About: Lefschetz fixed-point theorem

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In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space to itself by means of traces of the induced mappings on the homology groups of . 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).

Attributes	Values
rdf:type	Thing agent yago:WikicatContinuousMappings yago:WikicatMathematicalTheorems yago:WikicatTheoremsInAlgebraicTopology yago:WikicatTheoremsInTopology yago:Abstraction100002137 yago:Communication100033020 yago:Function113783816 yago:MathematicalRelation113783581 yago:Message106598915 yago:Proposition106750804 yago:Relation100031921 yago:Statement106722453 yago:Theorem106752293 yago:WikicatFixed-pointTheorems
rdfs:label	Teorema del punt fix de Lefschetz (ca) Fixpunktsatz von Lefschetz (de) Teorema del punto fijo de Lefschetz (es) Théorème du point fixe de Lefschetz (fr) Lefschetz fixed-point theorem (en) 렙셰츠 수 (ko) レフシェッツ不動点定理 (ja) Число Лефшеца (ru) Теорема Лефшеца про нерухому точку (uk)
rdfs:comment	En matemàtiques, el teorema del punt fix de Lefschetz és una fórmula que explica els punts fixos d'un mapatge continu a partir d'un espai topològic compacte a si mateix mitjançant traces de les assignacions induïdes en els de . S'anomena així en honor del matemàtic Salomon Lefschetz, el primer en definir-lo el 1926. El recompte està subjecte a una multiplicitat imputada en un punt fix anomenat índex del punt fix. Una versió feble del teorema és suficient per mostrar que un mapatge sense cap punt fix ha de tenir propietats topològiques especials (com ara una rotació d'un cercle). (ca) In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space to itself by means of traces of the induced mappings on the homology groups of . 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). (en) En mathématiques, le théorème du point fixe de Lefschetz est une formule qui compte le nombre de points fixes d'une application continue d'un espace compact X dans lui-même en utilisant les traces des endomorphismes qu'elle induit sur l'homologie de X. Il est nommé d'après Solomon Lefschetz qui l'a démontré en 1926. Chaque point fixe est compté avec sa multiplicité. Une version faible du théorème suffit à démontrer qu'une application qui n'a aucun point fixe doit vérifier certaines propriétés particulières (comme une rotation du cercle). (fr) 数学で、レフシェッツ不動点定理(Lefschetz fixed-point theorem)は、コンパクトな位相空間 X からそれ自身への連続写像の不動点の数を、X のホモロジー群の上の誘導された写像のトレースによって数える公式である。この名称はソロモン・レフシェッツ(Solomon Lefschetz)にちなみ、1926年に彼が最初に提唱した。数え上げの問題は、不動点と呼ばれる点での多重度も考慮して不動点を数える問題である。この定理の弱いバージョンは、全く不動点を持たない写像は、むしろ特別のトポロジー的（円の回転に似た）性質を持つことを示すことができる。 (ja) 일반위상수학에서 렙셰츠 수(Лефшец數, 영어: Lefschetz number)는 콤팩트 공간 위의 연속 자기 함수의 호모토피류에 대응되는 유리수 값의 불변량이다. 렙셰츠 수가 0이 아닌 경우, 렙셰츠 고정점 정리(Лефшец固定點定理, 영어: Lefschetz fixed-point theorem)에 따르면 함수는 고정점을 갖는다. (ko) Число Лефшеца — определённая целочисленная характеристика отображения топологического пространства в себя. (ru) Теорема Лефшеца про нерухому точку — результат у алгебричній топології про існування нерухомих точок неперервного відображення в себе для досить широких класів топологічних просторів. (uk) Beim Fixpunktsatz von Lefschetz handelt es sich um einen topologischen Satz, gemäß dem bei bestimmten stetigen Abbildungen die Existenz eines Fixpunkts gesichert ist. Grundlage des von Solomon Lefschetz 1926 bewiesenen Satzes ist die sogenannte Lefschetz-Zahl, bei der es sich um eine Kenngröße stetiger Abbildungen handelt, die mit Hilfe relativ abstrakter Konzepte der algebraischen Topologie definiert wird und eine Homotopie-Invariante ist. (de) En el ámbito de las matemáticas, el teorema del punto fijo de Lefschetz es una fórmula que permite contar el número de puntos fijos de un mapeo continuo desde un espacio topológico compacto X a sí mismo mediante el uso de trazas de los mapeos inducidos en los grupos homólogos de X. Su nombre hace honor a Solomon Lefschetz, quién fue el que lo descubrió en 1926. (es)
dcterms:subject	Theorems in algebraic topology Fixed-point theorems Continuous mappings
Wikipage page ID	582530 (xsd:integer)
Wikipage revision ID	1104450217 (xsd:integer)
Link from a Wikipage to another Wikipage	Multiplicity (mathematics) Algebraic stack Annals of Mathematics Betti number Compact space Continuous map Mathematics Frobenius endomorphism Equidimensionality Simplicial homology Homotopic Theorems in algebraic topology Trace (linear algebra) Poincaré duality Fixed-point theorem Fixed point (mathematics) Brouwer fixed-point theorem Lefschetz zeta function Fixed-point theorems Continuous mappings Cohomology Coincidence point Holomorphic Lefschetz fixed-point formula Simplicial approximation theorem Manifold CW complex Solomon Lefschetz Continuous function (topology) Identity function Rational number Singular homology Euler characteristic Poincaré–Hopf theorem Fixed-point index Topological space Simplicial map Unit disk Transactions of the American Mathematical Society Triangulable space Matrix trace Homology group
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