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Topological space
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
Loop_space
Mathematical group of loops in a Lie group
For compact G, smooth loop groups are modeled on nuclear Fréchet spaces. The free loop group of G is LG itself. The based loop group is Ω G = { γ ∈ L
Loop_group
Describes a periodicity in the homotopy groups of classical groups
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
Bott_periodicity_theorem
Topological path whose initial point is equal to its terminal point
In mathematics, a loop in a topological space X is a continuous function f from the unit interval I = [0,1] to X such that f(0) = f(1). In other words
Loop_(topology)
Theory of quantum gravity merging quantum mechanics and general relativity
of quantum space. Research has evolved in two directions: the more traditional canonical loop quantum gravity, and the newer covariant loop quantum gravity
Loop_quantum_gravity
Mathematical group of the homotopy classes of loops in a topological space
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
Fundamental_group
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
Infinite_loop_space_machine
On the loop space of a wedge of spheres
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
Hilton's_theorem
Concept in mathematics
{\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
Suspension_(topology)
hierarchy of higher homotopies. This concept is fundamental to the study of loop spaces and is a key tool in fields like symplectic geometry (through the Fukaya
A∞-operad
Topological space with only one nontrivial homotopy group
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
Eilenberg–MacLane_space
French-British singer
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
Space_Cowboy_(musician)
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
Free_loop
Type of textile fastener
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
Velcro
Proposed system for launching objects into orbit
launching humans for space tourism, space exploration and space colonization, and provides a relatively low 3g acceleration. Launch loops were described by
Launch_loop
Topological construction defined up to homotopy
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
Homotopy_fiber
Mathematical theory of topological spaces
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
Rational_homotopy_theory
2004 single by Asian Kung-Fu Generation
"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
Loop_&_Loop
Pathological embedding of the sphere in 3D space
three-dimensional space. Together with its inside, it is a topological 3-ball, the Alexander horned ball, and so is simply connected; i.e., every loop can be shrunk
Alexander_horned_sphere
Infinite loop space
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
E∞-operad
Manifold modelled on Hilbert spaces
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
Hilbert_manifold
Former corporate headquarters of Apple Inc. in California, U.S.
2). The campus is located at 1 Infinite Loop in Cupertino, California, and remains an Apple office and lab space. Its design resembles that of a university
Apple_Infinite_Loop_campus
Transforming a function in such a way that it only takes a single argument
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}
Currying
Mathematical construction used in homotopy theory
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}
Simplicial_set
Symplectic topology tool
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
Floer_homology
How spheres of various dimensions can wrap around each other
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
Homotopy_groups_of_spheres
Mathematical product in topology
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
Pontryagin_product
Mathematics glossary
homotopy groups and homology groups. infinite loop space infinite loop space machine Infinite loop space machine. infinite mapping telescope intersection
Glossary of algebraic topology
Glossary_of_algebraic_topology
Mathematical set with some added structure
product space Kolmogorov space Lp-space Lens space Liouville space Locally finite space Loop space Lorentz space Mapping space Measure space Metric space Minkowski
Space_(mathematics)
Concept in topology
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
H-space
theory, the homotopy groups of spheres, and conjectural index theories on loop spaces of manifolds. tmf was first constructed by Michael Hopkins and Haynes
Topological_modular_forms
Branch of topology
homology of free loop spaces. The field was started by Moira Chas and Dennis Sullivan (1999). While the singular cohomology of a space has always a product
String_topology
Assistive listening technology
the looped space which can be picked up by a telecoil-equipped hearing aid, a cochlear implant (CI) processor, or a specialized hand-held hearing loop receiver
