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being called a homotopy (/həˈmɒtəpiː/ hə-MOT-ə-pee; /ˈhoʊmoʊˌtoʊpiː/ HOH-moh-toh-pee) between the two functions. A notable use of homotopy is the definition...

In mathematical logic and computer science, homotopy type theory (HoTT) includes various lines of development of intuitionistic type theory, based on the...

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other....

In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic...

branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology...

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental...

In mathematics, especially algebraic topology, the homotopy dimension of a topological space does not have a fixed meaning. However, it can refer to the...

is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger...

In mathematics, algebraic homotopy is a research program on homotopy theory proposed by J.H.C. Whitehead in his 1950 ICM talk, where he described it as:...

of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter...

mathematics, A1 homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties...

In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the...

mathematics, the homotopy category K(A) of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences...

category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy-theoretically speaking, that the ∞-groupoids are spaces....

In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration...

topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored...

In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic"...

Since simplicial sets have a good homotopy theory, one can ask questions about the meaning of the various homotopy groups πn(N(C)). One hopes that the...

topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they...

In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L ∞ {\displaystyle L_{\infty }} -algebra) is a generalisation of...

{\displaystyle i\geq 2} , the i-th homotopy group with coefficients in an abelian group G of a based space X is the pointed set of homotopy classes of based maps from...

topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study...

The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear ordinary/partial differential equations. The homotopy analysis method...

{\displaystyle p\colon E\to B} satisfies the homotopy lifting property for a space X {\displaystyle X} if: for every homotopy h : X × [ 0 , 1 ] → B {\displaystyle...

In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain...

In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial...

In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting...

Homotopy to Marie is the fifth album by Nurse with Wound, released in 1982. Although Nurse with Wound had generated considerable interest across their...

a fiber-homotopy equivalence is a map over a space B that has homotopy inverse over B (that is if h t {\displaystyle h_{t}} is a homotopy between the...

computational method used in numerical algebraic geometry is homotopy continuation, in which a homotopy is formed between two polynomial systems, and the isolated...

technical difficulties, but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing...

In mathematics, especially category theory, the homotopy category of an ∞-category C is the category where the objects are those in C but the hom-set from...

In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized...

In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,pg 23 if f , g : X →...

mathematics, especially in algebraic geometry, the étale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties. Roughly...

topology, a rational homotopy n {\displaystyle n} -sphere is an n {\displaystyle n} -dimensional manifold with the same rational homotopy groups as the n {\displaystyle...

purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is...

category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as the...

the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension...

directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental...

of closed manifolds whose homeomorphism type is not determined by their homotopy type. J. W. Alexander in 1919 showed that the lens spaces L ( 5 ; 1 ) {\displaystyle...

as "Henry", was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in British India...

The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume...

In mathematics, more specifically in topology, the equivariant stable homotopy theory is a subfield of equivariant topology that studies a spectrum with...

For example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex. Let X be a topological...

deformations, such as the homeomorphism between a trefoil knot and a circle. Homotopy and isotopy are precise definitions for the informal concept of continuous...

at the University of California, Los Angeles, known for computations of homotopy groups of spheres. Xu earned both his B.S. and M.S. in mathematics from...

all Kan complexes and it furthermore models the homotopy theory of CW complexes up to weak homotopy equivalence, with the correspondence between simplicial...

surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while...

In mathematics, Tutte's homotopy theorem, introduced by Tutte (1958), generalises the concept of "path" from graphs to matroids, and states roughly that...

algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category Ho ( Top ) {\displaystyle...

particular homotopy theory, a continuous mapping between topological spaces i : A → X {\displaystyle i:A\to X} is a cofibration if it has the homotopy extension...

constructing a section of a bundle. The older meaning for obstruction theory in homotopy theory relates to the procedure, inductive with respect to dimension, for...

Blakers and William S. Massey, gave vanishing conditions for certain triad homotopy groups of spaces. This connectivity result may be expressed more precisely...

In algebraic topology, the Dold-Thom theorem states that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as...

already cellular on a subcomplex A of X, then we can furthermore choose the homotopy to be stationary on A. From an algebraic topological viewpoint, any map...

mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres. In 1931 Heinz Hopf used Clifford...

widely-known fundamental group; as such, it captures information about the homotopy type of a topological space. In terms of category theory, the fundamental...

the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism...

Haskell (/ˈhæskəl/) is a general-purpose, statically typed, purely functional programming language with type inference and lazy evaluation. Haskell pioneered...

the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by George...

configuration space of two points in R n {\displaystyle \mathbf {R} ^{n}} is homotopy equivalent to the sphere S n − 1 {\displaystyle S^{n-1}} . The configuration...

In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity...

September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to...

curves are distinguished as timelike. A timelike homotopy between two timelike curves is a homotopy such that each intermediate curve is timelike. No...

terms. A parallel development was the category of spectra in homotopy theory. The homotopy category of spectra and the derived category of a ring are both...

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...

Homology, Homotopy and Applications is a peer-reviewed delayed open access mathematics journal published by International Press. It was established in...

mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott (1957, 1959), which...

Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. Let G be a group and n a positive integer. A connected topological...

His research interests include the intersections of number theory and homotopy theory. Dustin Clausen completed his undergraduate studies at Harvard University...

Kuiper's theorem, is that this group is weakly contractible, ie. all its homotopy groups are trivial. This result has important uses in topological K-theory...

of San Diego who works in category theory and higher category theory, homotopy theory, logic as applied to set theory, and computer science. Shulman did...

associated to the site underlying a topos a pro-simplicial set (up to homotopy). (It's better to consider it in Ho(pro-SS); see Edwards) Using this inverse...

mathematics, simple homotopy theory is a homotopy theory (a branch of algebraic topology) that concerns with the simple-homotopy type of a space. It was...

also define paths and loops in pointed spaces, which are important in homotopy theory. If X {\displaystyle X} is a topological space with basepoint x...

and independent filmmaker. He is a specialist in homological algebra, homotopy theory, and group theory. He was an invited speaker at the 7th European...

University of Nottingham known for his research on logic, type theory, and homotopy type theory. Altenkirch was part of the 2012/2013 special year on univalent...

spaces. In the case of singular homology, a homotopy between continuous maps f, g : X → Y induces a chain homotopy between the chain maps corresponding to...

was a professor at Hebrew University of Jerusalem. He primarily works in homotopy theory, algebraic geometry, and number theory. In 2022 he won the Erdős...

regular homotopy is thus a homotopy through immersions. Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940s...

In algebraic topology, the homotopy excision theorem offers a substitute for the absence of excision in homotopy theory. More precisely, let ( X ; A ...

in glossary of topology are generally omitted. Abstract homotopy theory and motivic homotopy theory are also outside the scope. Glossary of category theory...

that generalizes various constructions in group theory and equivariant homotopy theory. Named after American mathematician George Mackey, these functors...

In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences'...

which classifies n {\displaystyle n} -dimensional manifolds within the homotopy type of X {\displaystyle X} . The basic idea is that in order to calculate...

