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...

subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain...

topology, a branch of mathematics, the lantern relation is a relation that appears between certain Dehn twists in the mapping class group of a surface. The...

problem is a question asked by Jakob Nielsen (1932, pp. 147–148) about whether finite subgroups of mapping class groups can act on surfaces, that was answered...

explicit matrices. The mapping class group of a genus 2 surface is also known to be linear. In some cases the fundamental group of a manifold can be shown...

group is important in the topology of surfaces because there is a connection provided by the Dehn–Nielsen theorem: the extended mapping class group of...

of William Thurston in the late 1970s, who introduced a geometric compactification which he used in his study of the mapping class group of a surface...

generators) for the mapping class group of a surface. The 3-manifold is the one that uses the word as the attaching map for a Heegaard splitting of the 3-manifold...

in the study of the mapping class group. Non-compact surfaces are more difficult to classify. As a simple example, a non-compact surface can be obtained...

(2004). "On the action of the mapping class group for Riemann surfaces of infinite type". Journal of the Mathematical Society of Japan. 56 (4): 1069–1086...

homeomorphisms of orientable surfaces of genus ≥ 2, but the type of a homeomorphism only depends on its associated element of the mapping class group Mod(S)....

turned out to be a fundamental tool in the study of the geometry of the Teichmüller space, of mapping class groups and of Kleinian groups. It was introduced...

automorphisms is the outer automorphism group of a free group, which is similar in some ways to the mapping class group of a surface. Jakob Nielsen (1924) showed...

quotient of Teichmüller space by the mapping class group. In this case it is the modular curve. In the remaining cases, X is a hyperbolic Riemann surface, that...

a. It is a theorem of Max Dehn that maps of this form generate the mapping class group of isotopy classes of orientation-preserving homeomorphisms of...

Other topics of interest Chiral knot Conjugacy problem Freiheitssatz Group isomorphism problem Lotschnittaxiom Mapping class group of a surface Non-Archimedean...

She has made contributions to the study of knots, 3-manifolds, mapping class groups of surfaces, geometric group theory, contact structures and dynamical...

are a special case of mapping tori. Here is the construction: take the Cartesian product of a surface with the unit interval. Glue the two copies of the...

for mapping class groups of closed surfaces. Nielsen transformations were introduced in (Nielsen 1921) to prove that every subgroup of a free group is...

space of a closed surface of genus g {\displaystyle g} is homeomorphic to a sphere of dimension 6 g − 7 {\displaystyle 6g-7} . The action of the mapping class...

particularly surfaces, the homeomorphism group is studied via this short exact sequence, and by first studying the mapping class group and group of isotopically...

a planar Riemann surface (or schlichtartig Riemann surface) is a Riemann surface sharing the topological properties of a connected open subset of the...

especially in F-theory. Elliptic surfaces form a large class of surfaces that contains many of the interesting examples of surfaces, and are relatively well understood...

to" Mapping class group – Group of isotopy classes of a topological automorphism group Permutation group – Group whose operation is composition of permutations...

PMID 17764440, S2CID 17402424 Mosher, Lee (1996), "A user's guide to the mapping class group: once-punctured surfaces", in Baumslag, Gilbert (ed.), Geometric and...

action of the mapping class group, and using the relationship between Dehn twists and the generalized Fenchel–Nielsen flows, he proved the ergodicity of the...

Roman surface Steiner surface Alexander horned sphere Klein bottle Mapping class group Dehn twist Nielsen–Thurston classification Moise's Theorem (see also...

distinguish between two mappings if one can be continuously deformed to the other; thus, only equivalence classes of mappings are summarized. An "addition"...

_{\operatorname {TOP} }(T^{n})\to 1.} The mapping class group of higher genus surfaces is much more complicated, and an area of active research. The torus's Heawood...

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...