equivariant index theorem
E391908
The equivariant index theorem is a generalization of the Atiyah–Singer index theorem that computes indices of elliptic operators while taking into account the action of a symmetry group.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Atiyah–Segal–Singer equivariant index theorem | 1 |
| equivariant index theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3821405 — 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: equivariant index theorem Context triple: [Atiyah–Singer index theorem, hasVariant, equivariant index theorem]
-
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.
Connes–Moscovici index theorem
The Connes–Moscovici index theorem is a fundamental result in noncommutative geometry that generalizes the classical Atiyah–Singer index theorem to the setting of foliations and noncommutative spaces.
-
C.
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.
-
D.
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.
-
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: equivariant index theorem Target entity description: The equivariant index theorem is a generalization of the Atiyah–Singer index theorem that computes indices of elliptic operators while taking into account the action of a symmetry group.
-
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.
Connes–Moscovici index theorem
The Connes–Moscovici index theorem is a fundamental result in noncommutative geometry that generalizes the classical Atiyah–Singer index theorem to the setting of foliations and noncommutative spaces.
-
C.
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.
-
D.
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.
-
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 (48)
| Predicate | Object |
|---|---|
| instanceOf |
generalization of the Atiyah–Singer index theorem
ⓘ
index theorem ⓘ mathematical theorem ⓘ |
| appliesTo |
compact Lie group actions
ⓘ
manifolds with group actions ⓘ |
| associatedWith |
Graeme Segal
ⓘ
Isadore Singer ⓘ Michael Atiyah ⓘ |
| assumes |
ellipticity of the operator
ⓘ
proper group action ⓘ |
| characterizes | index as an element of the representation ring ⓘ |
| computes |
equivariant index of an elliptic operator
ⓘ
virtual character of a group representation ⓘ |
| concerns |
invariance of index under equivariant deformations
ⓘ
localization at fixed points of the group action ⓘ |
| dealsWith |
elliptic differential operators
ⓘ
equivariant K-theory ⓘ equivariant elliptic operators ⓘ fixed point formulas ⓘ group actions ⓘ |
| domain |
compact smooth manifolds
ⓘ
elliptic complexes with group action ⓘ |
| field |
differential geometry
ⓘ
global analysis ⓘ representation theory ⓘ topology ⓘ |
| generalizes | Atiyah–Singer index theorem ⓘ |
| hasApplicationIn |
geometry of group actions
ⓘ
mathematical physics ⓘ representation theory of compact Lie groups ⓘ symplectic geometry ⓘ |
| hasVersion |
Atiyah–Segal–Singer equivariant index theorem
ⓘ
linked to:
equivariant index theorem
Lefschetz fixed point formula ⓘ
linked to:
Lefschetz fixed-point theorem
|
| implies |
character formulas for group representations
ⓘ
fixed point formulas for group actions ⓘ |
| motivation | study of symmetry in elliptic operator theory ⓘ |
| output | class in the representation ring of the group ⓘ |
| relatedTo |
Atiyah–Bott fixed point formula
ⓘ
linked to:
Atiyah–Bott fixed-point theorem
Lefschetz fixed point theorem ⓘ
linked to:
Lefschetz fixed-point theorem
Riemann–Roch theorem ⓘ equivariant Riemann–Roch theorem ⓘ |
| relates |
analytic index
ⓘ
topological index ⓘ |
| usesConcept |
Chern character
ⓘ
K-theory ⓘ Todd class ⓘ equivariant K-theory ⓘ
linked to:
K-theory
equivariant cohomology ⓘ |
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: equivariant index theorem Description of subject: The equivariant index theorem is a generalization of the Atiyah–Singer index theorem that computes indices of elliptic operators while taking into account the action of a symmetry group.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: equivariant index theorem