In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built...

created rational homotopy theory in the late 1960s and 1970s. It examines "rationalizations" of simply connected topological spaces with homotopy groups and...

In stable homotopy theory, a ring spectrum is a spectrum E together with a multiplication map μ: E ∧ E → E and a unit map η: S → E, where S is the sphere...

in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or...

abelian groups. A simplicial group is a Kan complex (in particular, its homotopy groups make sense). The Dold–Kan correspondence says that a simplicial...

The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological...

It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. CW complexes have better categorical properties than simplicial...

conjecture, asserting that a weak, algebraic notion of equivalence (namely, homotopy equivalence) should imply a stronger, topological notion (namely, homeomorphism)...

the obstruction to a homotopy equivalence f : X → Y {\displaystyle f\colon X\to Y} of finite CW-complexes being a simple homotopy equivalence is its Whitehead...

In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping f between CW complexes X and Y induces isomorphisms...

of nonabelian second relative homotopy groups, and in fact of homotopy 2-types. The second part applies a Higher Homotopy van Kampen Theorem for crossed...

is an active area of research, one direction being the development of homotopy type theory. The first computer proof assistant, called Automath, used...

connectivity is based on the homotopy groups of the space. A space is n-connected (or n-simple connected) if its first n homotopy groups are trivial. Homotopical...

symmetric product. This construction can easily be extended to give a homotopy functor. From an algebraic point of view, the infinite symmetric product...

orientable manifold to be a boundary. 1958 Frank Adams Adams spectral sequence to calculate, potentially, stable homotopy groups from cohomology groups....

Society "for contributions to homotopy theory and for exposition". Bousfield, Aldridge K.; Kan, Daniel M. (1972), Homotopy Limits, Completions and Localizations...

geometric realization functor from the homotopy category of the category Kan of Kan complexes to the homotopy category of the category Top of (reasonable)...

University. Hopkins' work concentrates on algebraic topology, especially stable homotopy theory. It can roughly be divided into four parts (while the list of topics...

which is explained by having the soliton belong to a different topological homotopy class or cohomology class than the base physical system. More simply: it...

American mathematician who has contributed to higher category theory and homotopy theory. Much of her work, including her PhD thesis, concerns model structures...

{\displaystyle 0<a<f(q),} then M a {\displaystyle M^{a}} is a disk, which is homotopy equivalent to a point (a 0-cell) which has been "attached" to the empty...

the fact that a common approach to such generalizations is via abstract homotopy theory, as in nonabelian algebraic topology, and in particular the theory...

{\displaystyle d} restricted to closed forms has a local inverse called a homotopy operator. Since it is also nilpotent, it forms a dual chain complex with...

A representation up to homotopy has several meanings. One of the earliest appeared in physics, in constrained Hamiltonian systems. The essential idea is...

or "homotopy"; the latter omit condition [TLH3]. So from now on we refer to homotopy (homotope) in the sense of theorem 2-1 as a tubular homotopy (resp...

forms by the Cartan formula (also known as the Cartan identity, Cartan homotopy formula or Cartan magic formula): L X ω = d ( ι X ω ) + ι X d ω = { d ...

group Θ n {\displaystyle \Theta _{n}} of h-cobordism classes of oriented homotopy n-spheres, which is finite and abelian. In dimension 4 almost nothing is...

In mathematics, Spanier–Whitehead duality is a duality theory in homotopy theory, based on a geometrical idea that a topological space X may be considered...

In mathematics, especially homotopy theory, the mapping cone is a construction in topology analogous to a quotient space and denoted C f {\displaystyle...

is the statement that the diffeomorphism group of the 3-sphere has the homotopy-type of its isometry group, the orthogonal group O(4). It was proved in...

be homotopy equivalent. Quillen's Theorem B gives a sufficient condition for a square consisting of classifying spaces of categories to be homotopy Cartesian...

the homotopy groups stabilize, and πk(O(n + 1)) = πk(O(n)) for n > k + 1: thus the homotopy groups of the stable space equal the lower homotopy groups...

within mathematics, the obstruction to a finitely dominated space X being homotopy-equivalent to a finite CW-complex is its Wall finiteness obstruction w(X)...

some desired property, in such a way that the effects on the homology, homotopy groups, or other invariants of the manifold are known. A relatively easy...

generalized Poincaré conjecture is a statement that a manifold that is a homotopy sphere is a sphere. More precisely, one fixes a category of manifolds:...

November 2010) was a British mathematician, noted for his contributions to homotopy theory and for code-breaking during World War II. He was born in Brondesbury...

Space compact connected Hausdorff metric uniform second-countable Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex...

before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent...

BG. When the definition of the classifying space takes place within the homotopy category of CW complexes, existence theorems for universal bundles arise...

mathematics at the University of Edinburgh. His research is centered around homotopy theory, algebraic K-theory, higher category theory, and related areas....

Look up product in Wiktionary, the free dictionary. Product may refer to: Product (business), an item that can be offered to a market to satisfy the desire...

In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e...

mathematics, an A∞-operad is a type of operad used in algebraic topology and homotopy theory to describe algebraic structures where the property of associativity...

other by higher order invertible morphisms (2-simplices thought of as "homotopies"). These higher order morphisms can also be composed, but again the composition...

In the mathematical disciplines of algebraic topology and homotopy theory, Eckmann–Hilton duality in its most basic form, consists of taking a given diagram...

{\displaystyle E_{\infty }} -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf...

In stable homotopy theory, a branch of mathematics, the sphere spectrum S is the monoidal unit in the category of spectra. It is the suspension spectrum...

{\displaystyle h:I\to \operatorname {Map} (X,Y)} in the mapping space is exactly a homotopy between the starting point and the end point. From the category theory...

first to systematically calculate the homotopy groups of spheres. He is also central to the study of stable homotopy theory, in particular making concrete...

a normal map on X endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, X has a...

differential topology, saying that the topological degree is the only homotopy invariant of continuous maps to spheres. Let M be an n-dimensional compact...

collapse reduces a simplicial complex (or more generally, a CW complex) to a homotopy-equivalent subcomplex. Collapses, like CW complexes themselves, were invented...

topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study...

22, 1945 in Chicago) is an American mathematician known for his work in homotopy theory, an area of algebraic topology. He is a Fellow of the American Mathematical...

that are associative only up to homotopy, and the A∞ structure keeps track of these homotopies, homotopies of homotopies, and so forth. They are ubiquitous...

an equivalence of the homotopy category with itself. The prototypical examples are the category of spectra in the stable homotopy theory and the category...

ring spectrum or A ∞ {\displaystyle A_{\infty }} -ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology...

topology that provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite...

American mathematician at Johns Hopkins University. Morava specialized in homotopy theory and is credited for Morava E-theory and Morava K-theory, both classes...

→ B having the same behaviour as a fibration regarding the (relative) homotopy groups of E, B and p−1(x). Equivalently, one can define a quasifibration...

with an action of a finite group G. The important notion is that of the homotopy fixed point set X h G {\displaystyle X^{hG}} . There is always X G → X...

category of elements of a simplicial set is fundamental in simplicial homotopy theory, a branch of algebraic topology. More generally, the category of...

morphism is an isomorphism. The homotopy hypothesis states that ∞-groupoids are equivalent to spaces up to homotopy. Alexander Grothendieck suggested...

monoid. This can then be used to prove the commutativity of the higher homotopy groups. The principle is named after Beno Eckmann and Peter Hilton, who...

