Lefschetz number
E904006
The Lefschetz number is a topological invariant, computed from the traces of induced maps on homology, that predicts the existence and number of fixed points of a continuous self-map on a topological space.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Lefschetz number canonical | 2 |
| Lefschetz number L(f) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11085957 — 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 number Context triple: [Lefschetz fixed-point theorem, defines, Lefschetz number]
-
A.
Lefschetz fixed-point theorem
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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B.
Lefschetz
Lefschetz is a surname most notably associated with Solomon Lefschetz, a pioneering mathematician in algebraic topology and geometry.
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C.
Lefschetz operator
The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
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D.
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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E.
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.
- 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 number Target entity description: The Lefschetz number is a topological invariant, computed from the traces of induced maps on homology, that predicts the existence and number of fixed points of a continuous self-map on a topological space.
-
A.
Lefschetz fixed-point theorem
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.
-
B.
Lefschetz
Lefschetz is a surname most notably associated with Solomon Lefschetz, a pioneering mathematician in algebraic topology and geometry.
-
C.
Lefschetz operator
The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
fixed point invariant
ⓘ
homotopy invariant ⓘ topological invariant ⓘ |
| appliesTo |
continuous self-map
ⓘ
topological space ⓘ |
| assumes | suitable compactness or finiteness conditions on X ⓘ |
| canBeDefinedUsing |
cohomology groups H^k(X)
ⓘ
trace of f^* on cohomology ⓘ |
| definedFor | map f : X → X ⓘ |
| definitionFormula | L(f) = Σ_k (-1)^k tr(f_* | H_k(X)) ⓘ |
| dependsOn |
induced maps on homology
ⓘ
traces of linear maps ⓘ |
| equals |
Euler characteristic χ(X) when f is identity map on X
ⓘ
sum of fixed point indices for isolated fixed points ⓘ |
| field |
algebraic topology
ⓘ
fixed point theory ⓘ |
| generalizes | Euler characteristic of a space ⓘ |
| hasVariant |
Lefschetz number in cohomology
ⓘ
Lefschetz number with local coefficients ⓘ equivariant Lefschetz number ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies | existence of fixed point if nonzero under Lefschetz fixed point theorem ⓘ |
| interpretsAs | algebraic count of fixed points under suitable hypotheses ⓘ |
| namedAfter | Solomon Lefschetz ⓘ |
| notation | L(f) ⓘ |
| property |
additive with respect to decomposition of space under suitable conditions
ⓘ
depends only on homotopy class of f ⓘ independent of choice of basis on homology ⓘ invariant under homotopy of maps ⓘ multiplicative under product of maps on product spaces ⓘ |
| relatedTo |
Lefschetz fixed point theorem
ⓘ
linked to:
Lefschetz fixed-point theorem
Lefschetz zeta function ⓘ Lefschetz–Hopf theorem ⓘ
linked to:
Lefschetz fixed-point theorem
Nielsen fixed point theory ⓘ Poincaré–Hopf index theorem ⓘ
linked to:
Poincaré–Hopf theorem
|
| requires | finite-dimensional homology groups for standard definition ⓘ |
| specialCaseOf | Reidemeister trace in some contexts ⓘ |
| sumsOver | all homological degrees k ⓘ |
| usedIn |
Morse theory
ⓘ
linked to:
Morse Theory
algebraic geometry ⓘ differential topology ⓘ dynamical systems ⓘ equivariant topology ⓘ |
| usedToStudy |
fixed points of iterates of a map
ⓘ
periodic points in dynamical systems ⓘ |
| uses |
homology with coefficients in a field or ring
ⓘ
singular homology ⓘ |
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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Lefschetz number Description of subject: The Lefschetz number is a topological invariant, computed from the traces of induced maps on homology, that predicts the existence and number of fixed points of a continuous self-map on a topological space.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.