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Loop (topology)
In mathematics, a loop in a topological space is a continuous function from the unit interval to such that In other words, it is a path whose initial point is equal to its terminal point.. A loop may also be seen as a continuous map from the pointed unit circle into , because may be regarded as a quotient of under the identification of 0 with 1. The set of all loops in forms a space called the loop space of . See also * Free loop * Loop group * Loop space * Loop algebra *Fundamental group *Quasigroup In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element pro ... References Topology es:Grupo fundamental#Lazo {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group Torus2
Fundamental may refer to: * Foundation of reality * Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental" * Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" ideas based on faith in a system of thought * '' Fundamentals: Ten Keys to Reality'', a 2021 popular science book by Frank Wilczek * ''The Fundamentals'', a set of books important to Christian fundamentalism * Any of a number of fundamental theorems identified in mathematics, such as: ** Fundamental theorem of algebra, a theorem regarding the factorization of polynomials ** Fundamental theorem of arithmetic, a theorem regarding prime factorization * Fundamental analysis, the process of reviewing and analyzing a company's financial statements to make better economic decisions Music * Fun-Da-Mental, a rap group * ''Fundamental'' (Bonnie Raitt album), 1998 * ''Fundamental'' (Pet Shop Boys album), 2006 * ''Fundamental'' (Puya album) or the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an ''A''∞-space. That is, the multiplication is homotopy-coherently associative. The set of path components of Ω''X'', i.e. the set of based-homotopy equivalence classes of based loops in ''X'', is a group, the fundamental group ''π''1(''X''). The iterated loop spaces of ''X'' are formed by applying Ω a number of times. There is an analogous construction for topological spaces without basepoint. The free loop space of a topological space ''X'' is the space of maps from the circle ''S''1 to ''X'' with the compact-open topology. The free loop space of ''X'' is often denoted by \mathcalX. As a functor, the free loop space construction is rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element properties are optional. In fact, a nonempty associative quasigroup is a group. A quasigroup that has an identity element is called a loop. Definitions There are at least two structurally equivalent formal definitions of quasigroup: * One defines a quasigroup as a set with one binary operation. * The other, from universal algebra, defines a quasigroup as having three primitive operations. The homomorphic image of a quasigroup that is defined with a single binary operation, however, need not be a quasigroup, in contrast to a quasigroup as having three primitive operations. We begin with the first definition. Algebra A quasigroup is a non-empty set with a binary operation (that is, a magma, indicating that a quasigroup has to sat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. 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 homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have Group isomorphism, isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface (mathematics), surface), and some point in it, and all the loops both starting and ending at this point—path (topology), paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then alo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Algebra
In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics. Definition For a Lie algebra \mathfrak over a field K, if K ,t^/math> is the space of Laurent polynomials, then L\mathfrak := \mathfrak\otimes K ,t^ with the inherited bracket \otimes t^m, Y\otimes t^n= ,Yotimes t^. Geometric definition If \mathfrak is a Lie algebra, the tensor product of \mathfrak with , the algebra of (complex) smooth functions over the circle manifold (equivalently, smooth complex-valued periodic functions of a given period), \mathfrak\otimes C^\infty(S^1), is an infinite-dimensional Lie algebra with the Lie bracket given by _1\otimes f_1,g_2 \otimes f_2 _1,g_2otimes f_1 f_2. Here and are elements of \mathfrak and and are elements of . This isn't precisely what would correspond to the direct product of infinitely many copies of \mathfrak, one for each point in , because of the smoothness restriction. Instead, it can be thought of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an ''A''∞-space. That is, the multiplication is homotopy-coherently associative. The set of path components of Ω''X'', i.e. the set of based-homotopy equivalence classes of based loops in ''X'', is a group, the fundamental group ''π''1(''X''). The iterated loop spaces of ''X'' are formed by applying Ω a number of times. There is an analogous construction for topological spaces without basepoint. The free loop space of a topological space ''X'' is the space of maps from the circle ''S''1 to ''X'' with the compact-open topology. The free loop space of ''X'' is often denoted by \mathcalX. As a functor, the free loop space construction is rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Group
In mathematics, a loop group (not to be confused with a loop) is a group of loops in a topological group ''G'' with multiplication defined pointwise. Definition In its most general form a loop group is a group of continuous mappings from a manifold to a topological group . More specifically, let , the circle in the complex plane, and let denote the space of continuous maps , i.e. :LG = \, equipped with the compact-open topology. An element of is called a ''loop'' in . Pointwise multiplication of such loops gives the structure of a topological group. Parametrize with , :\gamma:\theta \in S^1 \mapsto \gamma(\theta) \in G, and define multiplication in by :(\gamma_1 \gamma_2)(\theta) \equiv \gamma_1(\theta)\gamma_2(\theta). Associativity follows from associativity in . The inverse is given by :\gamma^:\gamma^(\theta) \equiv \gamma(\theta)^, and the identity by :e:\theta \mapsto e \in G. The space is called the free loop group on . A loop group is any subgroup of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Loop
In the mathematical field of topology, a free loop is a variant of the notion of a loop. Whereas a loop has a distinguished point on it, called its basepoint, a free loop lacks such a distinguished point. Formally, let X be a topological space. Then a free loop in X is an equivalence class of continuous functions from the circle S^1 to X. Two loops are equivalent if they differ by a reparameterization of the circle. That is, f \sim g if there exists a homeomorphism \psi : S^1 \rightarrow S^1 such that g = f\circ\psi. Thus, a free loop, as opposed to a based loop used in the definition of the fundamental group, is a map from the circle to the space without the basepoint-preserving restriction. Assuming the space is path-connected, free homotopy classes of free loops correspond to conjugacy classes in the fundamental group. Recently, interest in the space of all free loops LX has grown with the advent of string topology, i.e. the study of new algebraic structures on the homo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Space (topology)
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space. Intuitively speaking, the points of each equivalence class are or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space. Definition Let X be a topological space, and let \sim be an equivalence relation on X. The quotient set Y = X/ is the set of equivalence classes of elements of X. The e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as because it is a one-dimensional unit -sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation x^2 + y^2 = 1. Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of "dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointed Space
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains unchanged during subsequent discussion, and is kept track of during all operations. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e., a map f between a pointed space X with basepoint x_0 and a pointed space Y with basepoint y_0 is a based map if it is continuous with respect to the topologies of X and Y and if f\left(x_0\right) = y_0. This is usually denoted :f : \left(X, x_0\right) \to \left(Y, y_0\right). Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint. The pointed set concept is less important; it is anyway the case of a pointed discrete space. Pointed spaces are often taken a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
