Dehn–Lickorish theorem

E912779

mathematical theorem theorem in low-dimensional topology

The Dehn–Lickorish theorem is a fundamental result in low-dimensional topology stating that the mapping class group of a closed, orientable surface is generated by finitely many Dehn twists.

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Statements (44)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in low-dimensional topology ⓘ
about	Dehn twists NERFINISHED ⓘ closed orientable surfaces ⓘ mapping class group of a surface ⓘ
appliesTo	mapping class group of a closed orientable surface ⓘ mapping class group of a compact connected orientable surface ⓘ
assertsExistenceOf	finite generating set of Dehn twists for the mapping class group ⓘ
concerns	finite generation of mapping class groups ⓘ generation of mapping class groups ⓘ homeomorphisms of surfaces up to isotopy ⓘ
context	compact connected orientable 2-manifolds without boundary ⓘ orientation-preserving homeomorphisms of surfaces ⓘ
field	geometric topology ⓘ low-dimensional topology ⓘ topology ⓘ
hasConsequence	any mapping class can be expressed as a product of right and left Dehn twists ⓘ mapping class group is generated by Dehn twists about nonseparating curves and some separating curves ⓘ
historicalContributor	Max Dehn NERFINISHED ⓘ W. B. R. Lickorish NERFINISHED ⓘ
holdsFor	closed orientable surface of genus at least 1 ⓘ mapping class group of a surface of genus g ≥ 1 ⓘ
implies	every element of the mapping class group of a closed orientable surface can be written as a product of Dehn twists ⓘ the mapping class group of a closed orientable surface is finitely generated ⓘ
isFundamentalResultIn	surface topology ⓘ theory of mapping class groups ⓘ
isUsedIn	3-manifold topology via Heegaard splittings ⓘ construction of presentations of mapping class groups ⓘ study of moduli space of Riemann surfaces ⓘ symplectic topology of surfaces ⓘ theory of Lefschetz fibrations NERFINISHED ⓘ
namedAfter	Max Dehn NERFINISHED ⓘ William Bernard Raymond Lickorish NERFINISHED ⓘ
relatedTo	Dehn twist factorization of mapping classes ⓘ Humphries generators for the mapping class group ⓘ Nielsen–Thurston classification NERFINISHED ⓘ
statesThat	the mapping class group of a closed orientable surface is generated by finitely many Dehn twists ⓘ
typicalProofUses	cutting a surface along simple closed curves ⓘ decomposition of homeomorphisms into twists ⓘ surgery on curves on surfaces ⓘ
usesConcept	Dehn twist NERFINISHED ⓘ isotopy class of homeomorphisms ⓘ mapping class group NERFINISHED ⓘ simple closed curve on a surface ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Dehn twist → appearsIn → Dehn–Lickorish theorem ⓘ