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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
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
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
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
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
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 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
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
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)
Topological variant of the loop
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
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)
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
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
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
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)
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
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
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
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 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
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
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
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
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
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 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)
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
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
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
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)
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
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
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
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
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
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
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
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
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)
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
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
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
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
) {\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
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
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)
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
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
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
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 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)
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
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
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
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
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
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
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
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
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)
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
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
may occur in a space as small as a room or as large as a global hemisphere. The Hadley cell is an example of a global-scale thermal loop. Bartell, Patrick
Thermal_loop
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
Former corporate headquarters of Apple Inc. in California, U.S.
The Apple Infinite Loop campus (also known as Apple Campus 1 and by its address, 1 Infinite Loop) is an office complex owned and occupied by Apple Inc
Apple_Infinite_Loop_campus
. Simply connected spaces and simple spaces are (trivial) examples of nilpotent spaces; other examples are connected loop spaces. The homotopy fiber
Nilpotent_space
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
Concept in geometric topology
structure of an abelian group since the space G / O {\displaystyle G/O} is a loop space and in fact an infinite loop space so the normal invariants are a zeroth
Normal_invariant
Cloud of heated and ionized gas and dust in the constellation Cygnus
It constitutes the brightest parts of the visible portion of the Cygnus Loop, a supernova remnant, many portions of which have acquired their own individual
Veil_Nebula
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
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
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
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
Neighborhood in Chicago, Illinois, United States
South Loop is a Chicago neighborhood south of the Loop. The South Branch of the Chicago River flows through it. Chinatown is adjacent to the southwest
South_Loop
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
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
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
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
Canadian mathematician (born 1942)
University of Lille. His research deals with homotopy theory and homology of loop spaces with applications in geometry. He wrote a three-volume textbook on differential
Stephen_Halperin
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
Australian mathematician
homotopy theory in the past four decades, including an approach to infinite loop spaces. He was also a pioneer of elliptic cohomology, which is related to his
Graeme_Segal
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
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
Process to determine relationships among statements
In computer science, loop dependence analysis is a process which can be used to find dependencies within iterations of a loop with the goal of determining
Loop_dependence_analysis
2026 Indian film
Onnu (lit: Incident Chapter One) is a 2026 Indian Malayalam-language time loop mystery horror thriller film written and directed by Jithu Satheesan Mangalathu
Sambhavam_Adhyayam_Onnu
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)
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
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
Property in algebraic topology
of covering spaces. Roughly speaking, a topological space X satisfies the property if for each point x in X any sufficiently small loop going through
Semi-locally_simply_connected
Combination of pointed topological spaces
reduced suspension functor Σ {\displaystyle \Sigma } is left adjoint to the loop space functor Ω {\displaystyle \Omega } : M a p s ∗ ( Σ X , Y ) ≅ M a p s ∗
Smash_product
Communication technique in which every received message is confirmed by the sender
practice of closed-loop communication is still effectively used and has many benefits. Professional environments are spaces where closed-loop communication
Closed-loop_communication
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
2022 studio album by Loop
alternative rock band Loop. It was released in 2022 by Reactor Records and peaked at #89 in the UK Albums Chart. Sonancy was Loop's first album in 32 years
Sonancy
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
Three dimensional group structure
space and time. The basic operations which generate a loop braid group for n loops are exchanges of two adjacent loops, and passing one adjacent loop
Loop_braid_group
building and maintaining the space station. The Space Station Mobility Trainer rotates the trainee on a single axis in a small loop. Grip tape as used on skateboards
Space Station Mobility Trainer
Space_Station_Mobility_Trainer
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