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
StatementFor 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 Classification
Statement Form Connections
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
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Sources
1. Lefschetz fixed-point theorem (Wikipedia)
Wikimedia FoundationLefschetz fixed-point theorem, Weak version of theorem section
A simplest version of the Lefschetz fixed-point theorem states that if Λf ≠ 0, then f has a fixed point.
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.
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