About: Mapping class group of a surface

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In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves. The mapping class group of surfaces are related to various other groups, in particular braid groups and outer automorphism groups.

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rdfs:label	Mapping class group of a surface (en) Группа классов преобразований поверхности (ru)
rdfs:comment	Группа классов преобразований поверхности — это группа гомеоморфизмов с точностью до непрерывной деформации. Она естественно возникает при изучении трёхмерных многообразий и связана с другими группами, в частности с группами кос и группой внешних автоморфизмов группы. Группа классов отображений может быть определена для произвольных многообразий и для произвольных топологических пространств, но случай поверхностей является наиболее изученным в теории групп. (ru) In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves. The mapping class group of surfaces are related to various other groups, in particular braid groups and outer automorphism groups. (en)
dct:subject	Geometric group theory Geometric topology
Wikipage page ID	31060586 (xsd:integer)
Wikipage revision ID	1068759389 (xsd:integer)
Link from a Wikipage to another Wikipage	Properly discontinuous action Braid group Riemann surface Curve complex Intersection number Jakob Nielsen (mathematician) Geometric group theory Compact-open topology Countable set Counterclockwise Max Dehn Geometric group theory Out(Fn) Modular group Annulus (mathematics) Arithmetic group Closed manifold Identity component Surface (topology) Symplectic group Weil–Petersson metric Topology Torus William Thurston Lantern relation Locally finite space Open problem Algebraic geometry 3-manifold Nielsen realisation problem Outer automorphism group Diffeomorphism Free action Residually finite group Geometric topology Joan Birman Cohomological dimension Homeomorphism Teichmüller space Tits alternative Moduli space Manifold Mapping class group Free group Group theory Thurston boundary Singular homology Finitely generated group Finitely presented group Pair of pants (mathematics) Topological quantum field theory Linear representation Hurwitz automorphisms theorem Homology group
Link from a Wikipage to an external page	https://www.ams.org/journals/proc/2010-138-02/S0002-9939-09-10124-7/
sameAs	Mapping class group of a surface Mapping class group of a surface Mapping class group of a surface
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Cite_journal dbt:Efn dbt:Harvtxt dbt:Main dbt:Notelist dbt:Refbegin dbt:Refend dbt:Reflist dbt:Sfn dbt:Harvid dbt:Harvnb
has abstract	In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves. The mapping class group can be defined for arbitrary manifolds (indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in group theory. The mapping class group of surfaces are related to various other groups, in particular braid groups and outer automorphism groups. (en) Группа классов преобразований поверхности — это группа гомеоморфизмов с точностью до непрерывной деформации. Она естественно возникает при изучении трёхмерных многообразий и связана с другими группами, в частности с группами кос и группой внешних автоморфизмов группы. Группа классов отображений может быть определена для произвольных многообразий и для произвольных топологических пространств, но случай поверхностей является наиболее изученным в теории групп. (ru)
prov:wasDerivedFrom	wikipedia-en:Mapping_class_group_of_a_surface?oldid=1068759389&ns=0
page length (characters) of wiki page	30900 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Mapping_class_group_of_a_surface
is Link from a Wikipage to another Wikipage of	Riemann surface Dehn twist Max Dehn Fundamental polygon Linear group Scott core theorem Dan Margalit (mathematician) Joan Birman Automorphism group of a free group Thurston boundary Cannon–Thurston map Teichmuller modular group Teichmüller modular group Dehn-Nielsen theorem Dehn–Nielsen theorem
is Wikipage redirect of	Teichmuller modular group Teichmüller modular group

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