Lefschetz fixed-point theorem

E262120

mathematical theorem theorem in algebraic topology

The Lefschetz fixed-point theorem is a fundamental result in algebraic topology that relates the number of fixed points of a continuous map on a topological space to traces of the induced maps on its homology groups.

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All labels observed (11)

Label	Occurrences
Lefschetz fixed-point theorem canonical	9
Lefschetz trace formula	4
Lefschetz fixed point formula	2
Lefschetz fixed point theorem	2
Holomorphic Lefschetz fixed-point formula	1
Lefschetz fixed-point theorem for flows	1
Lefschetz fixed-point theorem in sheaf cohomology	1
Lefschetz fixed-point theorem in ℓ-adic cohomology	1
Lefschetz–Hopf fixed-point theorem	1
Lefschetz–Hopf theorem	1
topological fixed-point theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T2394220 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Lefschetz fixed-point theorem
Context triple: [Riemann–Hurwitz formula, relatedTo, Lefschetz fixed-point theorem]

A. Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
B. Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
C. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
D. Poincaré–Birkhoff fixed-point theorem
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
E. Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Lefschetz fixed-point theorem
Target entity description: The Lefschetz fixed-point theorem is a fundamental result in algebraic topology that relates the number of fixed points of a continuous map on a topological space to traces of the induced maps on its homology groups.

A. Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
B. Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
C. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
D. Poincaré–Birkhoff fixed-point theorem
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
E. Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in algebraic topology ⓘ
appliesTo	continuous self-maps of compact manifolds ⓘ continuous self-maps of compact triangulable spaces ⓘ continuous self-maps of finite CW-complexes ⓘ
assumes	finite-dimensional homology groups ⓘ suitable compactness conditions on the space ⓘ
conclusion	Nonzero Lefschetz number implies existence of a fixed point ⓘ
coreIdea	global topological data constrain existence of fixed points ⓘ
defines	Lefschetz number ⓘ
field	algebraic geometry ⓘ algebraic topology ⓘ topology ⓘ
formula	L(f) = Σ_k (-1)^k Tr(f_* \| H_k(X)) ⓘ
generalizationOf	Brouwer fixed-point theorem ⓘ Euler characteristic formula for fixed points ⓘ
hasExtension	Lefschetz fixed-point theorem for correspondences ⓘ Lefschetz fixed-point theorem in derived categories ⓘ Lefschetz fixed-point theorem in sheaf cohomology ⓘ linked to: Lefschetz fixed-point theorem
hasVariant	Lefschetz fixed-point theorem for flows ⓘ linked to: Lefschetz fixed-point theorem Lefschetz fixed-point theorem in ℓ-adic cohomology ⓘ linked to: Lefschetz fixed-point theorem Lefschetz–Hopf fixed-point theorem ⓘ linked to: Lefschetz fixed-point theorem equivariant Lefschetz fixed-point theorem ⓘ linked to: Atiyah–Bott fixed-point theorem holomorphic Lefschetz fixed-point formula ⓘ
historicalPeriod	20th-century mathematics ⓘ
implies	maps with no fixed points have Lefschetz number zero ⓘ
inspired	development of cohomological fixed-point formulas ⓘ
involves	Lefschetz number L(f) ⓘ linked to: Lefschetz number alternating sum of traces ⓘ induced maps on homology groups ⓘ
namedAfter	Solomon Lefschetz ⓘ
relatedTo	Atiyah–Bott fixed-point theorem ⓘ Grothendieck–Lefschetz trace formula ⓘ Poincaré–Hopf theorem ⓘ
relates	fixed points of a continuous map ⓘ traces of induced maps on homology groups ⓘ
statement	If the Lefschetz number of a continuous self-map is nonzero, then the map has at least one fixed point. ⓘ
usedIn	algebraic geometry over finite fields ⓘ dynamical systems ⓘ index theory ⓘ topological fixed-point theory ⓘ
usesConcept	Euler characteristic ⓘ fixed point ⓘ homology ⓘ trace of a linear map ⓘ

How these facts were elicited

Referenced by (24)

Full triples — surface form annotated when it differs from this entity's canonical label.

Riemann–Hurwitz formula → relatedTo → Lefschetz fixed-point theorem ⓘ

Solomon Lefschetz → knownFor → Lefschetz fixed-point theorem ⓘ

Schauder fixed-point theorem → category → topological fixed-point theorem ⓘ