with coordinates (i.e. is the geometric Frobenius). Thus the fixed points of are exactly the points of with coordinates in , notation for the set of these points: . The Lefschetz trace formula holds in this context and reads:This formula involves the trace of the Frobenius on the étale cohomology, with compact supports, of
with values in the field of -adic numbers, where is a prime coprime to .If
is smooth and equidimensional, this formula can be rewritten in terms of the arithmetic Frobenius , which acts as the inverse of on cohomology:This formula involves usual cohomology, rather than cohomology with compact supports.
The Lefschetz trace formula can also be generalized to algebraic stacks over finite fields.