Godbillon–Vey invariant
E911357
The Godbillon–Vey invariant is a characteristic class in differential topology that assigns a real number to certain codimension-one foliations of manifolds, capturing subtle geometric and dynamical properties of their leaf structure.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Godbillon–Vey class | 1 |
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
characteristic class
ⓘ
differential-topological invariant ⓘ foliation invariant ⓘ topological invariant ⓘ |
| appliesTo |
codimension-one foliations
ⓘ
foliations of 3-manifolds ⓘ oriented foliations ⓘ smooth foliations ⓘ |
| associatedWith | codimension-one integrable plane fields ⓘ |
| captures |
dynamical properties of foliations
ⓘ
geometric properties of foliations ⓘ global behavior of leaves ⓘ |
| codomain | real numbers ⓘ |
| cohomologyDegree | 3 ⓘ |
| definedFor | C^2 codimension-one foliations ⓘ |
| definedUsing |
Bott connection
NERFINISHED
ⓘ
cohomology class in degree three ⓘ differential forms ⓘ |
| dependsOn | foliation structure ⓘ |
| field |
differential topology
ⓘ
dynamical systems ⓘ foliation theory ⓘ |
| hasGeneralization | Godbillon–Vey class for higher codimension in some settings NERFINISHED ⓘ |
| independentOf | choice of defining 1-form up to cohomology ⓘ |
| introducedBy |
Claude Godbillon
NERFINISHED
ⓘ
Jacques Vey NERFINISHED ⓘ |
| involves |
a 1-form defining the foliation
ⓘ
a 3-form whose cohomology class is the invariant ⓘ a connection 1-form ⓘ |
| isHomotopyInvariant | for foliations under suitable conditions ⓘ |
| isRealValued | true ⓘ |
| namedAfter |
Claude Godbillon
NERFINISHED
ⓘ
Jacques Vey NERFINISHED ⓘ |
| nontrivialOn |
Reeb-type foliations
ⓘ
certain foliations of the 3-sphere ⓘ |
| relatedTo |
Bott–Haefliger cohomology
NERFINISHED
ⓘ
secondary characteristic classes ⓘ |
| studiedInContextOf |
ergodic theory of foliations
ⓘ
geometric topology of 3-manifolds ⓘ |
| targetCohomologyGroup | third real cohomology group of the manifold ⓘ |
| usedAs | measure of complexity of codimension-one foliations ⓘ |
| usedToDistinguish |
foliations with different dynamical complexity
ⓘ
non-cobordant foliations ⓘ |
| valueType | real number ⓘ |
| yearProposed | 1971 ⓘ |
| zeroFor | many simple product foliations ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Godbillon–Vey class