Atiyah–Singer index theorem

E84379

index theorem mathematical 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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All labels observed (4)

Label	Occurrences
Atiyah–Singer index theorem canonical	23
Atiyah–Singer index theorem for a single operator	1
Dirac operator	1
index theorems	1

How this entity was disambiguated

This entity first appeared as the object of triple T692432 — 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–Singer index theorem
Context triple: [Isadore Singer, notableWork, Atiyah–Singer index theorem]

A. Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
B. Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
C. Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
D. Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
E. Whitney approximation theorem
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
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–Singer index theorem
Target entity description: 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.

A. Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
B. Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
C. Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
D. Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
E. Whitney approximation theorem
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	index theorem ⓘ mathematical theorem ⓘ
appliesTo	compact manifolds ⓘ elliptic pseudodifferential operators ⓘ
concerns	Dirac operator ⓘ index of elliptic operators ⓘ
connects	analysis ⓘ geometry ⓘ topology ⓘ
coreStatement	analytic index equals topological index ⓘ
field	K-theory ⓘ algebraic topology ⓘ differential geometry ⓘ global analysis ⓘ mathematical physics ⓘ operator theory ⓘ topology ⓘ
generalizes	Gauss–Bonnet theorem ⓘ linked to: Gauss–Bonnet theorem (early form) Hirzebruch–Riemann–Roch theorem ⓘ Hopf index theorem ⓘ linked to: Poincaré–Hopf theorem Riemann–Roch theorem ⓘ
hasApplicationIn	gauge theory ⓘ noncommutative geometry ⓘ quantum field theory ⓘ representation theory ⓘ spectral geometry ⓘ string theory ⓘ
hasVariant	equivariant index theorem ⓘ families index theorem ⓘ index theorem for manifolds with boundary ⓘ
implies	integrality of certain characteristic numbers ⓘ
influenced	development of K-theory ⓘ development of modern differential topology ⓘ
namedAfter	Isadore Singer ⓘ Michael Atiyah ⓘ
publishedIn	Annals of Mathematics ⓘ
recognizedAs	landmark result in 20th-century mathematics ⓘ
relates	analytic index ⓘ elliptic differential operators ⓘ topological index ⓘ topological invariants ⓘ
usesConcept	Chern character ⓘ Fredholm operator ⓘ K-theory of vector bundles ⓘ Todd class ⓘ Â-genus ⓘ
yearProved	1963 ⓘ

How these facts were elicited

Referenced by (26)

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

Isadore Singer → notableWork → Atiyah–Singer index theorem ⓘ

Dirac equation → relatedTo → Dirac operator ⓘ

linked to: Atiyah–Singer index theorem

Michael Atiyah → knownFor → Atiyah–Singer index theorem ⓘ

Riemann–Roch theorem → relatedTo → Atiyah–Singer index theorem ⓘ

Poincaré–Hopf theorem → specialCaseOf → Atiyah–Singer index theorem ⓘ

Chern–Weil theory → relatedTo → Atiyah–Singer index theorem ⓘ

Witten index → mathematicallyRelatedTo → Atiyah–Singer index theorem ⓘ

Grothendieck–Riemann–Roch theorem → relatedTo → Atiyah–Singer index theorem ⓘ

K-theory → relatedTo → Atiyah–Singer index theorem ⓘ

Elliptic Operators and Compact Groups → topic → Atiyah–Singer index theorem ⓘ

subject linked to: “Elliptic Operators and Compact Groups” (with Raoul Bott)

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

Hirzebruch–Riemann–Roch theorem → relatedTo → Atiyah–Singer index theorem ⓘ

Hirzebruch–Riemann–Roch theorem → category → index theorems ⓘ

linked to: Atiyah–Singer index theorem

Connes–Moscovici index theorem → generalizes → Atiyah–Singer index theorem ⓘ

Chern character → usedIn → Atiyah–Singer index theorem ⓘ

Todd class → roleIn → Atiyah–Singer index theorem ⓘ

Dirac operator → centralTo → Atiyah–Singer index theorem ⓘ

families index theorem → generalizationOf → Atiyah–Singer index theorem ⓘ

families index theorem → relatedTo → Atiyah–Singer index theorem for a single operator ⓘ

linked to: Atiyah–Singer index theorem

equivariant index theorem → generalizes → Atiyah–Singer index theorem ⓘ

Hirzebruch genera → relatedTo → Atiyah–Singer index theorem ⓘ

Topological Methods in Algebraic Geometry → relatedWork → Atiyah–Singer index theorem ⓘ

Hirzebruch signature theorem → relatedTo → Atiyah–Singer index theorem ⓘ

Pontryagin classes → constrainedBy → Atiyah–Singer index theorem ⓘ

Bott periodicity → relatedTo → Atiyah–Singer index theorem ⓘ

Euler–Poincaré characteristic formula → relatedResult → Atiyah–Singer index theorem ⓘ

All labels observed (4)

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Statements (47)

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Referenced by (26)

How this entity was disambiguated

How these facts were elicited