In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn...

called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied...

In mathematics, and especially in homotopy theory, a crossed module consists of groups G {\displaystyle G} and H {\displaystyle H} , where G {\displaystyle...

field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and...

Nancy, France Collège de France, France "Achieved major results on the homotopy groups of spheres, especially in his use of the method of spectral sequences...

Homology is a topological invariant, and moreover a homotopy invariant: Two topological spaces that are homotopy equivalent have isomorphic homology groups. It...

1963) is a fluid mechanics and applied mathematics expert working in homotopy analysis method (HAM), nonlinear waves, nonlinear dynamics, and applied...

Mathematical Society. Since 2002, he has been Executive Editor of Homology, Homotopy and Applications. Davis has published in algebraic topology, differential...

representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed...

Fellowship. Lurie's research interests are algebraic geometry, topology, and homotopy theory. When he was a student in the Science, Mathematics, and Computer...

the Eilenberg–Maclane spaces of homotopy theory, in the sense that it has only one nonzero homology (rather than homotopy) group. The study of Moore spaces...

isomorphism of homotopy groups is described below, and the isomorphism of homotopy groups is a standard calculation in stable homotopy theory (which can...

conjectured that the homotopy category of geometric realizations of infinity groupoids is equivalent to the homotopy category of homotopy types. This is called...

is a homotopy equivalence, then the ∞-category of algebras over O in C is equivalent to the ∞-category of algebras over O' in C. En-ring Homotopy Lie algebra...

August 4, 2013) was a Dutch mathematician working in category theory and homotopy theory. He was a prolific contributor to both fields for six decades, having...

Kervaire–Milnor group is an abelian group defined as the h-cobordism classes of homotopy spheres with the connected sum as composition and the reverse orientation...

completion theorem is a theorem in mathematics about equivariant K-theory in homotopy theory. Let G be a compact Lie group and let X be a G-CW-complex. The theorem...

{\displaystyle g>1} have the homotopy-type of their mapping class groups (i.e. the components are contractible). The homotopy-type of the diffeomorphism...

ANR) is a "nice"[weasel words] topological space that is considered in homotopy theory; more specifically, in the theory of retracts. For a more general...

manifolds was first defined by Brouwer, who showed that the degree is homotopy invariant and used it to prove the Brouwer fixed point theorem. Less general...

will have a homotopy n {\displaystyle n} -group at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group π n {\displaystyle...

since all coverings have the homotopy lifting property, covering spaces are an important tool in the calculation of homotopy groups. A standard example...

category D ( Z ) {\displaystyle D(\mathbb {Z} )} of abelian groups in the homotopy category of spectra. In addition, these spectra can be used to construct...

Introduction to Category Theory, ISBN 978-0521283045. Simpson, Carlos (2010). Homotopy theory of higher categories. arXiv:1001.4071. Bibcode:2010arXiv1001.4071S...

is, the multiplication is homotopy-coherently associative. The set of path components of ΩX, i.e. the set of based-homotopy equivalence classes of based...

relation. It is weaker than isotopy, and stronger than homotopy: isotopy implies concordance implies homotopy. A link is a slice link if it is concordant to the...

In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space by filtering its...

space. It also follows from rational homotopy theory that H P n {\displaystyle \mathbb {HP} ^{n}} has infinite homotopy groups only in dimensions 4 and 4...

homotopy classes of maps, in particular on homotopy groups of spheres, named after Hiroshi Toda, who defined them and used them to compute homotopy groups...

set-theoretic continuum Pointless Algebraic combinatorial homology cohomology homotopy Differential Geometric low-dimensional knot Digital Key concepts Open set /...

projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y. A result of this kind was first stated...

1 ( A , ∅ ) {\displaystyle H_{i-1}(A,\varnothing )} ). The axioms are: Homotopy: Homotopic maps induce the same map in homology. That is, if g : ( X ,...

about 600 pages of research notes. The topic of the work is a generalized homotopy theory using higher category theory. The word "stacks" in the title refers...

Springer-Verlag, 1983 Sasho Kalajdzievski, An Illustrated Introduction to Topology and Homotopy, CRC Press, 2010, Chapter 10: Isotopy and Homotopy v t e...

higher algebraic K-theory in 1972. This new tool, formulated in terms of homotopy theory, proved to be successful in formulating and solving problems in...

restart algorithm. B. Curtis Eaves presented the homotopy method, based on the concept of homotopy. Given a function f, for which we want to find a fixed...

cohomology, which is in turn used for calculating the stable homotopy groups of spheres. Chromatic homotopy theory Adams-Novikov spectral sequence p-local spectrum...

cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that...

2026)) was a Japanese mathematician, who specialized in stable and unstable homotopy theory. He started publishing in 1952. Many of his early papers are concerned...

originally posed the conjecture in 1965. The Novikov conjecture concerns the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold...

group. The nth homotopy group π n ( X ) {\displaystyle \pi _{n}(X)} of a topological space X {\displaystyle X} is the group of homotopy classes of basepoint-preserving...

notably PL-manifolds and topological manifolds. There is also an analogue in homotopy theory for Poincaré spaces, the Spivak spherical fibration, named after...

but they may be thought of as spaces, with equal types corresponding to homotopy equivalent spaces and with equal elements of a type corresponding to points...

set-theoretic continuum Pointless Algebraic combinatorial homology cohomology homotopy Differential Geometric low-dimensional knot Digital Key concepts Open set /...

homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy...

certain restrictions). In homotopy theory, one assigns a group to each space X and positive integer p called the pth homotopy group of X. These groups...

K ( A , j ) {\displaystyle K(A,j)} whose j-th homotopy group is isomorphic to A and whose other homotopy groups are zero. Such a space is called an Eilenberg–MacLane...

fundamental group based at x0, denoted π1(X, x0). This is the group of homotopy classes of loops based at x0, with the group operation of concatenation...

Japanese mathematician. He was a leading member of the Japanese school of homotopy theory, following in the tradition of Hiroshi Toda. Nishida received his...

of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. It was introduced by Ralph Fox in 1945...

22 January 1965) is a French algebraic geometer and key developer of A¹ homotopy theory with Vladimir Voevodsky. Among his accomplishments is the proof...

but must be distinguished to consider their intersection, sum, etc. In homotopy theory, the fundamental group of a space X {\displaystyle X} at a point...

the chain complex. The resulting homology groups are the same for all homotopy equivalent spaces, which is the reason for their study. These constructions...

the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in (Whitehead 1941)...

4 {\displaystyle \dim M\geq 4} , this condition is not sufficient. The homotopy problem consists in describing and classifying the path-connected components...

quotient bundle. The higher homotopy groups of RPn are exactly the higher homotopy groups of Sn, via the long exact sequence on homotopy associated to a fibration...

