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Mapping class group, the Glossary

In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space.^[1]

75 relations: Ambient isotopy, Automorphism, Braid group, Bulletin of the American Mathematical Society, Category of topological spaces, Category theory, Cohomology, Compact space, Compact-open topology, Connected space, Cup product, Cyclic group, David Mumford, Degree of a continuous mapping, Dehn twist, Diffeomorphism, Differentiable manifold, Dihedral group, Direct limit, Direct sum, Discrete group, Eilenberg–MacLane space, Exact sequence, Genus (mathematics), Geometric topology, Homeomorphism, Homeomorphism group, Homeotopy, Homology (mathematics), Homotopy, Homotopy category, Homotopy group, Homotopy sphere, Hyperbolic group, Hyperbolic link, Ib Madsen, If and only if, Intersection (set theory), Inventiones Mathematicae, Inverse function, Klein bottle, Knot (mathematics), Lantern relation, Link (knot theory), Low-dimensional topology, Manifold, Mathematics, Möbius strip, Michael Weiss (mathematician), Moduli space, ... Expand index (25 more) »
Homeomorphisms

Ambient isotopy

In the mathematical subject of topology, an ambient isotopy, also called an h-isotopy, is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold.

See Mapping class group and Ambient isotopy

Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

See Mapping class group and Automorphism

Braid group

In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see); and in monodromy invariants of algebraic geometry.

See Mapping class group and Braid group

Bulletin of the American Mathematical Society

The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.

See Mapping class group and Bulletin of the American Mathematical Society

Category of topological spaces

In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps.

See Mapping class group and Category of topological spaces

Category theory

Category theory is a general theory of mathematical structures and their relations.

See Mapping class group and Category theory

Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex.

See Mapping class group and Cohomology

Compact space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.

See Mapping class group and Compact space

Compact-open topology

In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces.

See Mapping class group and Compact-open topology

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets.

See Mapping class group and Connected space

Cup product

In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H∗(X), called the cohomology ring.

See Mapping class group and Cup product

Cyclic group

In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently \Zn or Zn, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element.

See Mapping class group and Cyclic group

David Mumford

David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory.

See Mapping class group and David Mumford

Degree of a continuous mapping

In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping.

See Mapping class group and Degree of a continuous mapping

Dehn twist

In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold). Mapping class group and Dehn twist are geometric topology and homeomorphisms.

See Mapping class group and Dehn twist

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds.

See Mapping class group and Diffeomorphism

Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.

See Mapping class group and Differentiable manifold

Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.

See Mapping class group and Dihedral group

Direct limit

In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way.

See Mapping class group and Direct limit

Direct sum

The direct sum is an operation between structures in abstract algebra, a branch of mathematics.

See Mapping class group and Direct sum

Discrete group

In mathematics, a topological group G is called a discrete group if there is no limit point in it (i.e., for each element in G, there is a neighborhood which only contains that element).

See Mapping class group and Discrete group

Eilenberg–MacLane space

In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name.

See Mapping class group and Eilenberg–MacLane space

Exact sequence

An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.

See Mapping class group and Exact sequence

Genus (mathematics)

In mathematics, genus (genera) has a few different, but closely related, meanings. Mapping class group and genus (mathematics) are geometric topology.

See Mapping class group and Genus (mathematics)

Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

See Mapping class group and Geometric topology

Homeomorphism

In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Mapping class group and homeomorphism are homeomorphisms.

See Mapping class group and Homeomorphism

Homeomorphism group

In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation.

See Mapping class group and Homeomorphism group

Homeotopy

In algebraic topology, an area of mathematics, a homeotopy group of a topological space is a homotopy group of the group of self-homeomorphisms of that space. Mapping class group and homeotopy are homeomorphisms.

See Mapping class group and Homeotopy

Homology (mathematics)

In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces.

See Mapping class group and Homology (mathematics)

Homotopy

In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

See Mapping class group and Homotopy

Homotopy category

In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape.

See Mapping class group and Homotopy category

Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.

See Mapping class group and Homotopy group

Homotopy sphere

In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere.

See Mapping class group and Homotopy sphere

Hyperbolic group

In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry.

See Mapping class group and Hyperbolic group

Hyperbolic link

In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry.

See Mapping class group and Hyperbolic link

Ib Madsen

Ib Henning Madsen (born 12 April 1942, in Copenhagen), retrieved 3 February 2013.

