Atiyah–Bott fixed-point theorem

E258614

fixed-point theorem mathematical theorem result in equivariant cohomology

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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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]

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
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	Lefschetz fixed-point theorem ⓘ surface form: Holomorphic Lefschetz fixed-point formula 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 ⓘ Atiyah–Bott fixed-point theorem self-linksurface differs ⓘ surface form: Berline–Vergne localization formula 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

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Michael Atiyah → knownFor → Atiyah–Bott fixed-point theorem ⓘ

Atiyah–Bott fixed-point theorem → relatedTo → Atiyah–Bott fixed-point theorem self-linksurface differs ⓘ

this entity surface form: Berline–Vergne localization formula

Lefschetz fixed-point theorem → hasVariant → Atiyah–Bott fixed-point theorem ⓘ

this entity surface form: equivariant Lefschetz fixed-point theorem

Lefschetz fixed-point theorem → relatedTo → Atiyah–Bott fixed-point theorem ⓘ

equivariant index theorem → relatedTo → Atiyah–Bott fixed-point theorem ⓘ

this entity surface form: Atiyah–Bott fixed point formula

All labels observed (4)

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Statements (46)

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Referenced by (5)

How this entity was disambiguated

How these facts were elicited