Algebraic Topology (NAAT) "can be applied to determine homotopy invariants of spaces, and homotopy classification of maps, in cases which include some classical...

philosophy with a focus on methods from type theory, category theory, and homotopy theory. The nLab espouses the "n-point of view" (a deliberate pun on Wikipedia's...

application is also a continuous function in homotopy theory, and, indeed underpins the entire theory: it allows a homotopy deformation to be viewed as a continuous...

 A. (ed.). "Sequence A001676 (Number of h-cobordism classes of smooth homotopy n-spheres.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation...

{\displaystyle A'} are homotopy equivalent, then A ⋆ B {\displaystyle A\star B} and A ′ ⋆ B {\displaystyle A'\star B} are homotopy equivalent too. Given...

and Yan Soibelman, and others, after an initial input of construction of homotopy algebraic structures on the Hochschild complex. It is of importance in...

homeotopy group of a topological space is a homotopy group of the group of self-homeomorphisms of that space. The homotopy group functors π k {\displaystyle \pi...

y)=(0,-1)\end{cases}}} is a weak homotopy equivalence; that is, f {\displaystyle f} induces an isomorphism on all homotopy groups. It follows that f {\displaystyle...

March 10, 1976) is a mathematician focussing on higher category theory, homotopy theory, K-theory and algebraic geometry. In 2001, Cisinski model structures...

ring conjecture, or, more briefly, the Segal conjecture, is a theorem in homotopy theory, a branch of mathematics. The theorem relates the Burnside ring...

Space compact connected Hausdorff metric uniform second-countable Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex...

circle. The set of homotopy classes of maps from a circle to a topological space form a group, which is called the first homotopy group or fundamental...

ε-balls centered at points of X. By the nerve lemma, the Čech complex is homotopy equivalent to the union of the balls, also known as the offset filtration...

a computational interpretation to univalent foundations (also known as homotopy type theory). In cubical type theory, function extensionality and univalence...

complex topic and has been the subject of research, such as the field of homotopy type theory. The identity type is one of 2 different notions of equality...

sets has finite homotopy groups. Pro-simplicial sets show up in shape theory, in the study of localization and completion in homotopy theory, and in the...

conjectures are a set of mathematical conjectures in the field of stable homotopy theory posed by Douglas Ravenel at the end of a paper published in 1984...

addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval"...

the abelianization of homotopy (cf. Hurewicz theorem), then the nonabelian cohomology may be thought of as a dual of homotopy groups. See: Nonabelian...

algebraic topology that could distinguish between closed manifolds which are homotopy equivalent but not homeomorphic, and can thus be seen as the birth of geometric...

{\displaystyle \mathbf {Z} } . The Maslov index may be viewed as the corresponding homotopy invariant, assigning an integer to a loop in the Lagrangian Grassmannian...

set-theoretic continuum Pointless Algebraic combinatorial homology cohomology homotopy Differential Geometric low-dimensional knot Digital Key concepts Open set /...

In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values...

triangular geometry, algebraic geometry, modular representation theory, and homotopy theory. He is a professor of mathematics at the University of California...

Adams spectral sequence, which is in turn used for calculating the stable homotopy groups of spheres. The May spectral sequence is described in detail in...

Elizabeth Bergner is a mathematician specializing in algebraic topology, homotopy theory, and higher category theory. She is a professor of mathematics at...

set-theoretic continuum Pointless Algebraic combinatorial homology cohomology homotopy Differential Geometric low-dimensional knot Digital Key concepts Open set /...

simple homotopy equivalence is a finer invariant than homotopy equivalence by introducing an invariant called the torsion. The torsion of a homotopy equivalence...

in it is a fiber sequence if and only if it is a cofiber sequence. The homotopy category of a stable ∞-category is triangulated. A stable ∞-category admits...

In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories...

spectral sequence is a spectral sequence used for inductively calculating the homotopy groups of spheres localized at some prime p. It is described in more detail...

homotopy equivalence) if the inclusion maps M ↪ W and N ↪ W {\displaystyle M\hookrightarrow W\quad {\mbox{and}}\quad N\hookrightarrow W} are homotopy...

topological space that has a homotopy type of a CW complex and whose fundamental group is abelian and acts trivially on the homotopy and homology of the universal...

isomorphic in the homotopy category to the loop space of B G {\displaystyle BG} ; that implies various restrictions on the homotopy type of G. Some of...

specializing in algebraic topology, including stable homotopy theory, chromatic homotopy theory, equivariant homotopy theory, and applications of these theories...

on the set tmf n ⁡ ( X ) {\displaystyle \operatorname {tmf} ^{n}(X)} of homotopy classes of continuous maps from X to tmf n {\displaystyle \operatorname...

2π. Regular 3D immersions of the Klein bottle fall into three regular homotopy classes. The three are represented by: the "traditional" Klein bottle;...

Homotopy groups above π 1 {\displaystyle \pi _{1}} do not change under covers, so they agree with those of the orthogonal group. The lower homotopy groups...

) {\displaystyle N(C)} is a 2-simplex (without its interior) and it is homotopy-equivalent to the original circle. A nerve theorem (or nerve lemma) is...

Chow groups. One of the motivations for higher Chow groups comes from homotopy theory. In particular, if α , β ∈ Z ∗ ( X ) {\displaystyle \alpha ,\beta...

(for which a weak homotopy equivalence from the circle exists) is a rational homotopy 1 {\displaystyle 1} -sphere, which is not a homotopy 1 {\displaystyle...

Ph.D. in 2009. Her dissertation, Lower Central Series Obstructions To Homotopy Sections of Curves Over Number Fields, was supervised by Gunnar Carlsson...

"for contributions to algebraic topology, particularly equivariant stable homotopy theory, algebraic K-theory, and applied algebraic topology". In 2008, Carlsson...

and Mikhail Kapranov discovered that some duality phenomena in rational homotopy theory could be explained using Koszul duality of operads. Operads have...

of mathematics at the University of Manchester. Eccles specialises in homotopy theory and its applications to different topology. Eccles taught a wide...

algebraic topology, including algebraic K-theory and equivariant stable homotopy theory. She is an associate professor of mathematics at the University...

Kapranov published “ ∞ {\displaystyle \infty } -Groupoids as a Model for a Homotopy Category”, in which they claimed to provide a rigorous mathematical formulation...

type theories with the univalence axiom, this correspondence holds up to homotopy (propositional equality). M-types are dual to W-types, and represent coinductive...

Cartan and participated in seminars and watched Hurewicz's lectures on homotopy in the Collège de France in Paris. During his time in France, Lyra met...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

different p {\displaystyle p} are not homotopy equivalent. Moreover, classifications up to homeomorphism and homotopy equivalence are known, as follows....

In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack...

British mathematician working in the field of topology, particularly in homotopy theory. James was born in Croydon, Surrey, England, and was educated at...

and Miller proved that the higher indices of the signature operator are homotopy-invariant. Hirzebruch signature theorem Pontryagin class Friedrich Hirzebruch...

to the 3-sphere. As to the homotopy groups, we have π1(S3) = π2(S3) = 0 and π3(S3) is infinite cyclic. The higher-homotopy groups (k ≥ 4) are all finite...

