Continuous group action

Last updated March 21, 2019

In topology, a continuous group action on a topological space X is a group action of a topological group G that is continuous: i.e.,

In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

In mathematics, a group action is a formal way of interpreting the manner in which the elements of a group correspond to transformations of some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group.

G\times X\to X,\quad (g,x)\mapsto g\cdot x

is a continuous map. Together with the group action, X is called a G-space.

If $f:H\to G$ is a continuous group homomorphism of topological groups and if X is a G-space, then H can act on Xby restriction: $h\cdot x=f(h)x$ , making X a H-space. Often f is either an inclusion or a quotient map. In particular, any topological space may be thought of as a G-space via $G\to 1$ (and G would act trivially.)

Two basic operations are that of taking the space of points fixed by a subgroup H and that of forming a quotient by H. We write $X^{H}$ for the set of all x in X such that $hx=x$ . For example, if we write $F(X,Y)$ for the set of continuous maps from a G-space X to another G-space Y, then, with the action $(g\cdot f)(x)=gf(g^{-1}x)$ , $F(X,Y)^{G}$ consists of f such that $f(gx)=gf(x)$ ; i.e., f is an equivariant map. We write $F_{G}(X,Y)=F(X,Y)^{G}$ . Note, for example, for a G-space X and a closed subgroup H, $F_{G}(G/H,X)=X^{H}$ .

In mathematics, equivariance is a form of symmetry for functions from one symmetric space to another. A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation.

Related Research Articles

In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a topological invariant: homeomorphic topological spaces have the same fundamental group.

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

Quotient space (topology) topological space consisting of equivalence classes of points in another topological space

In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space. The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones. The quotient topology consists of all sets with an open preimage under the canonical projection map that maps each element to its equivalence class.

In mathematics, more specifically algebraic topology, a covering map is a continuous function p from a topological space C to a topological space X such that each point in X has an open neighbourhood evenly covered by p ; the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.

Fiber bundle continuous surjection satisfying a local triviality condition

In mathematics, and particularly topology, a fiber bundle is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space E and a product space $is defined using a continuous surjective map$

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product $X \times G$ of a space $X$ with a group $G$ . In the same way as with the Cartesian product, a principal bundle $P$ is equipped with

An action of $G$ on $P$ , analogous to $(x, g) h =$ for a product space.
A projection onto $X$ . For a product space, this is just the projection onto the first factor, $(x, g) \mapsto x$ .

In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup $H$ to a representation of the (whole) group $G$ itself. Given a representation of $H$ , the induced representation is, in a sense, the "most general" representation of $G$ that extends the given one. Since it is often easier to find representations of the smaller group $H$ than of $G$ , the operation of forming induced representations is an important tool to construct new representations.

In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively . An analogous definition holds in other categories, where, for example,

In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups $Intuitively, singular homology counts, for each dimension n, the n -dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.$

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. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of automorphism groups. Homeomorphism groups are topological invariants in the sense that the homeomorphism groups of homeomorphic topological spaces are isomorphic as groups.

In mathematics, equivariant cohomology is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space X with action of a topological group G is defined as the ordinary cohomology ring with coefficient ring $of the homotopy quotient :$

In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (small) categories, analogous to a quotient group or quotient space, but in the categorical setting.

In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods. That is, if x and y are points in X, and N_x is the set of all neighborhoods that contain x, and N_y is the set of all neighborhoods that contain y, then x and y are "topologically indistinguishable" if and only if N_x = N_y. (See Hausdorff's axiomatic neighborhood systems.)

In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G. Group (co)homology provides an important set of tools for studying general G-modules.

In differential geometry, a Lie group action on a manifold M is a group action by a Lie group G on M that is a differentiable map; in particular, it is a continuous group action. Together with a Lie group action by G, M is called a G-manifold. The orbit types of G form a stratification of M and this can be used to understand the geometry of M.

In mathematics, given an action $of a group scheme G on a scheme X over a base scheme S, an equivariant sheaf F on X is a sheaf of -modules together with the isomorphism of -modules$

References

John Greenlees, Peter May, Equivariant stable homotopy theory

Continuous group action

Related Research Articles

References

See also