Lefschetz fixed-point theorem

In mathematics, the Lefschetz fixed-point theorem is a formula that counts the number of fixed points of a continuous mapping from a topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named after Solomon Lefschetz, who first stated it in 1926.

Related Topics:
Mathematics - Fixed point - Topological space - Trace - Homology group - Solomon Lefschetz - 1926

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The counting is subject to some imputed multiplicity at a fixed point. 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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For a formal statement, let

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:f:X ightarrow X

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be a continuous map from a compact triangulable space X to itself. A point x of X is a fixed point of f if f(x)=x. Denote the Lefschetz number of f by

Related Topics:
Continuous map - Compact - Triangulable space

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:Lambda_f.,

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By definition this is

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:sum(-1)^kmathrm{Tr}(f_*|H_k(X,mathbb{Q})),

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the alternating (finite) sum of the matrix traces of the linear maps induced by f on the singular homology of X, with rational coefficients.

Related Topics:
Matrix trace - Singular homology - Rational

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Then the Lefschetz fixed-point theorem states that if

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:Lambda_f eq 0,

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then f has a fixed point. In fact, since the Lefschetz number has been defined at the homology level, our conclusion can be extended to say that any map homotopic to f has a fixed point.

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