M_{f}} together with a homotopy equivalence between them. The construction serves to replace any map of topological spaces by a homotopy equivalent cofibration...

algebraic topology, including algebraic K-theory and equivariant stable homotopy theory. Gerhardt studied mathematics as an undergraduate at Stanford University...

usual cohomology. For a path connected base, all the different fibers are homotopy equivalent. In particular, their cohomology is isomorphic, so the choice...

 1)-categories. Baues 1995, p. 8 Bergner 2007 Baues, Hans Joachim (1995), "Homotopy types", in James, I. M. (ed.), Handbook of Algebraic Topology, Amsterdam:...

Freyd and A. Heller showed that the shift map on F induces an unsplittable homotopy idempotent on the Eilenberg–MacLane space K(F,1) and that this is universal...

converging to stable homotopy groups of spheres Federer spectral sequence converging to homotopy groups of a function space. Homotopy fixed point spectral...

This group is important because of its relation to Galois theory, étale homotopy theory, and the ring of adeles. In addition, it provides a basic tractable...

where d ≥ 3, then this homotopy class only has one element. If M is ⁠ R 2 {\displaystyle \mathbb {R} ^{2}} ⁠, then this homotopy class has countably many...

In homotopy theory, phantom maps are continuous maps f : X → Y {\textstyle f:X\to Y} of CW-complexes for which the restriction of f {\textstyle f} to any...

January 1989) was a British mathematician, one of the major contributors to homotopy theory. He was born in Woolwich, a suburb in south-east London, and attended...

Massachusetts Institute of Technology. He is best known for introducing higher homotopy groups, proving the Hurewicz theorem. Witold Hurewicz was born on 29 June...

oriented diffeomorphism group of the two-dimensional sphere has the same homotopy type as the special orthogonal group of 3 × 3 matrices. Smale's theorem...

least ρ ( n ) − 1 {\displaystyle \rho (n)-1} such fields. Adams applied homotopy theory and topological K-theory to prove that no more independent vector...

Categories and Homology, Homotopy and Applications, which he helped found. Since 2006, he has been involved with Journal of Homotopy and Related Structures...

deformation of the homeomorphisms themselves called homotopies. We define the mapping class group by taking homotopy classes of homeomorphisms, and inducing the...

topological space ( X , x ) {\displaystyle (X,x)} is defined as the group of homotopy classes of loops based at x {\displaystyle x} . This definition works well...

In mathematics, more specifically in chromatic homotopy theory, the redshift conjecture states, roughly, that algebraic K-theory K ( R ) {\displaystyle...

nilpotent group; π {\displaystyle \pi } acts nilpotently on the higher homotopy groups π i ( X ) , i ≥ 2 {\displaystyle \pi _{i}(X),i\geq 2} , i.e., there...

different from the limits in the sense of enriched category theory. The homotopy coherent nerve of a simplicially enriched category is a simplicial set...

{\displaystyle EG\times _{G}X} is known as the Borel construction, or homotopy quotient. Projection onto the first factor gives it the structure of a...

In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology...

groups. A later application of the theory has been in the field of stable homotopy theory, with the construction of a particular extraordinary cohomology...

philosophy of mathematics. He is one of the originators of the field of homotopy type theory. He was a member of the School of Mathematics at the Institute...

physics. In the 1960s he wrote fundamental papers on higher homotopy theory and homotopy algebras. He introduced A ∞ {\displaystyle A_{\infty }} , Stasheff...

where S 0 {\displaystyle S^{0}} is a discrete space with two points. 5. In Homotopy type theory, S X {\displaystyle SX} is defined as a higher inductive type...

"nilpotence and stable homotopy". Devinatz, Ethan S.; Hopkins, Michael J.; Smith, Jeffrey H. (1988). "Nilpotence and stable homotopy theory. I". Annals of...

{\displaystyle \phi } is a homotopy equivalence; thus, the above decomposition says that any map is a fibration up to homotopy equivalence. If f {\displaystyle...

functoriality of the pushforward.) A main result about the push-forward is the homotopy invariance: if two maps f , g : X → Y {\displaystyle f,g\colon X\rightarrow...

Space compact connected Hausdorff metric uniform second-countable Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex...

to all higher homotopies". (An operad that describes a multiplication that is associative but not necessarily commutative "up to homotopy" is called an...

Evanston, Illinois. Much of Mahowald's most important works concerns the homotopy groups of spheres, especially using the Adams spectral sequence at the...

{\displaystyle n} th homotopy group of the corresponding simplicial abelian group, and a chain homotopy corresponds to a simplicial homotopy. (In fact, the...

fundamental group π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} , the covering homotopy property shows they define the same action γ ⋅ x ~ = γ ′ ⋅ x ~ {\displaystyle...

conjecture, which he formulated. He has made many other contributions to homotopy theory in the past four decades, including an approach to infinite loop...

Junhyong Kim. Kim has made contributions to the application of arithmetic homotopy theory to the study of Diophantine problems, especially to finiteness theorems...

unitary operator from the 0-end of the homotopy to the first cycle, and a unitary operator from the 1-end of the homotopy to the second cycle. The KK-group...

contractible if all of its homotopy groups are trivial. Equivalently, a space is weakly contractible if it is weakly homotopy equivalent to a point. Every...

homology-local theory, in the sense of Bousfield localization in homotopy theory. Chain homotopy equivalence Derived category Gelfand, Sergei I., Manin, Yuri...

} the inclusion map yields an isomorphism between all homotopy groups (that is, it is a homotopy equivalence). Inclusion maps in geometry come in different...

required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of simple homotopy theory. The use of the term geometric...

are all ∞-categories and it furthermore models the homotopy theory of CW complexes up to homotopy equivalence, with the correspondence between simplicial...

Whitehead, and his doctoral dissertation was on Finite Computability of the Homotopy Groups of Finite Groups. In 1962–63, he visited the Institute for Advanced...

Q. A quasigroup homomorphism is just a homotopy for which the three maps are equal. An isotopy is a homotopy for which each of the three maps (α, β,...

Belleville, Canada) is a Canadian mathematician working in the fields of homotopy theory, category theory, and number theory. Jardine obtained his Ph.D....

point defects, line defects, planar defects, bulk defects. Topological homotopy establishes a mathematical method of characterization. Point defects are...

Christos Papakyriakopoulos (1957) gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres. One example...

In mathematics, specifically in homotopy theory in the context of a model category M, a fibrant object A of M is an object that has a fibration to the...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

Lifts are ubiquitous; for example, the definition of fibrations (see Homotopy lifting property) and the valuative criteria of separated and proper maps...

topology, a branch of mathematics, a connective spectrum is a spectrum whose homotopy sets π k {\displaystyle \pi _{k}} of negative degrees are zero. Kuku, Aderemi...

S2CID 119711633. Milnor, John W. (1958), Differentiable manifolds which are homotopy spheres (PDF) Bishop, Richard L.; Goldberg, Samuel I. (1968), Tensor Analysis...

