Theorem

Lefschetz Fixed-Point Theorem

Topology

Gives a criterion, in terms of a computable trace invariant on homology called the Lefschetz number, for a continuous self-map of a compact space to have a fixed point. Named for Solomon Lefschetz, it generalizes the Brouwer fixed-point theorem.

Facts

Statement

For a continuous map from a compact triangulable space to itself, the Lefschetz fixed-point theorem gives a criterion for the map to have a fixed point: if the Lefschetz number, the alternating sum of the traces of the maps the function induces on the rational homology groups of the space, is nonzero, the map must have a fixed point. 1

Proof Year

1926 1

Classification

Statement Form

Identity or Equation 1

Connections

Source Lefschetz fixed-point theorem (Wikipedia)

Sources

1. Lefschetz fixed-point theorem (Wikipedia)

Wikimedia Foundation

Historical context
Lefschetz presented his fixed-point theorem in his 1926 paper about mappings on manifolds.
Lefschetz fixed-point theorem, lead section
It is named after Solomon Lefschetz, who first stated it in 1926 but in different way involving coincidence points of functions.
Proved By: Solomon Lefschetz, Lead
It is named after Solomon Lefschetz, who first stated it in 1926
In Group: Fixed-Point Theorems, Article text
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact triangulable topological space X {\displaystyle X} to itself by means of traces of the induced mappings on the homology groups of X {\displaystyle X} .

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Lefschetz Fixed-Point Theorem

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