Editing Lefschetz fixed-point theorem

{{Short description|Counts the fixed points of a continuous mapping from a compact topological space to itself}}
{{More footnotes needed|date=March 2022}}
In [[mathematics]], the '''Lefschetz fixed-point theorem'''<ref name="Lef1926">{{cite journal | first=Solomon|last= Lefschetz|authorlink=Solomon Lefschetz | title=Intersections and transformations of complexes and manifolds | journal=[[Transactions of the American Mathematical Society]] | year=1926 | volume=28 | pages=1–49 | doi=10.2307/1989171 | issue=1|jstor= 1989171|mr=1501331 | doi-access=free }}</ref> is a formula that counts the [[fixed point (mathematics)|fixed point]]s of a [[continuous function (topology)|continuous mapping]] from a [[compact space|compact]] [[topological space]] <math>X</math> to itself  by means of [[trace (linear algebra)|trace]]s of the induced mappings on the [[homology group]]s of <math>X</math>. It is named after [[Solomon Lefschetz]], who first stated it in 1926.

The counting is subject to an imputed [[Multiplicity (mathematics)|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).

==Formal statement==
For a formal statement of the theorem, let

:<math>f\colon X \rightarrow X\,</math>

be a [[continuous map]] from a compact [[triangulable space]] <math>X</math> to itself. Define the '''Lefschetz number'''{{r|EncLNumbers}} <math>\Lambda_f</math> of <math>f</math> by

:<math>\Lambda_f:=\sum_{k\geq 0}(-1)^k\mathrm{tr}(H_k(f,\Q)),</math>

the alternating (finite) sum of the [[matrix trace]]s of the linear maps [[Singular homology#Functoriality|induced]] by <math>f</math> on <math>H_k(X,\Q)</math>, the [[singular homology]] groups of <math>X</math> with [[rational number|rational]] coefficients.

A simple version of the Lefschetz fixed-point theorem states: if

:<math>\Lambda_f \neq 0\,</math>

then <math>f</math> has at least one fixed point, i.e., there exists at least one <math>x</math> in <math>X</math> such that <math>f(x) = x</math>.  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 <math>f</math> has a fixed point as well.

Note however that the converse is not true in general: <math>\Lambda_f</math> may be zero even if <math>f</math> 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]], one shows that if <math>f</math> has no fixed points, then (possibly after subdividing <math>X</math>) <math>f</math> is homotopic to a fixed-point-free [[simplicial map]] (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 homology|simplicial chain complex]] of <math>X</math> 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#Topological definition|Euler characteristic has a definition in terms of homology groups]]; see [[#Relation to the Euler characteristic|below]] 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 <math>f</math> has only finitely many fixed points, then

:<math>\sum_{x \in \mathrm{Fix}(f)} \mathrm{ind}(f,x) = \Lambda_f,</math>

where <math>\mathrm{Fix}(f)</math> is the set of fixed points of <math>f</math>, and <math>\mathrm{ind}(f,x)</math> denotes the [[fixed-point index|index]] of the fixed point <math>x</math>.<ref>{{Cite book | last=Dold | first=Albrecht | authorlink=Albrecht Dold| title=Lectures on algebraic topology | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd  | isbn=978-3-540-10369-1 |mr=606196 | year=1980 | volume=200 }}, Proposition VII.6.6.</ref> From this theorem one may deduce the [[Poincaré–Hopf theorem]] for vector fields as follows. Any [[vector field]] on a compact manifold induces a [[Flow (mathematics)|flow]] <math>\varphi(x,t)</math> in a natural way, and for every <math>t</math> the map <math>\varphi(x,t)</math> is homotopic to the identity (thus having the same Lefschetz number); moreover, for sufficiently small <math>t</math> the fixed points of the flow and the zeroes of the vector field have the same indices.

==Relation to the Euler characteristic==
The Lefschetz number<ref name="EncLNumbers">{{cite web|access-date=2025-01-11 |title=Lefschetz number - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Lefschetz_number |website=encyclopediaofmath.org}}<!-- auto-translated from Polish by Module:CS1 translator --></ref> of the [[identity function|identity map]] on a finite [[CW complex]] can be easily computed by realizing that each <math>f_\ast</math> 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]] <math>\chi(X)</math>. Thus we have
:<math>\Lambda_{\mathrm{id}} = \chi(X).\ </math>

==Relation to the Brouwer fixed-point theorem==
The Lefschetz fixed-point theorem generalizes the [[Brouwer fixed-point theorem]],<ref name="brouwer-1910">{{cite journal | last1 = Brouwer | first1 = L. E. J. | author-link = Luitzen Egbertus Jan Brouwer | year = 1911| title = Über Abbildungen von Mannigfaltigkeiten | url = http://resolver.sub.uni-goettingen.de/purl?GDZPPN002264021 | journal = [[Mathematische Annalen]] | volume = 71 | pages = 97–115 | doi = 10.1007/BF01456931 | s2cid = 177796823 | language = de }}</ref> which states that every continuous map from the <math>n</math>-dimensional [[unit disk|closed unit disk]] <math>D^n</math> to <math>D^n</math> must have at least one fixed point.