See Mapping class group and Ib Madsen

If and only if

In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements.

See Mapping class group and If and only if

Intersection (set theory)

In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A.

See Mapping class group and Intersection (set theory)

Inventiones Mathematicae

Inventiones Mathematicae is a mathematical journal published monthly by Springer Science+Business Media.

See Mapping class group and Inventiones Mathematicae

Inverse function

In mathematics, the inverse function of a function (also called the inverse of) is a function that undoes the operation of.

See Mapping class group and Inverse function

Klein bottle

In mathematics, the Klein bottle is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Mapping class group and Klein bottle are geometric topology.

See Mapping class group and Klein bottle

Knot (mathematics)

In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of which takes one knot to the other.

See Mapping class group and Knot (mathematics)

Lantern relation

In geometric 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. Mapping class group and lantern relation are geometric topology and homeomorphisms.

See Mapping class group and Lantern relation

Link (knot theory)

In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together.

See Mapping class group and Link (knot theory)

Low-dimensional topology

In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Mapping class group and low-dimensional topology are geometric topology.

See Mapping class group and Low-dimensional topology

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

See Mapping class group and Manifold

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Mapping class group and Mathematics

Möbius strip

In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist.

See Mapping class group and Möbius strip

Michael Weiss (mathematician)

Michael Weiss (born 14 December 1955) is a German mathematician and an expert in algebraic and geometric topology.

See Mapping class group and Michael Weiss (mathematician)

Moduli space

In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects.

See Mapping class group and Moduli space

Mumford conjecture

There are several conjectures in mathematics by David Mumford.

See Mapping class group and Mumford conjecture

Nielsen–Thurston classification

In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface. Mapping class group and Nielsen–Thurston classification are geometric topology and homeomorphisms.

See Mapping class group and Nielsen–Thurston classification

Open set

In mathematics, an open set is a generalization of an open interval in the real line.

See Mapping class group and Open set

Orientability

In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise".

See Mapping class group and Orientability

Piecewise linear manifold

In mathematics, a piecewise linear manifold (PL manifold) is a topological manifold together with a piecewise linear structure on it. Mapping class group and piecewise linear manifold are geometric topology.

See Mapping class group and Piecewise linear manifold

Princeton University Press

Princeton University Press is an independent publisher with close connections to Princeton University.

See Mapping class group and Princeton University Press

Real projective plane

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. Mapping class group and real projective plane are geometric topology.

See Mapping class group and Real projective plane

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold.

See Mapping class group and Riemann surface

Split exact sequence

In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way.

See Mapping class group and Split exact sequence

Subgroup

In group theory, a branch of mathematics, given a group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗.

See Mapping class group and Subgroup

Surface (topology)

In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Mapping class group and surface (topology) are geometric topology.

See Mapping class group and Surface (topology)

Surface bundle

In mathematics, a surface bundle is a bundle in which the fiber is a surface. Mapping class group and surface bundle are geometric topology.

See Mapping class group and Surface bundle

Symplectic geometry

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.

See Mapping class group and Symplectic geometry

Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer n and field F (usually C or R).

See Mapping class group and Symplectic group

Teichmüller space

In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism.

See Mapping class group and Teichmüller space

Topological manifold

In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space.

See Mapping class group and Topological manifold

Topological pair

In mathematics, more specifically algebraic topology, a pair (X,A) is shorthand for an inclusion of topological spaces i\colon A \hookrightarrow X. Sometimes i is assumed to be a cofibration.

See Mapping class group and Topological pair

Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.

See Mapping class group and Topological space

Torelli theorem

In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety.

See Mapping class group and Torelli theorem

References

[1] https://en.wikipedia.org/wiki/Mapping_class_group

Also known as Torelli group.

, Mumford conjecture, Nielsen–Thurston classification, Open set, Orientability, Piecewise linear manifold, Princeton University Press, Real projective plane, Riemann surface, Split exact sequence, Subgroup, Surface (topology), Surface bundle, Symplectic geometry, Symplectic group, Teichmüller space, Topological manifold, Topological pair, Topological space, Torelli theorem, Torus knot, Union (set theory), W. B. R. Lickorish, William Thurston, Y-homeomorphism, 3-manifold.

Mapping class group, the Glossary

Table of Contents

See also

Homeomorphisms

References