Thurston norm
E518459
The Thurston norm is a topological invariant on the second homology of a 3-manifold that measures the minimal complexity (via Euler characteristic) of embedded surfaces representing a given homology class.
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
seminorm
ⓘ
topological invariant ⓘ |
| associatedWith |
Thurston’s geometrization program
NERFINISHED
ⓘ
Thurston’s work on norm on homology ⓘ |
| boundaryCondition | surfaces considered are properly embedded ⓘ |
| codomain | nonnegative real numbers ⓘ |
| definedFor | compact 3-manifolds with boundary ⓘ |
| definedOn |
H_2(M; \mathbb{R})
ⓘ
H_2(M; \mathbb{Z}) ⓘ second homology group of a 3-manifold ⓘ |
| definedUsing | minimal value of -\chi(S) over embedded surfaces S ⓘ |
| dependsOn | topology of the 3-manifold ⓘ |
| domain | homology classes in H_2(M; \mathbb{Z}) ⓘ |
| generalizes | notion of genus for knots to higher-dimensional homology classes ⓘ |
| ignores |
disk components of surfaces
ⓘ
spherical components of surfaces ⓘ |
| introducedBy | William P. Thurston NERFINISHED ⓘ |
| introducedInContext | study of hyperbolic 3-manifolds ⓘ |
| introducedInField | 3-manifold topology ⓘ |
| invariantOf | oriented compact 3-manifolds ⓘ |
| invariantUnder | homeomorphisms of the 3-manifold ⓘ |
| mathematicsSubjectClassification |
57M27
ⓘ
57N10 ⓘ |
| measures |
complexity of embedded surfaces
ⓘ
minimal complexity of a homology class ⓘ |
| minimizes | sum of max(0, -\chi(S_i)) over components S_i ⓘ |
| property |
homogeneous on rays
ⓘ
is a seminorm, not necessarily a norm ⓘ satisfies triangle inequality ⓘ subadditive ⓘ |
| relatedInvariant |
Gromov norm
NERFINISHED
ⓘ
simplicial volume ⓘ |
| relatedTo |
fibered faces of the unit ball
ⓘ
fibrations of 3-manifolds over the circle ⓘ |
| unitBall | a finite-sided rational polyhedron in H_2(M; \mathbb{R}) ⓘ |
| unitBallProperty | symmetric about the origin ⓘ |
| usedIn |
classification of 3-manifolds
ⓘ
study of mapping tori ⓘ |
| usedToStudy |
fibered 3-manifolds
ⓘ
hyperbolic structures on 3-manifolds ⓘ taut foliations ⓘ |
| uses | Euler characteristic ⓘ |
| vanishesOn | classes represented by unions of embedded spheres and tori ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.