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 (1)
| Label | Occurrences |
|---|---|
| Hirzebruch genera canonical | 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" NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| invariantUnder |
cobordism equivalence
ⓘ
diffeomorphism of manifolds ⓘ |
| mapsTo |
ring of complex numbers
ⓘ
ring of rational numbers ⓘ |
| namedAfter | Friedrich Hirzebruch NERFINISHED ⓘ |
| property |
cobordism invariant
ⓘ
determined by characteristic power series ⓘ multiplicative with respect to cartesian product of manifolds ⓘ |
| relatedTo |
Atiyah–Singer index theorem
NERFINISHED
ⓘ
Chern classes ⓘ Hirzebruch–Riemann–Roch theorem NERFINISHED ⓘ L-genus ⓘ Pontryagin classes NERFINISHED ⓘ Todd class NERFINISHED ⓘ 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
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Instruction
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.