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| In [[mathematics]], the '''Lefschetz fixed-point theorem''' 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]] ''X'' to itself by means of [[trace (linear algebra)|trace]]s of the induced mappings on the [[homology group]]s of ''X''. It is named after [[Solomon Lefschetz]], who first stated it in 1926.
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| 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).
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| ==Formal statement==
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| For a formal statement of the theorem, let
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| :<math>f: X \rightarrow X\,</math>
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| be a [[continuous map]] from a compact [[triangulable space]] ''X'' to itself. Define the '''Lefschetz number''' Λ<sub>''f''</sub> of ''f'' by
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| :<math>\Lambda_f:=\sum_{k\geq 0}(-1)^k\mathrm{Tr}(f_*|H_k(X,\mathbb{Q})),</math>
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| the alternating (finite) sum of the [[matrix trace]]s of the linear maps [[Singular homology#Functoriality|induced]] by ''f'' on the H<sub>''k''</sub>(''X'','''Q'''), the [[singular homology]] of ''X'' with [[rational number|rational]] coefficients. | |
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| A simple version of the Lefschetz fixed-point theorem states: if
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| :<math>\Lambda_f \neq 0\,</math>
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| 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.
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| Note however that the converse is not true in general: Λ<sub>''f''</sub> may be zero even if ''f'' has fixed points.
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| == Sketch of a proof ==
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| First, by applying the [[simplicial approximation theorem]], one shows that if ''f'' has no fixed points, then (possibly after subdividing ''X'') ''f'' 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 ''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#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.
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| == Lefschetz-Hopf theorem ==
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| A stronger form of the theorem, also known as the '''Lefschetz-Hopf theorem''', states that, if ''f'' has only finitely many fixed points, then
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| :<math>\sum_{x \in \mathrm{Fix}(f)} i(f,x) = \Lambda_f,</math>
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| where Fix(''f'') is the set of fixed points of ''f'', and ''i''(''f'',''x'') denotes the [[fixed point index|index]] of the fixed point ''x''.<ref>{{Cite book | last1=Dold | first1=Albrecht | title=Lectures on algebraic topology | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-3-540-10369-1 | id={{MathSciNet | id = 606196}} | year=1980 | volume=200 | postscript=<!--None-->}}, Proposition VII.6.6.</ref>
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| ==Relation to the Euler characteristic==
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| The Lefschetz number of the [[identity function|identity map]] on a finite [[CW complex]] can be easily computed by realizing that each <math>\scriptstyle 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]] χ(''X''). Thus we have
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| :<math>\Lambda_{\mathrm{id}} = \chi(X).\ </math>
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| ==Relation to the Brouwer fixed point theorem==
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| The Lefschetz fixed point theorem generalizes the [[Brouwer fixed point theorem]], which states that every continuous map from the ''n''-dimensional [[unit disk|closed unit disk]] ''D''<sup>''n''</sup> to ''D''<sup>''n''</sup> must have at least one fixed point.
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| This can be seen as follows: ''D''<sup>''n''</sup> is compact and triangulable, all its homology groups except H<sub>0</sub> are 0, and every continuous map ''f'' : ''D''<sup>''n''</sup> → ''D''<sup>''n''</sup> induces a non-zero homomorphism ''f''<sub>*</sub> : H<sub>0</sub>(''D''<sup>''n''</sup>, '''Q''') → H<sub>0</sub>(''D''<sup>''n''</sup>, '''Q'''); all this together implies that Λ<sub>''f''</sub> is non-zero for any continuous map ''f'' : ''D''<sup>''n''</sup> → ''D''<sup>''n''</sup>.
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| ==Historical context==
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| 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 point]]s of mappings.
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| 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
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| :<math>\Lambda_{f,g} = \sum (-1)^k \mathrm{Tr}( D_X \circ g^* \circ D_Y^{-1} \circ f_*),</math> | |
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| where ''f''<sub>∗</sub> is as above, ''g''<sup>∗</sup> is the mapping induced by ''g'' on the [[cohomology]] groups with rational coefficients, and ''D''<sub>''X''</sub> and ''D''<sub>''Y''</sub> are the [[Poincaré duality]] isomorphisms for ''X'' and ''Y'', respectively.
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| 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.
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| == Frobenius ==
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| 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 lift of <math>X\,</math> to the algebraic closure of <math>k</math>. The '''[[Frobenius endomorphism]]''' (often just ''the Frobenius''), notation <math>F_q</math>, of <math>\bar X</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> (i.e. <math>F_q</math> is the ''geometric Frobenius''). Thus the fixed points of <math>F_q</math> are exactly the points of <math>X</math> with coordinates in <math>k</math>, notation for the set of these points: <math>X(k)</math>. The Lefschetz trace formula holds in this context and reads:
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| :<math>\#X(k)=\sum_i (-1)^i \mathop{\rm tr} F_q| H^i_c(\bar X,{\Bbb Q}_\ell).</math>
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| 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 <math>\ell</math>-adic numbers, where <math>\ell</math> is a prime coprime to <math>q</math>.
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| If <math>X</math> is smooth and 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: | |
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| :<math>\#X(k)=q^{\dim X}\sum_i (-1)^i \mathop{\rm tr} \Phi_q^{-1}| H^i(\bar X,{\Bbb Q}_\ell).</math>
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| This formula involves usual cohomology, rather than cohomology with compact supports.
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| The Lefschetz trace formula can also be generalized to [[algebraic stack]]s over finite fields.
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| ==See also==
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| *[[Fixed point theorem]]s
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| *[[Lefschetz zeta function]]
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| *[[Holomorphic Lefschetz fixed-point formula]]
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| ==Notes==
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| <references/>
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| ==References==
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| * {{cite journal | author=Solomon Lefschetz | title=Intersections and transformations of complexes and manifolds | journal=[[Transactions of the American Mathematical Society|Trans. Amer. Math. Soc.]] | year=1926 | volume=28 | pages=1–49 | doi=10.2307/1989171 | issue=1 }}
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| * {{cite journal | author=Solomon Lefschetz | title=On the fixed point formula | journal=[[Annals of Mathematics|Ann. of Math.]] | year=1937 | volume=38 | pages=819–822 | doi=10.2307/1968838 | issue=4 }}
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| ==External links==
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| * {{springer|title=Lefschetz formula|id=p/l057980}}
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| [[Category:Fixed-point theorems]]
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| [[Category:Continuous mappings]]
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| [[Category:Theorems in algebraic topology]]
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