Audio_induction_loop
Control flow construct for executing code repeatedly
programming, a loop is a control flow construct that allows code to be executed repeatedly, usually with minor alterations between repetitions. Loops can be used
Loop_(statement)
Set of functions between two fixed sets
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
Function_space
) {\displaystyle (X,*)} ; thus it is the loop space of ( X , ∗ ) {\displaystyle (X,*)} . The free path space of X, that is, Map ( I , X ) = X I {\displaystyle
Path_space_fibration
Relationship between two functors abstracting many common constructions
the space [X, ΩY] of homotopy classes of maps from X to the loop space ΩY of Y. The suspension functor is therefore left adjoint to the loop space functor
Adjoint_functors
American mathematician
on algebraic K-theory. His results include a proof that all infinite loop space machines are in some sense equivalent, and progress on the Quillen–Lichtenbaum
Robert_Wayne_Thomason
Internet aesthetic capturing empty and often transitional places
aesthetics, liminal spaces are empty or abandoned places that appear eerie, forlorn, uncanny, and often surreal. Liminal spaces are commonly places of
Liminal_space
Continuous function whose domain is a closed unit interval
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
Path_(topology)
Generalization of associativity properties
in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1968 and by J. Peter May
Operad
Type of group in mathematics
dimensions of the vector spaces over the two points, so π0(KO) = Z is the dimension. Using concrete descriptions of the loop spaces in Bott periodicity, one
Orthogonal_group
English rock band
Loop are an English rock band, formed in 1986 by Robert Hampson in Croydon. The group topped the UK independent charts with their albums Fade Out (1989)
Loop_(band)
Type of radio antenna consisting of a loop or coil
A loop antenna is a radio antenna consisting of a loop or coil of wire, tubing, or other electrical conductor, that for transmitting is usually fed by
Loop_antenna
Establishes the concept of stabilization of homotopy groups
isomorphism if k ≤ 2n and an epimorphism if k = 2n + 1. A basic result on loop spaces gives the relation π k ( Ω ( Σ X ) ) ≅ π k + 1 ( Σ X ) {\displaystyle
Freudenthal suspension theorem
Freudenthal_suspension_theorem
Japanese mathematician
University. Nishida's earliest work grew out of the study of infinite loop spaces; his first paper (in 1968, on what came eventually to be known as the
Goro_Nishida
Space which has no holes through it
group is trivial. A topological space X {\displaystyle X} is called simply connected if it is path-connected and any loop in X {\displaystyle X} defined
Simply_connected_space
Concept in algebraic topology
& Wilkerson (1994), making precise earlier notions of a mod p finite loop space. A p-compact group has many Lie-like properties like maximal tori and
P-compact_group
Self-reproducing cellular automaton patterns
particular property of the loops' "pseudopodia", they are unable to reproduce into the space occupied by another loop. Thus, once a loop is surrounded, it is
Langton's_loops
Branch of engineering and mathematics
placement can be performed mathematically using a state space representation of the open-loop system and calculating a feedback matrix assigning poles
Control_theory
Loop transformation technique
as space–time tradeoff. The transformation can be undertaken manually by the programmer or by an optimizing compiler. On modern processors, loop unrolling
Loop_unrolling
Topics referred to by the same term
see path space fibration#Moore's path space. See also: loop space, the space of loops in a topological space This disambiguation page lists mathematics
Path_space
American theoretical physicist
conformal and exactly integrable field theories and string theory in loop space. In solid state physics, he is primarily concerned with highly correlated
Natan_Andrei
Increasing execution speed and reducing the overheads associated with loops
In compiler theory, loop optimization is the process of increasing execution speed and reducing the overheads associated with loops. It plays an important
Loop_optimization
Algebraic invariant of topological spaces
differential operators on free loop spaces. In some sense it can be seen as an approximation to the K-theory of the free loop space. Call a cohomology theory
Elliptic_cohomology
Rail loop for turning vehicles
A balloon loop, turning loop, or reversing loop (North American Terminology) allows a rail vehicle or train to reverse direction without having to shunt
Balloon_loop
Concept in math
category of spaces. For example, the suspension ΣX and the loop space ΩX are defined for a pointed space X and produce another pointed space. Also, the
Homotopy_category
