Pontryagin classes
E627990
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Pontryagin classes in algebraic topology | 1 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
characteristic classes
ⓘ
cohomology class ⓘ topological invariant ⓘ |
| appliedTo | tangent bundle to define Pontryagin classes of a manifold ⓘ |
| appliesTo | real vector bundles ⓘ |
| are | homotopy invariants for topological manifolds in many cases ⓘ |
| associatedTo | real vector bundle E ⓘ |
| captures | topological information about curvature ⓘ |
| component |
first Pontryagin class
ⓘ
fourth Pontryagin class ⓘ second Pontryagin class ⓘ third Pontryagin class ⓘ |
| constrainedBy |
Atiyah–Singer index theorem
NERFINISHED
ⓘ
Hirzebruch signature theorem NERFINISHED ⓘ |
| constructedFrom | curvature of a connection ⓘ |
| constructedUsing | Chern–Weil theory NERFINISHED ⓘ |
| definedFor | oriented real vector bundles ⓘ |
| definedOn | smooth manifolds ⓘ |
| field | mathematics ⓘ |
| firstNontrivialDegree | degree 4 ⓘ |
| forComplexBundle | expressible in terms of Chern classes of complexification ⓘ |
| helpDistinguish | homeomorphic but not diffeomorphic manifolds ⓘ |
| invariantUnder | bundle isomorphism ⓘ |
| livesIn | H^{4i}(B;Z) ⓘ |
| namedAfter | Lev Pontryagin NERFINISHED ⓘ |
| naturalWithRespectTo | pullback of bundles ⓘ |
| notation | p_i(E) ⓘ |
| parityProperty | lie in degrees divisible by 4 ⓘ |
| relatedTo |
Chern classes
NERFINISHED
ⓘ
Euler class NERFINISHED ⓘ Stiefel–Whitney classes NERFINISHED ⓘ tangent bundle of a manifold ⓘ |
| satisfies | Whitney sum formula NERFINISHED ⓘ |
| stableUnder | Whitney sum with trivial bundles ⓘ |
| subfield | topology ⓘ |
| takesValuesIn | even-degree cohomology ⓘ |
| usedIn |
algebraic topology
ⓘ
classification of smooth manifolds ⓘ classification of vector bundles over spheres ⓘ cobordism theory ⓘ differential geometry ⓘ differential topology ⓘ surgery theory ⓘ |
| usedToDefine |
Hirzebruch signature theorem
NERFINISHED
ⓘ
L-genus ⓘ Pontryagin numbers NERFINISHED ⓘ |
| vanishFor | contractible base space ⓘ |
| WhitneySumFormula | p(E⊕F)=p(E)·p(F) ⓘ |
| zeroFor | all bundles over a point ⓘ |
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Pontryagin classes in algebraic topology