Hirzebruch genera
E586792
Hirzebruch genera are topological invariants in algebraic topology and differential geometry that generalize characteristic classes to classify and study manifolds.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.