where Ω {\displaystyle \Omega } stands for a loop space and the (2) is localization of a topological space at the prime 2. This gives a spectral sequence
EHP_spectral_sequence
Mathematical object
{\to n}{\operatorname {colim} {}}}\Omega ^{n}E_{n}} called the infinite loop space of the spectrum. For a CW complex X {\displaystyle X} Ω ∞ Σ ∞ X = colim
Spectrum_(topology)
Community area in Chicago, Illinois
The Loop is Chicago's central business district and one of the 77 municipally-designated community areas of Chicago in Illinois, United States. Located
Chicago_Loop
Concept in algebraic topology
paths. The fiber is denoted by Ω B {\displaystyle \Omega B} and is called loop space. The fibers p − 1 ( b ) {\displaystyle p^{-1}(b)} over b ∈ B {\displaystyle
Fibration
Mathematical concept
spaces of infinite dimension. In particular, this is the case of iterated loop spaces. The structure of a fractal object is reiterated in its magnifications
Infinity
Cohomology class
of loop spaces. A loop space ΩX has a multiplication Ω X × Ω X → Ω X {\displaystyle \Omega X\times \Omega X\to \Omega X} by composition of loops. Here
Highly structured ring spectrum
Highly_structured_ring_spectrum
Branch of mathematics
CW complex, then so does its loop space Ω X {\displaystyle \Omega X} . Given a topological group G, the classifying space for principal G-bundles ("the"
Homotopy_theory
Aspect of astrophysics history
special relativity) replaces the group of rotational symmetries of space. (Later, loop quantum gravity inherited this geometric interpretation of gravity
History of loop quantum gravity
History_of_loop_quantum_gravity
the James reduced product J(X) has the same homotopy type as ΩΣX, the loop space of the suspension of X. The commutative analogue of the James reduced
James_reduced_product
Theoretical physicist and mathematician
Duisburg-Essen in 2005 with a thesis supervised by Robert Graham and titled From Loop Space Mechanics to Nonabelian Strings. Schreiber's research fields include the
Urs_Schreiber
In computer science, a loop variant is a mathematical function defined on the state space of a computer program whose value is monotonically decreased
Loop_variant
Process where information about current status is used to influence future status
as inputs as part of a chain of cause and effect that forms a circuit or loop. The system can then be said to feed back into itself. The notion of cause-and-effect
Feedback
Algebraic theorem
}(\Omega X;\mathbb {Q} ),} where Ω X {\displaystyle \Omega X} denotes the loop space of X, compare with Theorem 21.5 from Félix, Halperin & Thomas (2001).
Milnor–Moore_theorem
Theater in Chicago, Illinois
Nederlander Theatre is a theater located at 24 West Randolph Street in the Loop area of downtown Chicago, Illinois. It opened in 1926, named the Oriental
Nederlander_Theatre_(Chicago)
Modular space station in low Earth orbit
The International Space Station (ISS) is a space station in low Earth orbit (LEO). It is the product of the International Space Station program and is
International_Space_Station
Spectral sequence in algebraic topology
base and total space, so our intuition tells us that the Serre spectral sequence should be able to tell us the homology of the loop space. This is an example
Serre_spectral_sequence
Classification in abstract algebra
relevant loop spaces are modeled by successive quotients of the classical groups, which are compact symmetric spaces. In stable group theory, loop spaces enter
Classification of Clifford algebras
Classification_of_Clifford_algebras
Possibility of a consistent definition of "clockwise" in a mathematical space
some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed
Orientability
Two-phase heat transfer device
miniature LHPs (micro-loop heat pipe) have been developed and successfully employed in a wide sphere of applications both ground and space-based applications
Loop_heat_pipe
1986 breakup of American orbiter
On January 28, 1986, Space Shuttle Challenger broke apart 73 seconds into its flight, killing all seven crew members. The spacecraft disintegrated about
Space Shuttle Challenger disaster
Space_Shuttle_Challenger_disaster
Finite, symmetry-reduced model of loop quantum gravity
effects of loop quantum gravity (LQG). In particular, quantum geometry creates a brand new repulsive force which is totally negligible at low space-time curvature
Loop_quantum_cosmology
Gauge field loop operator
field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops. They encode all gauge
Wilson_loop
is that residues that make up the beginning and end of the loop are close together in space with no intervening lengths of regular secondary structural
Omega_loop
Topological space with a distinguished point
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
Pointed_space
2024 video game
to hack the Grand Reset, limiting its effect and saving the current loop's space-time. Grendel Shaddai, meanwhile, prevents Van's group from stopping
