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]
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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.
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.
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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.
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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.
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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: 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.
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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.
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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 |
equivariant index theorem
self-linksurface differs
ⓘ
surface form:
Atiyah–Segal–Singer equivariant index theorem
Lefschetz fixed-point theorem ⓘ
surface form:
Lefschetz fixed point formula
|
| 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 theorem
ⓘ
surface form:
Atiyah–Bott fixed point formula
Lefschetz fixed-point theorem ⓘ
surface form:
Lefschetz fixed point theorem
Riemann–Roch theorem ⓘ equivariant Riemann–Roch theorem ⓘ |
| relates |
analytic index
ⓘ
topological index ⓘ |
| usesConcept |
Chern character
ⓘ
K-theory ⓘ Todd class ⓘ K-theory ⓘ
surface form:
equivariant K-theory
equivariant cohomology ⓘ |
How these facts were elicited
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Instruction
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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.
this entity surface form:
Atiyah–Segal–Singer equivariant index theorem