topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from...

G.} The space LG is called the free loop group on G. A loop group is any subgroup of the free loop group LG. An important example of a loop group is...

describes the twofold loop space, Ω 2 B U {\displaystyle \Omega ^{2}BU} of BU. Here, Ω {\displaystyle \Omega } is the loop space functor, right adjoint...

identification of 0 with 1. The set of all loops in X forms a space called the loop space of X. Free loop Loop group Loop space Loop algebra Fundamental group Quasigroup...

{\displaystyle X} a loop in Y {\displaystyle Y} (an element of the loop space Ω Y {\displaystyle \Omega Y} ), and the trivial loop should be associated...

idea is sketched later as well. The loop space of an Eilenberg–MacLane space is again an Eilenberg–MacLane space: Ω K ( G , n ) ≅ K ( G , n − 1 ) {\displaystyle...

distinguished point. Formally, let X {\displaystyle X} be a topological space. Then a free loop in X {\displaystyle X} is an equivalence class of continuous functions...

of quantum space. Research has evolved in two directions: the more traditional canonical loop quantum gravity, and the newer covariant loop quantum gravity...

launching humans for space tourism, space exploration and space colonization, and provides a relatively low 3g acceleration. Launch loops were described by...

fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about...

homotopy (in a nice way), an infinite loop space machine produces a group completion of X together with infinite loop space structure. For example, one can...

and }}\gamma (1)=x_{0}\right\}} which is the loop space Ω X {\displaystyle \Omega X} . See also: Path space fibration. Given a universal covering π : X...

they are En-spaces.) Conversely, any connected En-space X is an n-fold loop space on some other space (called BnX, the n-fold classifying space of X). The...

A_{\infty }} -space X is a loop space. Third, the group completion of a possibly disconnected A ∞ {\displaystyle A_{\infty }} -space is a loop space. The importance...

Peter Hilton (1955), states that the loop space of a wedge of spheres is homotopy-equivalent to a product of loop spaces of spheres. John Milnor (1972) showed...

groups of an H-space. The fundamental group of an H-space is abelian. To see this, let X be an H-space with identity e and let f and g be loops at e. Define...

one wants to prove that a space is a loop space. The basic idea is that if G {\displaystyle G} is a group with classifying space B G {\displaystyle BG} ...

used rational homotopy theory to show that the Betti numbers of the free loop space of X are unbounded. The theorem then follows from a 1969 result of Detlef...

free loop space of a symplectic manifold with the symplectic action functional. For the (instanton) version for three-manifolds, it is the space of SU(2)-connections...

turn the interval into a loop. The curried form then maps the space X {\displaystyle X} to the space of functions from loops into Z {\displaystyle Z}...

example is the compact-open topology, e.g. loop space. Also available is the product topology on the space of set theoretic functions (i.e. not necessarily...

including: DJ Supreme, Kings of Rhythm, Loop Da Loop, and DJ Chrome. In 2002, Dresti changed his stage name to Space Cowboy and did a cover of Prince's song...

spaces to itself. This functor is left adjoint to the functor Ω {\displaystyle \Omega } taking a pointed space X {\displaystyle X} to its loop space Ω...

topology of the free loop space, especially in the field of string topology. We can do an analogous Sobolev construction for the loop space, making it a codimension...

of a space X {\displaystyle X} is often denoted π 0 ( X ) . {\displaystyle \pi _{0}(X).} One can also define paths and loops in pointed spaces, which...

"Loop & Loop" (ループ&ループ, Rūpu&Rūpu) is a song by the Japanese rock band Asian Kung-Fu Generation. It was the second single released from their second full-length...

homology groups of its repeated loop spaces. The Serre spectral sequence relates the homology of a space to that of its loop space, so can sometimes be used...

an abelian group, the Pontryagin product on an H-space, and the Pontryagin product on a loop space. In order to define the Pontryagin product we first...

astronauts maneuver in and out of bulky space suits. However, this use reinforced the view among the populace that hook-and-loop was something with very limited...

product space Kolmogorov space Lp-space Lens space Liouville space Locally finite space Loop space Lorentz space Mapping space Measure space Metric space Minkowski...