van Kampen diagram
E911230
A van Kampen diagram is a planar, combinatorial 2-complex used in combinatorial group theory to visually represent relations in a group presentation and to prove that a word equals the identity.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
combinatorial 2-complex
ⓘ
planar diagram ⓘ tool in combinatorial group theory ⓘ topological object ⓘ |
| basedOn |
group presentation
ⓘ
set of generators of a group ⓘ set of relators of a group ⓘ |
| boundaryRepresents |
element of the group
ⓘ
word in the generators ⓘ |
| canBeEmbeddedIn | plane ⓘ |
| canBeViewedAs | map of a 2-disc into a presentation complex ⓘ |
| encodes |
factorization of a word into conjugates of relators
ⓘ
proof that a word is equal to the identity ⓘ |
| field |
combinatorial group theory
ⓘ
geometric group theory ⓘ low-dimensional topology ⓘ |
| hasAlternativeName | disc diagram ⓘ |
| hasPart |
2-cells labeled by relators
ⓘ
basepoint on the boundary ⓘ boundary cycle ⓘ edges ⓘ edges labeled by generators and their inverses ⓘ faces ⓘ vertices ⓘ |
| hasProperty |
combinatorial
ⓘ
finite 2-dimensional CW-complex ⓘ planar ⓘ simply connected ⓘ |
| namedAfter | Egbert van Kampen NERFINISHED ⓘ |
| relatedTo |
Cayley 2-complex
NERFINISHED
ⓘ
Dehn diagram NERFINISHED ⓘ Dehn function NERFINISHED ⓘ isoperimetric function of a group ⓘ presentation complex ⓘ word problem for groups ⓘ |
| satisfies |
each 2-cell boundary label is a relator or its inverse
ⓘ
edge labels are consistent on adjacent faces ⓘ outer boundary label is the given word ⓘ |
| topologicallyEquivalentTo | disc diagram ⓘ |
| usedFor |
constructing null-homotopies of loops in a 2-complex
ⓘ
proving that a word equals the identity in a group ⓘ representing relations in a group presentation ⓘ studying the word problem in groups ⓘ visualizing relations among generators of a group ⓘ |
| usedIn |
combinatorial curvature arguments
ⓘ
hyperbolic group theory ⓘ small cancellation theory ⓘ |
| usedInProofOf |
equivalence of words in a group presentation
ⓘ
van Kampen theorem in algebraic topology NERFINISHED ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.