Atiyah–Bott fixed-point theorem
E258614
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Atiyah–Bott fixed-point theorem canonical | 2 |
| Atiyah–Bott fixed point formula | 1 |
| Berline–Vergne localization formula | 1 |
| equivariant Lefschetz fixed-point theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2314482 — 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: Atiyah–Bott fixed-point theorem Context triple: [Michael Atiyah, knownFor, Atiyah–Bott fixed-point theorem]
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A.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
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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.
Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
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D.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
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E.
Chern–Weil theory
Chern–Weil theory is a framework in differential geometry that constructs characteristic classes of vector bundles from curvature forms, linking topology and geometry through invariant polynomials.
- 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: Atiyah–Bott fixed-point theorem Target entity description: 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.
-
A.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
-
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.
Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
-
D.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
-
E.
Chern–Weil theory
Chern–Weil theory is a framework in differential geometry that constructs characteristic classes of vector bundles from curvature forms, linking topology and geometry through invariant polynomials.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
fixed-point theorem
ⓘ
mathematical theorem ⓘ result in equivariant cohomology ⓘ |
| appliesTo |
compact Lie group actions
ⓘ
elliptic complexes ⓘ elliptic differential operators ⓘ |
| assumes |
compactness of the manifold
ⓘ
properness of the group action ⓘ |
| context |
compact complex manifolds
ⓘ
smooth manifolds with group actions ⓘ |
| describes | expression of global invariants in terms of local fixed-point data ⓘ |
| field |
algebraic topology
ⓘ
differential geometry ⓘ equivariant cohomology ⓘ index theory ⓘ representation theory ⓘ |
| generalizes |
Holomorphic Lefschetz fixed-point formula
ⓘ
linked to:
Lefschetz fixed-point theorem
Lefschetz fixed-point theorem ⓘ |
| gives | sum over fixed points formula for equivariant indices ⓘ |
| hasVersion |
equivariant index formula version
ⓘ
holomorphic version ⓘ topological version ⓘ |
| influenced |
development of modern localization techniques
ⓘ
equivariant index theory ⓘ |
| introducedBy |
Michael Atiyah
ⓘ
Raoul Bott ⓘ |
| involves |
equivariant characteristic classes
ⓘ
fixed-point set of a group action ⓘ local contributions from each fixed component ⓘ normal bundle to fixed-point components ⓘ |
| namedAfter |
Michael Atiyah
ⓘ
Raoul Bott ⓘ |
| relatedTo |
Atiyah–Singer index theorem
ⓘ
Berline–Vergne localization formula ⓘ
linked to:
Atiyah–Bott fixed-point theorem
Duistermaat–Heckman formula ⓘ |
| relates | indices of elliptic operators to fixed-point contributions ⓘ |
| usedFor |
applications in algebraic geometry
ⓘ
applications in symplectic geometry ⓘ computing characters of group representations ⓘ computing indices of elliptic operators ⓘ localization computations in equivariant cohomology ⓘ |
| usesConcept |
Chern character
ⓘ
Todd class ⓘ equivariant cohomology ⓘ localization in equivariant cohomology ⓘ |
| yearProposed | 1960s ⓘ |
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: Atiyah–Bott fixed-point theorem Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Atiyah–Bott fixed-point theorem
linked to: Atiyah–Bott fixed-point theorem
linked to: Atiyah–Bott fixed-point theorem