Oswald Veblen Prize in Geometry. Ravenel's main area of work is stable homotopy theory. Two of his most famous papers are Periodic phenomena in the Adams–Novikov...

are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an...

is left proper and combinatorical. For any model category, there is a homotopy category associated to it by formally inverting all weak equivalences....

the title of his dissertation was Dual Homotopy Invariants of G-Foliations. Hurder, Steven (1981). Dual homotopy invariants of G-foliations, Topology,...

is called the n-th homotopy group of M {\displaystyle M} , noted π n ( M ) {\displaystyle \pi _{n}(M)} . The relevant homotopy groups for two-band systems...

that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear group G. For the homotopy theory the relevant...

_{n+1}(BX)} between the homotopy groups of a space X and the homotopy groups of its classifying space BX, combined with the homotopy type of the circle U(1)...

the categories of right A-modules and left A-modules and D refers to the homotopy category (i.e., derived category). By definition, it is the left derived...

then A = B {\displaystyle A=B} , where ≃ {\displaystyle \simeq } denotes homotopy equivalence. Depending on the chosen foundation, some extensionality principles...

\mathbb {C} ^{n},} or H n ; {\displaystyle \mathbb {H} ^{n};} this is homotopy equivalent to the more restrictive definition, as the compact Stiefel manifold...

\partial )\simeq H_{*}({\mathcal {A}},\Delta ),} and similarly for the homotopy groups. Discrete Morse theory finds its application in molecular shape...

point which is close to a solution. Therefore, it is a basic tool for the homotopy continuation method described below. Optimization is rarely used for solving...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

known as Steve Hu, was a Chinese-American mathematician, specializing in homotopy theory. Hu received his B.S. from the National Central University in Nanking...

space K(G, 1), its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); all homotopy equivalences of the...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

{\displaystyle X} , such that this action behaves well with respect to the stable homotopy category, i.e., there is an isomorphism E ∗ ( E ) ⊗ π ∗ ( E ) E ∗ ( X )...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

Society. He died in 2016 at the age of 92. Moore, John C. (May 1954). "On Homotopy Groups of Spaces with a Single Non-Vanishing Homology Group". Annals of...

paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to homotopy. A particular application are principal U(1)-bundles. This space with its...

continuous maps to the category of sets and functions. They are dual to the homotopy groups, but less studied. The p-th cohomotopy set of a pointed topological...

parallelizable manifoldspg 75. Milnor fibers are special because they have the homotopy type of a bouquet of spherespg 78. The number of these spheres is the Milnor...

category, LS-category) of a topological space X {\displaystyle X} is the homotopy invariant defined to be the smallest integer number k {\displaystyle k}...

functor from the category of G {\displaystyle G} -complexes with equivariant homotopy maps to the category of abelian groups together with the connecting homomorphism...

(killing) homotopy groups of increasing order. This is done by constructing short exact sequences starting with an Eilenberg–MacLane space for the homotopy group...

known for his construction of the first rigorously correct model of the homotopy category of spectra. He received his PhD from the University of Cambridge...

far, published when he was completing his PhD. "On ex-homotopy theory and generalized homotopy products". The University of Hong Kong. 1978. Archived...

Ekedahl, Torsten (2002), "On minimal models in integral homotopy theory", Homology, Homotopy and Applications, 4 (2): 191–218, arXiv:math/0107004, doi:10...

induces a filtration on the chain complex of each theory, whose chain homotopy type is a knot invariant. (Their homologies satisfy similar formal properties...

object. In the theory of function spaces, such as in functional analysis or homotopy theory, one is commonly interested in continuous functions between topological...

compatible. Such objects have applications in deformation theory and rational homotopy theory. A differential graded Lie algebra is a graded vector space L =...

series and their rationality. A special issue of the journal Homology, Homotopy and Applications ("The Roos Festschrift volume") was published in 2002...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

Diophantine geometry Topology General Algebraic Differential Geometric Homotopy theory Applied Engineering mathematics Mathematical biology Mathematical...

The Bott periodicity theorem describes the eightfold periodicity of the homotopy groups of the direct limit of the orthogonal groups O(n). This has many...

Diophantine geometry Topology General Algebraic Differential Geometric Homotopy theory Applied Engineering mathematics Mathematical biology Mathematical...

duality Change of rings May, J.P.; Sigurdsson, J. (2006). Parametrized Homotopy Theory. A.M.S. p. 253. ISBN 0-8218-3922-5. Bourbaki, Nicolas (1989), Elements...

equivalence may refer to: Weak equivalence of categories Weak equivalence (homotopy theory) Weak equivalence between simplicial sets Weak equivalence (formal...

Alexander Grothendieck. He also collaborated with Barry Mazur to define étale homotopy theory which has become an important tool in algebraic geometry, and applied...

employed to ease the formalization of mathematics. Homotopy type theory is a synthetic approach to homotopy theory using the language of univalent foundations...

mathematician working in the fields of algebraic topology, category theory, homotopy theory, and the foundational aspects of spectra. He is known, in particular...

the study of manifolds as simplicial complexes and CW complexes play in homotopy theory, allowing one to analyze a space in terms of individual pieces and...

presheaf F with transfers is said to be A 1 {\displaystyle \mathbb {A} ^{1}} -homotopy invariant if F ( X ) ≃ F ( X × A 1 ) {\displaystyle F(X)\simeq F(X\times...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

{\displaystyle S} . The Conley index h ( S ) {\displaystyle h(S)} is the homotopy type of a space built from a certain pair ( N 1 , N 2 ) {\displaystyle...

the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. It was originally introduced by Yevsey...

Diophantine geometry Topology General Algebraic Differential Geometric Homotopy theory Applied Engineering mathematics Mathematical biology Mathematical...

varieties f : X → Y {\displaystyle f:X\to Y} . From here we can form the homotopy category K b ( S m C o r ) {\displaystyle K^{b}({\mathcal {SmCor}})} of...

Statistics and Computer Science from 2014 to 2022. Her research concerns homotopy theory and homological algebra. Shipley graduated from Harvard University...

{\text{Hom}}(\pi ,\mathbb {Z} )\end{aligned}}} From this information a basic homotopy theoretic classification of 3-manifolds can be found. Note from the Postnikov...

both from Rice University. His Ph.D. thesis, entitled Stable tangential homotopy equivalences, was written under the supervision of Morton L. Curtis. He...

Homotopy with fixed endpoints is necessary for the monodromy theorem to hold....

theorem: If X {\displaystyle X} is a compact topological space or has the homotopy type of a CW complex, then every (real or complex) Hilbert space bundle...

given homeomorphisms is an isotopy of the two given homeomorphisms in homotopy Regular isotopy of a link diagram, an equivalence relation in knot theory...

group is the rank of its maximal torus. The p-completion, in the sense of homotopy theory, of (the classifying space of) a compact connected Lie group defines...

size homotopy group is analogous to the one existing between homology groups and homotopy groups. In size theory, size functions and size homotopy groups...

more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is...