This can be seen as follows: <math>D^n</math> is compact and triangulable, all its homology groups except <math>H_0</math> are zero, and every continuous map <math>f\colon D^n \to D^n</math> induces the identity map <math>f_* \colon H_0(D^n, \Q) \to H_0(D^n, \Q)</math>, whose trace is one; all this together implies that <math>\Lambda_f</math> is non-zero for any continuous map <math>f\colon D^n \to D^n</math>.

==Historical context==
Lefschetz presented his fixed-point theorem in {{r|Lef1926}}. Lefschetz's focus was not on fixed points of maps, but rather on what are now called [[coincidence point]]s of maps.

Given two maps <math>f</math> and <math>g</math> from an orientable [[manifold]] <math>X</math> to an orientable manifold <math>Y</math> of the same dimension, the ''Lefschetz coincidence number'' of <math>f</math> and <math>g</math> is defined as

:<math>\Lambda_{f,g} = \sum (-1)^k \mathrm{tr}( D_X \circ g^* \circ D_Y^{-1} \circ f_*),</math>

where <math>f_*</math> is as above, <math>g^*</math> is the homomorphism induced by <math>g</math> on the [[cohomology]] groups with rational coefficients, and <math>D_X</math> and <math>D_Y</math> are the [[Poincaré duality]] isomorphisms for <math>X</math> and <math>Y</math>, respectively.

Lefschetz proved that if the coincidence number is nonzero, then <math>f</math> and <math>g</math> have a coincidence point. He noted in his paper that letting <math>X= Y</math> and letting <math>g</math> be the identity map gives a simpler result, which is now known as the fixed-point theorem.

==Frobenius==
Let <math>X</math> be a variety defined over the [[finite field]] <math>k</math> with <math>q</math> elements and let <math>\bar X</math> be the base change of <math>X</math> to the [[algebraic closure]] of <math>k</math>. The '''[[Frobenius endomorphism]]''' of <math>\bar X</math> (often the ''geometric Frobenius'', or just ''the Frobenius''), denoted by  <math>F_q</math>, maps a point with coordinates <math>x_1,\ldots,x_n</math> to the point with coordinates <math>x_1^q,\ldots,x_n^q</math>. Thus the fixed points of <math>F_q</math> are exactly the points of <math>X</math> with coordinates in <math>k</math>; the set of such points is denoted by <math>X(k)</math>.  The Lefschetz trace formula holds in this context, and reads:

:<math>\#X(k)=\sum_i (-1)^i \mathrm{tr}(F_q^*| H^i_c(\bar{X},\Q_{\ell})).</math>

This formula involves the trace of the Frobenius on the [[étale cohomology]], with compact supports, of <math>\bar X</math> with values in the field of [[p-adic number|<math>\ell</math>-adic numbers]], where <math>\ell</math> is a prime coprime to <math>q</math>.

If <math>X</math> is smooth and [[equidimensionality|equidimensional]], this formula can be rewritten in terms of the ''arithmetic Frobenius'' <math>\Phi_q</math>, which acts as the inverse of <math>F_q</math> on cohomology:

:<math>\#X(k)=q^{\dim X}\sum_i (-1)^i \mathrm{tr}((\Phi_q^{-1})^*| H^i(\bar X,\Q_\ell)).</math>

This formula involves usual cohomology, rather than cohomology with compact supports.

The Lefschetz trace formula can also be generalized to [[algebraic stack]]s over finite fields.

==See also==
*[[Fixed-point theorem]]s
*[[Lefschetz zeta function]]
*[[Holomorphic Lefschetz fixed-point formula]]<ref name="Lef1937">{{cite journal | first=Solomon|last= Lefschetz|authorlink=Solomon Lefschetz| title=On the fixed point formula | journal=[[Annals of Mathematics]] | year=1937 | volume=38 | pages=819–822 | doi=10.2307/1968838 | issue=4|jstor= 1968838|mr=1503373 }}</ref><ref name="Lef1994">{{springer|title=Lefschetz formula|id=p/l057980}}</ref>

==References==
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[[Category:Fixed-point theorems]]
[[Category:Theory of continuous functions]]
[[Category:Theorems in algebraic topology]]