The geometry of four-manifolds
E613809
The Geometry of Four-Manifolds is a foundational monograph in differential geometry that develops the theory of smooth four-dimensional manifolds using gauge theory and Yang–Mills instantons.
All labels observed (1)
| Label | Occurrences |
|---|---|
| The geometry of four-manifolds canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6709028 — 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: The geometry of four-manifolds Context triple: [Simon Donaldson, notableWork, The geometry of four-manifolds]
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A.
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.
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B.
Foliations of Three-Manifolds Which Are Circle Bundles
"Foliations of Three-Manifolds Which Are Circle Bundles" is William Thurston’s influential 1972 doctoral dissertation in geometric topology, where he developed foundational ideas about the structure and classification of foliations on 3-manifolds.
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C.
Donaldson invariants
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.
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D.
Hirzebruch signature theorem
The Hirzebruch signature theorem is a fundamental result in differential topology that expresses the signature of a smooth, compact, oriented 4k-dimensional manifold as a polynomial in its Pontryagin classes.
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E.
Topological Methods in Algebraic Geometry
Topological Methods in Algebraic Geometry is a foundational mathematical monograph by Friedrich Hirzebruch that applies topological techniques, particularly characteristic classes and cobordism theory, to problems in algebraic geometry.
- 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: The geometry of four-manifolds Target entity description: The Geometry of Four-Manifolds is a foundational monograph in differential geometry that develops the theory of smooth four-dimensional manifolds using gauge theory and Yang–Mills instantons.
-
A.
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.
-
B.
Foliations of Three-Manifolds Which Are Circle Bundles
"Foliations of Three-Manifolds Which Are Circle Bundles" is William Thurston’s influential 1972 doctoral dissertation in geometric topology, where he developed foundational ideas about the structure and classification of foliations on 3-manifolds.
-
C.
Donaldson invariants
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.
-
D.
Hirzebruch signature theorem
The Hirzebruch signature theorem is a fundamental result in differential topology that expresses the signature of a smooth, compact, oriented 4k-dimensional manifold as a polynomial in its Pontryagin classes.
-
E.
Topological Methods in Algebraic Geometry
Topological Methods in Algebraic Geometry is a foundational mathematical monograph by Friedrich Hirzebruch that applies topological techniques, particularly characteristic classes and cobordism theory, to problems in algebraic geometry.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
academic book
ⓘ
mathematics book ⓘ monograph ⓘ nonfiction book ⓘ |
| academicDiscipline |
geometric topology
ⓘ
global analysis ⓘ mathematics ⓘ |
| contribution |
detailed exposition of Donaldson’s results on smooth 4-manifolds
ⓘ
foundational reference for the study of smooth four-manifolds ⓘ systematic development of gauge-theoretic methods in 4-manifold topology ⓘ |
| countryOfPublication | United Kingdom ⓘ |
| field |
Yang–Mills theory
ⓘ
differential geometry ⓘ four-manifold theory ⓘ gauge theory ⓘ topology ⓘ |
| focusesOn |
Donaldson invariants
ⓘ
differential-topological invariants ⓘ smooth category of 4-manifolds ⓘ structure of definite and indefinite intersection forms ⓘ |
| hasAuthor |
Peter Kronheimer
NERFINISHED
ⓘ
Simon Donaldson NERFINISHED ⓘ |
| hasPublisher | Oxford University Press NERFINISHED ⓘ |
| hasSectionOn |
anti-self-dual Yang–Mills connections
ⓘ
applications to the classification of smooth 4-manifolds ⓘ compactness and bubbling for instantons ⓘ intersection forms and their constraints ⓘ |
| influenced |
applications of gauge theory in geometry
ⓘ
research in 4-manifold topology ⓘ |
| language | English ⓘ |
| mainSubject |
Donaldson theory
NERFINISHED
ⓘ
Yang–Mills instantons NERFINISHED ⓘ gauge-theoretic invariants of 4-manifolds ⓘ intersection forms of 4-manifolds ⓘ moduli spaces of anti-self-dual connections ⓘ smooth four-dimensional manifolds ⓘ smooth structures on 4-manifolds ⓘ |
| series | Oxford Mathematical Monographs NERFINISHED ⓘ |
| targetAudience |
graduate students in mathematics
ⓘ
research mathematicians ⓘ |
| title | The Geometry of Four-Manifolds NERFINISHED ⓘ |
| usesTool |
Yang–Mills instantons
NERFINISHED
ⓘ
elliptic partial differential equations ⓘ gauge theory ⓘ moduli space techniques ⓘ |
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: The geometry of four-manifolds Description of subject: The Geometry of Four-Manifolds is a foundational monograph in differential geometry that develops the theory of smooth four-dimensional manifolds using gauge theory and Yang–Mills instantons.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.