In mathematics, especially homotopy theory, a minimal fibration is used to approximate fibrations between presheaves. A minimal fibration has a defining...

informal discussion of braid groups on firm ground, one needs to use the homotopy concept of algebraic topology, defining braid groups as fundamental groups...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

Computer graphics Institutions Brown University Thesis Invariants of Regular Homotopy and Bordism of Low Dimensional Immersions (1982) Doctoral advisor Robion...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

univalent foundations and related to it homotopy type theory. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties...

number of edges), which remains invariant under contraction of edges (homotopy). The first property is that the Euler characteristic of a bouquet of s...

the i-th K-group of C with coefficients in a group G is defined as the homotopy group with coefficients: K i ( C ; G ) = π i ( B + C ; G ) {\displaystyle...

Stong (1965) and Akio Hattori (1966), gives an isomorphism between the stable homotopy of a Thom spectrum and the primitive elements of its K-homology. Hattori...

{\displaystyle X} and iteratively resolve with other spectra that are in the homotopy kernel of a map resolving the cohomology classes in H ∗ ( X ; Z / p ) {\displaystyle...

Adams–Novikov spectral sequence for calculating p {\displaystyle p} -local homotopy groups of spheres. BP {\displaystyle {\text{BP}}} is the universal example...

571–579. Bibcode:2011SHPSA..42..571C. doi:10.1016/j.shpsa.2011.09.013. Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy, Oxford University...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

space [SX, Y] of homotopy classes of maps from the suspension SX of X to Y is naturally isomorphic to the space [X, ΩY] of homotopy classes of maps from...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

objects are given by a space only up to homotopy. Since the notion of injection and surjection are not homotopy invariant notions (consider an interval...

be embarrassing not to develop parallel implementations of polynomial homotopy continuation methods." The term is first found in the literature in a 1986...

for homotopy groups of spheres. Computational methods for solving systems of polynomial equations. Brown has an algorithm to compute the homotopy groups...

Guozhen; Xu, Zhouli (2017). "The triviality of the 61-stem in the stable homotopy groups of spheres". Annals of Mathematics. 186 (2): 501–580. arXiv:1601...

{\displaystyle H_{Nis}^{1}(Spec(A),GL_{r}).} Positive results about the homotopy invariance of H N i s 1 ( U , G ) {\displaystyle H_{Nis}^{1}(U,G)} of isotropic...

Theory and Rational Homotopy Type, Memoirs of the A. M. S., vol. 179, 1976. Hinich, Vladimir (1997-02-11). "Homological algebra of homotopy algebras". arXiv:q-alg/9702015...

fiber is C× ≅ S1, so up to homotopy, GL → PGL is a circle bundle. The higher homotopy groups of the circle vanish, so the homotopy groups of GL(n, C) and...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily...

Moreover, any homotopy equivalence of closed surfaces deforms to a homeomorphism. Example 2. If a closed manifold Mn (n ≠ 3) is homotopy-equivalent to...

cofibrations and weak equivalences. Typically one is interested in the underlying homotopy category obtained by localizing against the weak equivalences. A Quillen...

conference}}: Cite uses deprecated parameter |citeseerx= (help) George Whitehead; Fifty years of homotopy theory Haynes Miller; The origin of sheaf theory...

disk D 2 {\displaystyle D^{2}} is contractible, the solid torus has the homotopy type of a circle, S 1 {\displaystyle S^{1}} . Therefore the fundamental...

Soviet and Russian mathematician known for his work on algebraic topology, homotopy theory, and mathematical physics. Buchstaber's first research work was...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

Meeting... had been dedicated to Luigi Russolo, and the following album Homotopy to Marie was listed as being inspired by Franz Kamin. "Alvin's Funeral...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

tangent bundle) which was used by Lev Pontryagin in 1950 to compute the homotopy group π n + 2 ( S n ) = Z / 2 Z {\displaystyle \pi _{n+2}(S^{n})=\mathbb...

perfect field is isomorphic to the motivic stable homotopy group of spheres π0,0(S0,0) (see "A¹ homotopy theory"). Two fields are said to be Witt equivalent...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

nontrivial homotopy group, preserved by the differential equations. Thus, the differential equation solutions can be classified into homotopy classes. No...

_{2})} which twists an odd number of times. The key fact is that up to homotopy there are two choices of trivialisation of a trivial 3-plane bundle over...

Geometric measure theory Institutions Princeton University Thesis The Homotopy Groups of the Integral Cycle Groups (1962) Doctoral advisor Herbert Federer...

Langlands and the Langlands program. The IAS is a main center of research for homotopy type theory, a modern approach to the foundations of mathematics which...

special case of the homotopy invariance of de Rham cohomology; in fact, it is common to establish the lemma by showing the homotopy invariance or at least...

spaces can refer to any of several results and conjectures prompted by homotopy theory work of Dennis Sullivan. A basic theme and motivation concerns the...

\Lambda (n)} the fundamental group may be inferred from the long exact homotopy sequence: π 1 ( Λ ( n ) ) = Z . {\displaystyle \pi _{1}(\Lambda (n))=\mathbb...

invertible complex matrices, which have the homotopy type of a circle. From the perspective of homotopy theory, a real line bundle therefore behaves...

Space compact connected Hausdorff metric uniform second-countable Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex...

uniqueness of graph drawings obtained from systems of idealized springs Tutte homotopy theorem, on the composition of generalized paths in matroids Hanani–Tutte...

equivalence of categories. (Unfortunately this conflicts with terminology from homotopy theory.) There is also a close relation to the concept of adjoint functors...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

complex (i.e., an ∞-category). The Duskin nerve is an instance of the homotopy coherent nerve. By definition, a functor is simply a structure-preserving...

In mathematics, Joyal's theorem is a theorem in homotopy theory that provides necessary and sufficient conditions for the solvability of a certain lifting...

uses of groupoids in group theory and topology. Muro, Fernando (2016). "Homotopy units in A-infinity algebras". Trans. Amer. Math. Soc. 368: 2145–2184....

{G}})=\operatorname {rank} H_{1}({\tilde {G}}).} It can also be computed via homotopy. If a (connected) control-flow graph is considered a one-dimensional CW...

implications: for example the existence of this bundle shows that the higher homotopy groups of spheres are not trivial in general. It also provides a basic...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

truncation (a type former that truncates a type down to a mere proposition in homotopy type theory): for any a : A {\displaystyle a:A} (read "term a {\displaystyle...

manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether two homotopy equivalent manifolds...

covering Simply connected Semi-locally simply connected Path (topology) Homotopy Homotopy lifting property Pointed space Wedge sum Smash product Cone (topology)...

coincides with the notion of a quotient monoid or a quotient group. The homotopy category of topological spaces hTop is a quotient category of Top, the...

Wilson is a mathematician based in Johns Hopkins University specializing in homotopy theory. Wilson received his Ph.D. from Massachusetts Institute of Technology...

Nice, and in 1995 at the University of Lille. His research deals with homotopy theory and homology of loop spaces with applications in geometry. He wrote...

homology of dimension one. Similar to the fundamental group or the higher homotopy groups of a space, homology groups are important topological invariants...

space that occurs in algebraic geometry. It is used in calculating the homotopy properties of a simplicial group. Srinivas, Vasudevan (2013). Algebraic...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

: M → X {\displaystyle (f,b)\colon M\to X} is normally cobordant to a homotopy equivalence if and only if the image θ ( f , b ) = 0 {\displaystyle \theta...

