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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 modern mathematics ...
, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a
continuous mapping In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
from a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
X to itself by means of
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...
s of the induced mappings on the
homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
s of X. It is named after
Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...
, who first stated it in 1926. The counting is subject to an imputed
multiplicity Multiplicity may refer to: In science and the humanities * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (philosophy), a philosophical concept * Multiplicity (psychology), having or using mult ...
at a fixed point called the
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. The index can be easily defined in the ...
. 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, let :f\colon X \rightarrow X\, be a
continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
from a compact
triangulable space In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangu ...
X to itself. Define the Lefschetz number \Lambda_f of f by :\Lambda_f:=\sum_(-1)^k\mathrm(f_*, H_k(X,\Q)), the alternating (finite) sum of the
matrix trace In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace o ...
s of the linear maps induced by f on H_k(X,\Q), the
singular homology In algebraic topology, 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(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...
groups of X with
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
coefficients. A simple version of the Lefschetz fixed-point theorem states: if :\Lambda_f \neq 0\, 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 In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to f has a fixed point as well. Note however that the converse is not true in general: \Lambda_f may be zero even if f has fixed points, as is the case for the identity map on odd-dimensional spheres.


Sketch of a proof

First, by applying the
simplicial approximation theorem In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise of the simplest kind. It applies t ...
, one shows that if f has no fixed points, then (possibly after subdividing X) f is homotopic to a fixed-point-free
simplicial map A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of a simplex always span a simplex. Simplicial maps can be used to approximate continuous functions ...
(i.e., it sends each simplex to a different simplex). This means that the diagonal values of the matrices of the linear maps induced on the simplicial chain complex of X must be all be zero. Then one notes that, in general, the Lefschetz number can also be computed using the alternating sum of the matrix traces of the aforementioned linear maps (this is true for almost exactly the same reason that the Euler characteristic has a definition in terms of homology groups; see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) Bottom may refer to: Anatomy and sex * Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
for the relation to the Euler characteristic). In the particular case of a fixed-point-free simplicial map, all of the diagonal values are zero, and thus the traces are all zero.


Lefschetz–Hopf theorem

A stronger form of the theorem, also known as the Lefschetz–Hopf theorem, states that, if f has only finitely many fixed points, then :\sum_ i(f,x) = \Lambda_f, where \mathrm(f) is the set of fixed points of f, and i(f,x) denotes the
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
of the fixed point x., Proposition VII.6.6. From this theorem one deduces the
Poincaré–Hopf theorem In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincar ...
for vector fields.


Relation to the Euler characteristic

The Lefschetz number of the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on a finite
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
can be easily computed by realizing that each f_\ast 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 In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s of the space, which in turn is equal to the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
\chi(X). Thus we have :\Lambda_ = \chi(X).\


Relation to the Brouwer fixed-point theorem

The Lefschetz fixed-point theorem generalizes the
Brouwer fixed-point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simplest ...
, which states that every continuous map from the n-dimensional closed unit disk D^n to D^n must have at least one fixed point. This can be seen as follows: D^n is compact and triangulable, all its homology groups except H_0 are zero, and every continuous map f\colon D^n \to D^n induces the identity map f_* \colon H_0(D^n, \Q) \to H_0(D^n, \Q), whose trace is one; all this together implies that \Lambda_f is non-zero for any continuous map f\colon D^n \to D^n.


Historical context

Lefschetz presented his fixed-point theorem in . Lefschetz's focus was not on fixed points of maps, but rather on what are now called
coincidence point In mathematics, a coincidence point (or simply coincidence) of two functions is a point in their common domain having the same image. Formally, given two functions :f,g \colon X \rightarrow Y we say that a point ''x'' in ''X'' is a ''coinciden ...
s of maps. Given two maps f and g from an orientable
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
X to an orientable manifold Y of the same dimension, the ''Lefschetz coincidence number'' of f and g is defined as :\Lambda_ = \sum (-1)^k \mathrm( D_X \circ g^* \circ D_Y^ \circ f_*), where f_* is as above, g_* is the homomorphism induced by g on the
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
groups with rational coefficients, and D_X and D_Y are the
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. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
isomorphisms for X and Y, respectively. Lefschetz proved that if the coincidence number is nonzero, then f and g have a coincidence point. He noted 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 X be a variety defined over the finite field k with q elements and let \bar X be the base change of X to the algebraic closure of k. The
Frobenius endomorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism ma ...
of \bar X (often the ''geometric Frobenius'', or just ''the Frobenius''), denoted by F_q, maps a point with coordinates x_1,\ldots,x_n to the point with coordinates x_1^q,\ldots,x_n^q. Thus the fixed points of F_q are exactly the points of X with coordinates in k; the set of such points is denoted by X(k). The Lefschetz trace formula holds in this context, and reads: :\#X(k)=\sum_i (-1)^i \mathrm(F_q^*, H^i_c(\bar,\Q_)). This formula involves the trace of the Frobenius on the étale cohomology, with compact supports, of \bar X with values in the field of \ell-adic numbers, where \ell is a prime coprime to q. If X is smooth and equidimensional, this formula can be rewritten in terms of the ''arithmetic Frobenius'' \Phi_q, which acts as the inverse of F_q on cohomology: :\#X(k)=q^\sum_i (-1)^i \mathrm((\Phi_q^)^*, H^i(\bar X,\Q_\ell)). This formula involves usual cohomology, rather than cohomology with compact supports. The Lefschetz trace formula can also be generalized to
algebraic stack In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's repr ...
s over finite fields.


See also

*
Fixed-point theorem In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors cla ...
s *
Lefschetz zeta function In mathematics, the Lefschetz zeta function, zeta-function is a tool used in topological periodic and fixed point (mathematics), fixed point theory, and dynamical systems. Given a continuous map f\colon X\to X, the zeta-function is defined as the fo ...
* Holomorphic Lefschetz fixed-point formula


Notes


References

* *


External links

* {{Authority control Fixed-point theorems Theory of continuous functions Theorems in algebraic topology