Donaldson invariants
E508544
Donaldson invariants are sophisticated topological invariants of smooth four-dimensional manifolds derived from moduli spaces of anti-self-dual connections, central to the study of 4-manifold differential topology.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Donaldson invariants canonical | 5 |
| Donaldson polynomials | 1 |
| applications of instanton gauge theory to topology | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5273894 — 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: Donaldson invariants Context triple: [topological quantum field theory, producesInvariant, Donaldson invariants]
-
A.
Seiberg–Witten theory
Seiberg–Witten theory is a framework in quantum field theory and string theory that uses supersymmetry to exactly analyze strongly coupled gauge theories, leading to profound insights into dualities and four-dimensional topology.
-
B.
Lefschetz fibration
A Lefschetz fibration is a smooth map from a higher-dimensional manifold to a lower-dimensional one whose singularities are modeled on complex Morse-type critical points, playing a central role in symplectic and complex geometry.
-
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.
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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E.
Dehn surgery
Dehn surgery is a fundamental operation in 3-manifold topology that modifies a 3-dimensional manifold by cutting out a solid torus and gluing it back in a different way, playing a central role in the classification and study of 3-manifolds.
- 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: Donaldson invariants Target entity description: Donaldson invariants are sophisticated topological invariants of smooth four-dimensional manifolds derived from moduli spaces of anti-self-dual connections, central to the study of 4-manifold differential topology.
-
A.
Seiberg–Witten theory
Seiberg–Witten theory is a framework in quantum field theory and string theory that uses supersymmetry to exactly analyze strongly coupled gauge theories, leading to profound insights into dualities and four-dimensional topology.
-
B.
Lefschetz fibration
A Lefschetz fibration is a smooth map from a higher-dimensional manifold to a lower-dimensional one whose singularities are modeled on complex Morse-type critical points, playing a central role in symplectic and complex geometry.
-
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.
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.
-
E.
Dehn surgery
Dehn surgery is a fundamental operation in 3-manifold topology that modifies a 3-dimensional manifold by cutting out a solid torus and gluing it back in a different way, playing a central role in the classification and study of 3-manifolds.
- F. None of above. chosen
Statements (54)
| Predicate | Object |
|---|---|
| instanceOf |
4-manifold invariant
ⓘ
gauge-theoretic invariant ⓘ smooth structure invariant ⓘ topological invariant ⓘ |
| constructedFrom |
cohomology classes on moduli space of anti-self-dual connections
ⓘ
evaluation of cohomology classes on fundamental class of moduli space ⓘ |
| definedOn |
simply connected 4-manifold
ⓘ
smooth closed oriented 4-manifold ⓘ |
| dependsOn | smooth structure of the 4-manifold ⓘ |
| domain | second homology of the 4-manifold ⓘ |
| field |
4-manifold theory
ⓘ
differential topology ⓘ gauge theory ⓘ geometric topology ⓘ |
| generalizationOf | intersection form invariants ⓘ |
| hasVariant |
Donaldson polynomial
ⓘ
Donaldson series ⓘ polynomial Donaldson invariants ⓘ |
| implies | intersection form of a smooth simply connected definite 4-manifold is diagonalizable over the integers ⓘ |
| independentOf |
Riemannian metric up to deformation
ⓘ
choice of generic perturbations ⓘ |
| inspired |
applications of gauge theory to low-dimensional topology
ⓘ
development of Seiberg–Witten theory ⓘ |
| introducedBy | Simon Donaldson ⓘ |
| namedAfter | Simon Donaldson ⓘ |
| relatedTo |
Floer homology
ⓘ
linked to:
Floer theory
Seiberg–Witten invariants ⓘ Witten’s topological quantum field theory ⓘ
linked to:
Donaldson–Witten theory
instanton Floer homology ⓘ |
| requires |
compactification of moduli space via ideal instantons
ⓘ
generic metric on the 4-manifold ⓘ transversality for moduli space ⓘ |
| takesValuesIn |
cohomology ring of the moduli space
ⓘ
polynomial ring over the rationals ⓘ |
| usedTo |
constrain intersection forms of smooth 4-manifolds
ⓘ
detect exotic smooth structures on 4-manifolds ⓘ distinguish homeomorphic but non-diffeomorphic 4-manifolds ⓘ |
| usesConcept |
Fredholm operator
ⓘ
SU(2) connections ⓘ Uhlenbeck compactness ⓘ Yang–Mills theory ⓘ anti-self-dual connection ⓘ cohomology ⓘ compactification of moduli space ⓘ elliptic partial differential equation ⓘ homology ⓘ index theory ⓘ instanton ⓘ intersection form ⓘ moduli space of anti-self-dual connections ⓘ orientation of moduli space ⓘ principal G-bundle ⓘ smooth four-dimensional manifold ⓘ |
| yearIntroducedApprox | 1980s ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Donaldson invariants Description of subject: Donaldson invariants are sophisticated topological invariants of smooth four-dimensional manifolds derived from moduli spaces of anti-self-dual connections, central to the study of 4-manifold differential topology.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Donaldson invariants
linked to: Donaldson invariants