Rognes at the workshop Modern foundations for stable homotopy theory, Oberwolfach 2005...

2009-06-04 at the Wayback Machine Nonabelian Algebraic Topology: Higher homotopy groupoids of filtered spaces Brown, Ronald; Higgins, Philip; Sivera, Rafael...

limits and colimits and then we can not develop the usual homotopy constructions like loops, homotopy limits and colimits, etc. An answer to this problem is...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

1999.0043. S2CID 168002. Rezk, Charles (2001). "A model for the homotopy theory of homotopy theory". Transactions of the American Mathematical Society. 353...

system that result in this same configuration. This ambiguity is called the homotopy class of the gradual rotation. The belt trick (shown, in which both ends...

In category theory and homotopy theory the Burnside category of a finite group G is a category whose objects are finite G-sets and whose morphisms are...

property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting...

from X to Y is universal for (homotopy classes of) maps from X to A-local CW complexes. This space Y is unique up to homotopy equivalence, and is called...

Y)/(X\vee Y).} The smash product shows up in homotopy theory, a branch of algebraic topology. In homotopy theory, one often works with a different category...

In rational homotopy theory, the Halperin conjecture concerns the Serre spectral sequence of certain fibrations. It is named after the Canadian mathematician...

groupoid of X {\displaystyle X} ): two loops (under equivalence relation of homotopy) may not have the same base point so they cannot be multiplied with each...

complex models. In any dimension, the medial axis of a bounded open set is homotopy equivalent to the given set. If S is given by a unit speed parametrisation...

groupoid object in X is effective. Mathematics portal Bousfield localization Homotopy hypothesis – Hypothesis in mathematical category theory ∞-groupoid – Abstract...

methods such as Euler's method and Runge–Kutta methods can be used. The homotopy analysis method (HAM) has also been reported for obtaining approximate...

topological spaces or schemes, or to be even more general: any object of a homotopy category to associated rings; these rings reflect some aspects of the structure...

mathematics, especially algebraic topology and homotopy theory, the Hopf–Whitney theorem is a result relating the homotopy classes between a CW complex and a multiply...

C^{*}(X\times Y)} which are also homotopy equivalences, as witnessed by the duals of the preceding equations, using the dual homotopy H ∗ {\displaystyle H^{*}}...

Blakers–Massey theorem (homotopy theory) Bott periodicity theorem (homotopy theory) Brown's representability theorem (homotopy theory) Cellular approximation...

"geometric object" that underlies the chromatic approach to the stable homotopy theory, a branch of algebraic topology. Currently, it is not known whether...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

In mathematics, especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a...

Technology, under the supervision of Daniel Quillen, with thesis Abstract Homotopy Theory and Generalized Sheaf Cohomology. Hired by Cornell University in...

\mathbb {R} } -graph X homotopy equivalent to a bouquet of n circles together with a certain choice of a free homotopy class of a homotopy equivalence from...

defining the n + 1 {\displaystyle n+1} st homotopy or homology group in terms of indeterminacy in null-homotopies/null-homologies of n-dimensional maps/chains...

Institute of Technology University of Chicago Johns Hopkins University Thesis Homotopy Colimits in Cat(+ Category of Small Categories) with Applications to Algebraic...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

American mathematician Dennis Sullivan developed a DGA to encode the rational homotopy type of topological spaces. Let A ∙ = ⨁ i ∈ Z A i {\displaystyle A_{\bullet...

Polytechnique Fédérale de Lausanne (EPFL) since 1999. She is known for her work on homotopy theory, category theory, and algebraic topology, both pure and applied...

when René Thom showed that cobordism groups could be computed by means of homotopy theory, via the Thom complex construction. Cobordism theory became part...

schemes. Let C be the category of CW-complexes with morphisms given by homotopy classes of continuous functions. For each natural number n there is a contravariant...

"Inductive types in homotopy type theory". arXiv:1201.3898 [math.LO]. Awodey, Steve; Gambino, Nicola; Sojakova, Kristina (2015-04-21). "Homotopy-initial algebras...

a recent reawakening as methods have been discovered to apply motivic homotopy theory to the problems. The problem of Apollonius is an early example of...

-algebras, it classifies the Fredholm modules over an algebra. An operator homotopy between two Fredholm modules ( H , F 0 , Γ ) {\displaystyle ({\mathcal...

In mathematics, and in particular homotopy theory, a hypercovering (or hypercover) is a simplicial object that generalises the Čech nerve of a cover. For...

the other hand, numerical methods typically use algebraically founded homotopy continuation, with a base field of the complex numbers. Algebraic equation...

work other models with non-trivial identity types were studied, including homotopy type theory which has been proposed as a foundation for mathematics in...

link, up to link homotopy. In other words, each component of the extended link is allowed to move through regular homotopy (homotopy through immersions)...

that a curve is homotopic to a constant curve if there exists a smooth homotopy (within U {\displaystyle U} ) from the curve to the constant curve. Intuitively...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

Bott, for his papers "The space of loops on a Lie group", and "The stable homotopy of the classical groups". 1966 Stephen Smale "for his contributions to...

numbers Jordan–Schur theorem Jordan–Schönflies theorem Bounded variation Homotopy group k-edge-connected graph Total variation Scientific career Fields Mathematics...

In mathematics, especially homotopy theory, a cartesian fibration is, roughly, a map so that every lift exists that is a final object among all lifts....

homotopy equivalent to the infinite real projective space. Let M be a Riemannian manifold. Almgren isomorphism theorem asserts that the m-th homotopy...

{\mathit {MF}}[f]=\min\{\#\mathrm {Fix} (g)\,|\,g\sim f\},} where ~ indicates homotopy of mappings, and #Fix(g) indicates the number of fixed points of g. The...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

supervision of Edgar H. Brown, Jr. His thesis was titled On Odd Primary Stable Homotopy Theory and was published in the Memoirs of the American Mathematical Society...

theory Game theory Graph theory Group theory Hodge theory Homology theory Homotopy theory Ideal theory Index theory Information theory Intersection theory...

1 and 2, were experimental and contained features such as support for homotopy type theory-based foundations that were later dropped. Lean 3 (first released...

category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram n-categories...

{\displaystyle S^{2}} . So, the superselection sectors are classified by the second homotopy group of Σ {\displaystyle \Sigma } , π 2 ( Σ ) {\displaystyle \pi _{2}(\Sigma...

invariant under a regular homotopy of a curve: it is the degree of the Gauss map. However, it is not invariant under homotopy: passing through a kink (cusp)...

example be used to explicitly state its factorizations or to search for weak homotopy equivalences. Using the subdivision of simplicial sets, the extension of...

with critical points of index at most n, and so V {\displaystyle V} is homotopy equivalent to a CW complex of real dimension at most n. Consequently, if...

due to the appearance of the fundamental group or a higher-dimensional homotopy group in the description of the problem, quite often because the boundary...