Hirzebruch genera
E586792
Hirzebruch genera are topological invariants in algebraic topology and differential geometry that generalize characteristic classes to classify and study manifolds.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hirzebruch genera canonical | 1 |
| Hirzebruch genus | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6337357 — 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 genera Context triple: [Friedrich Hirzebruch, knownFor, Hirzebruch genera]
-
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.
Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
-
C.
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.
-
D.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
E.
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.
- 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 genera Target entity description: Hirzebruch genera are topological invariants in algebraic topology and differential geometry that generalize characteristic classes to classify and study manifolds.
-
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.
Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
-
C.
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.
-
D.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
generalized cohomology invariant
ⓘ
multiplicative genus ⓘ topological invariant ⓘ |
| appearsIn |
Hirzebruch’s book "Neue topologische Methoden in der algebraischen Geometrie"
ⓘ
linked to:
Topological Methods in Algebraic Geometry
|
| characterizedBy |
formal group law associated to complex cobordism
ⓘ
formal power series in Chern roots ⓘ |
| context | stable homotopy theory ⓘ |
| definedOn |
oriented manifolds
ⓘ
smooth manifolds ⓘ stably almost complex manifolds ⓘ |
| dependsOn |
characteristic classes of the tangent bundle
ⓘ
tangent bundle of the manifold ⓘ |
| field |
algebraic topology
ⓘ
differential geometry ⓘ |
| formalism | expressed via characteristic power series Q(x) in one variable ⓘ |
| generalizes |
arithmetic genus
ⓘ
classical characteristic numbers ⓘ signature of a manifold ⓘ |
| hasExample |
L-genus
ⓘ
Todd genus ⓘ elliptic genus ⓘ Â-genus ⓘ |
| introducedBy | Friedrich Hirzebruch ⓘ |
| invariantUnder |
cobordism equivalence
ⓘ
diffeomorphism of manifolds ⓘ |
| mapsTo |
ring of complex numbers
ⓘ
ring of rational numbers ⓘ |
| namedAfter | Friedrich Hirzebruch ⓘ |
| property |
cobordism invariant
ⓘ
determined by characteristic power series ⓘ multiplicative with respect to cartesian product of manifolds ⓘ |
| relatedTo |
Atiyah–Singer index theorem
ⓘ
Chern classes ⓘ Hirzebruch–Riemann–Roch theorem ⓘ L-genus ⓘ Pontryagin classes ⓘ Todd class ⓘ Todd genus ⓘ complex cobordism ⓘ elliptic genera ⓘ oriented cobordism ⓘ Â-genus ⓘ |
| use |
classification of manifolds
ⓘ
cobordism theory ⓘ index theory ⓘ study of characteristic classes ⓘ |
| usedFor |
computing indices of elliptic operators
ⓘ
distinguishing non-cobordant manifolds ⓘ relating topology of manifolds to algebraic geometry ⓘ |
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: Hirzebruch genera Description of subject: Hirzebruch genera are topological invariants in algebraic topology and differential geometry that generalize characteristic classes to classify and study manifolds.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hirzebruch genera