Hirzebruch signature theorem
E587785
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Hirzebruch signature theorem canonical | 3 |
| Hirzebruch L-polynomial | 1 |
| signature theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6337356 — 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: Hirzebruch signature theorem Context triple: [Friedrich Hirzebruch, knownFor, Hirzebruch signature theorem]
-
A.
Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
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B.
Hirzebruch genera
Hirzebruch genera are topological invariants in algebraic topology and differential geometry that generalize characteristic classes to classify and study manifolds.
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C.
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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D.
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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E.
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.
- 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: Hirzebruch signature theorem Target entity description: 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.
-
A.
Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
-
B.
Hirzebruch genera
Hirzebruch genera are topological invariants in algebraic topology and differential geometry that generalize characteristic classes to classify and study manifolds.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appliesTo |
4k-dimensional manifolds
ⓘ
compact manifolds ⓘ oriented manifolds ⓘ smooth manifolds ⓘ |
| assumes |
compactness of the manifold
ⓘ
orientation on the manifold ⓘ smooth structure on the manifold ⓘ |
| codomain | integers ⓘ |
| consequence |
constraints on intersection forms of 4k-manifolds
ⓘ
topological invariance of certain Pontryagin numbers ⓘ |
| defines | L-genus as characteristic number ⓘ |
| dimensionCondition | dimension is a multiple of 4 ⓘ |
| domain | oriented cobordism ring ⓘ |
| expresses |
L-polynomial in terms of Pontryagin classes
ⓘ
signature of a manifold as a polynomial in Pontryagin classes ⓘ |
| field |
algebraic topology
ⓘ
differential topology ⓘ global analysis ⓘ |
| formulaType | characteristic number formula ⓘ |
| generalizationOf | Gauss–Bonnet theorem for 2-dimensional case (in spirit) ⓘ |
| gives | ring homomorphism from oriented cobordism ring to integers via signature ⓘ |
| hasInvariant |
Hirzebruch L-polynomial
ⓘ
linked to:
Hirzebruch signature theorem
|
| historicalPeriod | 20th century mathematics ⓘ |
| implies | signature is a cobordism invariant for oriented manifolds ⓘ |
| inspired | development of index theory ⓘ |
| involves |
intersection form on middle-dimensional cohomology
ⓘ
rational cohomology ⓘ |
| mathematicsSubjectClassification |
57R20
ⓘ
57R75 ⓘ |
| namedAfter | Friedrich Hirzebruch ⓘ |
| relatedTo |
Atiyah–Singer index theorem
ⓘ
Hirzebruch–Riemann–Roch theorem ⓘ Riemann–Roch theorem ⓘ |
| relatesConcept |
L-genus
ⓘ
Pontryagin classes ⓘ characteristic classes ⓘ signature of a manifold ⓘ |
| states | signature of a smooth compact oriented 4k-manifold equals its L-genus ⓘ |
| usedIn |
classification of 4-manifolds
ⓘ
cobordism theory ⓘ study of smooth structures on manifolds ⓘ |
| uses |
Pontryagin classes of the tangent bundle
ⓘ
rational Pontryagin classes ⓘ |
| yearProved | 1950s ⓘ |
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
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Input
Subject: Hirzebruch signature theorem Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hirzebruch signature theorem
linked to: Hirzebruch signature theorem