Lefschetz fixed-point theorem
E262120
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.
All labels observed (10)
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]
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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.
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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.
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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.
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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.
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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 self-linksurface differs ⓘ
surface form:
Lefschetz fixed-point theorem in sheaf cohomology
|
| hasVariant |
Lefschetz fixed-point theorem
self-linksurface differs
ⓘ
surface form:
Lefschetz fixed-point theorem for flows
Lefschetz fixed-point theorem self-linksurface differs ⓘ
surface form:
Lefschetz fixed-point theorem in ℓ-adic cohomology
Lefschetz fixed-point theorem self-linksurface differs ⓘ
surface form:
Lefschetz–Hopf fixed-point theorem
Atiyah–Bott fixed-point theorem ⓘ
surface form:
equivariant Lefschetz 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
ⓘ
surface form:
Lefschetz number L(f)
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Lefschetz fixed-point theorem Description of subject: 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.
Referenced by (17)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
topological fixed-point theorem
this entity surface form:
Lefschetz trace formula
this entity surface form:
Lefschetz trace formula
this entity surface form:
Lefschetz fixed point formula
subject surface form:
SGA 5
this entity surface form:
Lefschetz trace formula
this entity surface form:
Holomorphic Lefschetz fixed-point formula
Lefschetz fixed-point theorem
→
hasVariant
→
Lefschetz fixed-point theorem
self-linksurface differs
ⓘ
this entity surface form:
Lefschetz–Hopf fixed-point theorem
Lefschetz fixed-point theorem
→
hasVariant
→
Lefschetz fixed-point theorem
self-linksurface differs
ⓘ
this entity surface form:
Lefschetz fixed-point theorem for flows
Lefschetz fixed-point theorem
→
hasVariant
→
Lefschetz fixed-point theorem
self-linksurface differs
ⓘ
this entity surface form:
Lefschetz fixed-point theorem in ℓ-adic cohomology
Lefschetz fixed-point theorem
→
hasExtension
→
Lefschetz fixed-point theorem
self-linksurface differs
ⓘ
this entity surface form:
Lefschetz fixed-point theorem in sheaf cohomology
this entity surface form:
Lefschetz fixed point formula
this entity surface form:
Lefschetz fixed point theorem
subject surface form:
Solomon Lefschetz