The Legend of Heroes: Trails Beyond the Horizon
The_Legend_of_Heroes:_Trails_Beyond_the_Horizon
Concept in mathematical category theory
{\displaystyle E_{\infty }} space, so its group completion is an infinite loop space. A dagger symmetric monoidal category is a symmetric monoidal category
Symmetric_monoidal_category
List of concrete topologies and topological spaces
lists named topologies of uniform convergence. Compact-open topology Loop space Interlocking interval topology Modes of convergence (annotated index)
List_of_topologies
Specification of a derivative along a tangent vector of a manifold
Chen, Kuo-Tsai (1973). "Iterated Integrals of Differential Forms and Loop Space Homology". Annals of Mathematics. 97 (2): 217–246. doi:10.2307/1970846
Covariant_derivative
State highway loops in Texas are owned and maintained by the Texas Department of Transportation (TxDOT). Loop 300 was a proposed route in Snyder. It was
List of state highway loops in Texas (300–399)
List_of_state_highway_loops_in_Texas_(300–399)
Type of topology
{\displaystyle \Omega (X,x_{0})=\{f:I\to X\mid f(0)=f(1)=x_{0}\}} , the loop space of X {\displaystyle X} at x 0 {\displaystyle x_{0}} , E ( X , x 0 , x
Compact-open_topology
American infrastructure and tunneling company
California. In 2021, TBC completed the Las Vegas Convention Center (LVCC) Loop, a three-station transportation system with 1.7 miles (2.7 km) of tunnels
The_Boring_Company
Y^{I}:\omega (0)=y_{0}{\mbox{ and }}\omega (1)=f(x)\}} Observe that the loop space Ω Y {\displaystyle \Omega Y} injects into the mapping fibre: Ω Y → M f
Puppe_sequence
2025. Williscroft, Robert G. (August 21, 2015). "The Space Elevator, Space Launch Loop, Space Fountain, and Mass Driver in Science Fiction". sribd.com
Space_elevators_in_fiction
fiction Category:Time loop television series Category:Time loop television episodes Category:Video games about time loops Category:Time loop novels Matthew Jones;
List of films featuring time loops
List_of_films_featuring_time_loops
Void between celestial bodies
Outer space, or simply space, is the expanse that exists beyond Earth's atmosphere and between celestial bodies. It contains ultra-low levels of particle
Outer_space
American science fiction drama TV series
of light and space all do justice to Stålenhag's retrofuturistic vision". According to The Verge's Joshua Rivera, "Tales from the Loop is so pretty it
Tales_from_the_Loop
Construct in mathematics
abélienne, Springer, ISBN 3-540-05307-7. Brylinski, Jean-Luc (1993), Loop space, characteristic classes and geometric quantization, Birkhäuser Verlag
Gerbe
Aspect of loop quantum gravity
multiverse, and loop quantum gravity could have various general covariant principle results. Because loop quantum gravity can model universes, space-gravity theories
Lorentz invariance in loop quantum gravity
Lorentz_invariance_in_loop_quantum_gravity
Embedding of Cantor set in 3-dimensional Euclidean space
3-dimensional Euclidean space, whose complement is not simply connected. It also serves as a counterexample to the claim that all Cantor spaces are ambiently homeomorphic
Antoine's_necklace
inclined loop, also known as an oblique loop, is a 360° loop that has been tilted at an angle. It is not entered vertically, like a vertical loop, or horizontally
List of roller coaster elements
List_of_roller_coaster_elements
Logical error that can often be found in programming
mathematical context. It also often occurs in computer programming when a loop iterates one time too many or too few, usually caused by the use of non-strict
Off-by-one_error
Subject area in mathematics
category are the homotopy groups of ΩBQC, the loop space of the geometric realization (taking the loop space corrects the indexing). Quillen additionally
Algebraic_K-theory
Place on a railway where trains can pass each other
A passing loop (UK usage) or passing siding (North America) (also called a crossing loop, crossing place, refuge loop or, colloquially, a hole) is a place
Passing_loop
LOOP SPACE
LOOP SPACE
Surname or Lastname
North German
North German : habitational name from any of several places called Loose or Loosey.North German : from a short form of Nikolaus, German form of Nicholas.Dutch : nickname from the adjective loos ‘cunning’, ‘artful’, ‘guileful’.English : variant spelling of Loose.
Surname or Lastname
English (Somerset)
English (Somerset) : habitational name from Look in Puncknowle, Dorset, named in Old English with lūce ‘enclosure’.English : possibly a variant of Luck 3.Northern English and Scottish : from a vernacular pet form of Lucas.Dutch (van Look) : topographic name from look ‘enclosure’ or habitational name from a place named with this word.Thomas Look (b. c. 1622) was in Lynn, MA, by 1646. His son, also called Thomas (b. 1646), moved to Martha’s Vineyard about 1670.
