The geometry of four-manifolds

E613809

academic book mathematics book monograph nonfiction book

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.

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All labels observed (2)

Label	Occurrences
The Geometry of Four-Manifolds	1
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]

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
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 ⓘ Simon Donaldson ⓘ
hasPublisher	Oxford University Press ⓘ
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 ⓘ Yang–Mills instantons ⓘ linked to: Yang–Mills theory 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 ⓘ
targetAudience	graduate students in mathematics ⓘ research mathematicians ⓘ
title	The Geometry of Four-Manifolds ⓘ linked to: The geometry of four-manifolds
usesTool	Yang–Mills instantons ⓘ linked to: Yang–Mills theory elliptic partial differential equations ⓘ gauge theory ⓘ moduli space techniques ⓘ