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.
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the mappingclassgroupofasurface, sometimes called the modular group or Teichmüller modular group, is the groupof homeomorphisms of the surface viewed...
subfield of geometric topology, the mappingclassgroup is an important algebraic invariant ofa topological space. Briefly, the mappingclassgroup is a certain...
group is known not to be linear for n at least 4. In contrast with the case of braid groups, it is an open question whether the mappingclassgroup of...
topology, a branch of mathematics, the lantern relation is a relation that appears between certain Dehn twists in the mappingclassgroupofasurface. The...
automorphisms is the outer automorphism groupofa free group, which is similar in some ways to the mappingclassgroupofasurface. Jakob Nielsen (1924) showed...
generators) for the mappingclassgroupofasurface. The 3-manifold is the one that uses the word as the attaching map for a Heegaard splitting of the 3-manifold...
space ofa closed surfaceof genus g {\displaystyle g} is homeomorphic to a sphere of dimension 6 g − 7 {\displaystyle 6g-7} . The action of the mapping class...
Other topics of interest Chiral knot Conjugacy problem Freiheitssatz Group isomorphism problem Lotschnittaxiom Mappingclassgroupofasurface Non-Archimedean...
are a special case ofmapping tori. Here is the construction: take the Cartesian product ofasurface with the unit interval. Glue the two copies of the...
homology of the infinite symmetric group agrees with mapping spaces of spheres. This can also be stated as a relation between the plus construction of BS ∞...
particularly surfaces, the homeomorphism group is studied via this short exact sequence, and by first studying the mappingclassgroup and groupof isotopically...
}(\mathbb {T} ^{n})\to 1.} The mappingclassgroupof higher genus surfaces is much more complicated, and an area of active research. The torus's Heawood...
Roman surface Steiner surface Alexander horned sphere Klein bottle Mappingclassgroup Dehn twist Nielsen–Thurston classification Moise's Theorem (see also...
She has made contributions to the study of knots, 3-manifolds, mappingclassgroupsofsurfaces, geometric group theory, contact structures and dynamical...
Liouville's theorem sharply limits the conformal mappings to a few types. The notion of conformality generalizes in a natural way to maps between Riemannian or...
Fermat surfaceof points (w : x : y : z) in P3 satisfying w5 + x5 + y5 + z5 = 0 by mapping (w : x : y : z) to (w:ρx:ρ2y:ρ3z) where ρ is a fifth root of 1....
algorithm to produce a presentation of the mappingclassgroupofa closed, orientable surface. Their work relied on the notion ofa cut system and moves...
especially in F-theory. Elliptic surfaces form a large classofsurfaces that contains many of the interesting examples ofsurfaces, and are relatively well understood...
a planar Riemann surface (or schlichtartig Riemann surface) is a Riemann surface sharing the topological properties ofa connected open subset of the...
the mappingclassgroup. It is known (for compact, orientable S) that this is isomorphic with the automorphism groupof the fundamental groupof S. This...
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