Boy/Male
Hindu, Indian
Look
Boy/Male
Dutch, German, Hebrew
God will Multiply; God will Add
Boy/Male
Bengali, Indian
Loop; Autumn
Boy/Male
British, English
Barrel Maker
Male
Dutch
, Jehovah's gift (or grace).
Surname or Lastname
English
English : metonymic occupational name for a cooper, from Middle English coupe ‘tub’, ‘container’ (see Cooper). In some cases the surname may have been derived from a pub or house sign.Dutch : from koop ‘purchase’, ‘bargain’, hence a nickname for a haggler or a metonymic occupational name for a merchant.
Surname or Lastname
English
English : possibly from the Old Norse personal name Tópi, Túpi, a short form of a personal name formed with þórr, name of the Norse god of thunder (see Thor) + a second element with initial b-, for example björn ‘bear’, ‘warrior’. On the other hand, the name is found mainly in Dorset and Devon, which are far from areas of Scandinavian settlement.
Girl/Female
Arabic, Muslim
Look
Male
French
French form of Latin Lupus, LOUP means "wolf."
Girl/Female
Hindu
Look, Blessed with beauty, Shape, Beauty
Surname or Lastname
Dutch
Dutch : from a short form of the Germanic personal name Robrecht.Altered spelling of German Rupp.English : variant spelling of Roope.
Boy/Male
Hindu, Indian, Rajasthani, Sindhi, Traditional
Look; Beauty; Appearance
Girl/Female
Gujarati, Hindu, Indian
Look
Boy/Male
Hebrew
God will multiply.
Boy/Male
Arabic
The Biblical Lot is the English Language Equivalent
Boy/Male
Hindu
Flower
Girl/Female
Tamil
Look, Blessed with beauty, Shape, Beauty
Boy/Male
Indian, Sanskrit
Natural; Original; Innate; Simply; Loop
LOOP SPACE
LOOP SPACE
Girl/Female
Indian
Queen
Boy/Male
Tamil
Hemaprasad | ஹேமாபà¯à®°à®¸à®¾à®¤
King of gold
Girl/Female
French American English
Nobility; strength.
Girl/Female
Indian
Prosperous, Self-possessed, River, Ocean, River
Boy/Male
Hindu, Indian
Loving or God Gifted
Boy/Male
Hindu
Lord of the mind, God of mind
Boy/Male
Indian, Sanskrit
Moon of Gold
Boy/Male
Indian, Sanskrit
Brilliant; Shining
Girl/Female
Slavic
At peace.
Girl/Female
Australian, British, Christian, English, Swedish, Welsh
White and Smooth; Soft; White; Fair; Smooth
LOOP SPACE
LOOP SPACE
LOOP SPACE
LOOP SPACE
LOOP SPACE
v. t.
To make a loop of or in; to fasten with a loop or loops; -- often with up; as, to loop a string; to loop up a curtain.
v. t.
To beat in the game of loo by winning every trick.
v. t.
To bind or fasten with hoops; as, to hoop a barrel or puncheon.
n.
A curve of any kind in the form of a loop.
v. t.
To confine in a coop; hence, to shut up or confine in a narrow compass; to cramp; -- usually followed by up, sometimes by in.
n.
Any one of several aquatic, wed-footed, northern birds of the genus Urinator (formerly Colymbus), noted for their expertness in diving and swimming under water. The common loon, or great northern diver (Urinator imber, or Colymbus torquatus), and the red-throated loon or diver (U. septentrionalis), are the best known species. See Diver.
v. t.
To look at; to turn the eyes toward.
n.
Expression of the eyes and face; manner; as, a proud or defiant look.
v. t.
To let hang down; as, to lop the head.
v. t.
To break over the poop or stern, as a wave.
v. i.
To seem; to appear; to have a particular appearance; as, the patient looks better; the clouds look rainy.
v. i.
To direct the attention (to something); to consider; to examine; as, to look at an action.
n.
See 1st Loop.
n.
Hence; Appearance; aspect; as, the house has a gloomy look; the affair has a bad look.
n.
See Loon, the bird.
v. t.
To express or manifest by a look.
v. t.
To influence, overawe, or subdue by looks or presence as, to look down opposition.
n.
The act of looking; a glance; a sight; a view; -- often in certain phrases; as, to have, get, take, throw, or cast, a look.
n.
A ring; a circular band; anything resembling a hoop, as the cylinder (cheese hoop) in which the curd is pressed in making cheese.
pl.
of Trou-de-loup