Intopology, twocontinuous functions from onetopological space to another are calledhomotopic (fromAncient Greek:ὁμόςhomós'same, similar' andτόποςtópos'place') if one can be "continuously deformed" into the other, such a deformation being called ahomotopy (/həˈmɒtəpiː/[1]hə-MOT-ə-pee;/ˈhoʊmoʊˌtoʊpiː/[2]HOH-moh-toh-pee) between the two functions. A notable use of homotopy is the definition ofhomotopy groups andcohomotopy groups, importantinvariants inalgebraic topology.[3]
In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work withcompactly generated spaces,CW complexes, orspectra.
Formally, a homotopy between twocontinuous functionsf andg from a topological spaceX to a topological spaceY is defined to be acontinuous function from theproduct of the spaceX with theunit interval [0, 1] toY such that and for all.
If we think of the secondparameter ofH as time thenH describes acontinuous deformation off intog: at time 0 we have the functionf and at time 1 we have the functiong. We can also think of the second parameter as a "slider control" that allows us to smoothly transition fromf tog as the slider moves from 0 to 1, and vice versa.
An alternative notation is to say that a homotopy between two continuous functions is a family of continuous functions for such that and, and themap is continuous from to. The two versions coincide by setting. It is not sufficient to require each map to be continuous.[4]
The animation that is looped above right provides an example of a homotopy between twoembeddings,f andg, of the torus intoR3.X is the torus,Y isR3,f is some continuous function from the torus toR3 that takes the torus to the embedded surface-of-a-doughnut shape with which the animation starts;g is some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape. The animation shows the image ofht(X) as a function of the parametert, wheret varies with time from 0 to 1 over each cycle of the animation loop. It pauses, then shows the image ast varies back from 1 to 0, pauses, and repeats this cycle.
Continuous functionsf andg are said to be homotopic if and only if there is a homotopyH takingf tog as described above. Being homotopic is anequivalence relation on the set of all continuous functions fromX toY. This homotopy relation is compatible withfunction composition in the following sense: iff1,g1 :X →Y are homotopic, andf2,g2 :Y →Z are homotopic, then their compositionsf2 ∘ f1 andg2 ∘ g1 :X →Z are also homotopic.
Given two topological spacesX andY, ahomotopy equivalence betweenX andY is a pair of continuousmapsf :X →Y andg :Y →X, such thatg ∘ f is homotopic to theidentity map idX andf ∘ g is homotopic to idY. If such a pair exists, thenX andY are said to behomotopy equivalent, or of the samehomotopy type. This relation of homotopy equivalence is often denoted.[6] Intuitively, two spacesX andY are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. Spaces that are homotopy-equivalent to a point are calledcontractible.
Ahomeomorphism is a special case of a homotopy equivalence, in whichg ∘ f is equal to the identity map idX (not only homotopic to it), andf ∘ g is equal to idY.[7]: 0:53:00 Therefore, if X and Y are homeomorphic then they are homotopy-equivalent, but the opposite is not true. Some examples:
A function is said to benull-homotopic if it is homotopic to a constant function. (The homotopy from to a constant function is then sometimes called anull-homotopy.) For example, a map from theunit circle to any space is null-homotopic precisely when it can be continuously extended to a map from theunit disk to that agrees with on the boundary.
It follows from these definitions that a space is contractible if and only if the identity map from to itself—which is always a homotopy equivalence—is null-homotopic.
Homotopy equivalence is important because inalgebraic topology many concepts arehomotopy invariant, that is, they respect the relation of homotopy equivalence. For example, ifX andY are homotopy equivalent spaces, then:
An example of an algebraic invariant of topological spaces which is not homotopy-invariant iscompactly supported homology (which is, roughly speaking, the homology of thecompactification, and compactification is not homotopy-invariant).
In order to define thefundamental group, one needs the notion ofhomotopy relative to a subspace. These are homotopies which keep the elements of the subspace fixed. Formally: iff andg are continuous maps fromX toY andK is asubset ofX, then we say thatf andg are homotopic relative toK if there exists a homotopyH :X × [0, 1] →Y betweenf andg such thatH(k, t) =f(k) =g(k) for allk ∈K andt ∈ [0, 1]. Also, ifg is aretraction fromX toK andf is the identity map, this is known as a strongdeformation retract ofX toK.WhenK is a point, the termpointed homotopy is used.
When two given continuous functionsf andg from the topological spaceX to the topological spaceY areembeddings, one can ask whether they can be connected 'through embeddings'. This gives rise to the concept ofisotopy, which is a homotopy,H, in the notation used before, such that for each fixedt,H(x, t) gives an embedding.[9]
A related, but different, concept is that ofambient isotopy.
Requiring that two embeddings be isotopic is a stronger requirement than that they be homotopic. For example, the map from the interval [−1, 1] into the real numbers defined byf(x) = −x isnot isotopic to the identityg(x) =x. Any homotopy fromf to the identity would have to exchange the endpoints, which would mean that they would have to 'pass through' each other. Moreover,f has changed the orientation of the interval andg has not, which is impossible under an isotopy. However, the maps are homotopic; one homotopy fromf to the identity isH: [−1, 1] × [0, 1] → [−1, 1] given byH(x, y) = 2yx − x.
Two homeomorphisms (which are special cases of embeddings) of the unit ball which agree on the boundary can be shown to be isotopic usingAlexander's trick. For this reason, the map of theunit disc in defined byf(x, y) = (−x, −y) is isotopic to a 180-degreerotation around the origin, and so the identity map andf are isotopic because they can be connected by rotations.
Ingeometric topology—for example inknot theory—the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots,K1 andK2, in three-dimensional space. A knot is anembedding of a one-dimensional space, the "loop of string" (or the circle), into this space, and this embedding gives a homeomorphism between the circle and its image in the embedding space. The intuitive idea behind the notion of knot equivalence is that one candeform one embedding to another through a path of embeddings: a continuous function starting att = 0 giving theK1 embedding, ending att = 1 giving theK2 embedding, with all intermediate values corresponding to embeddings. This corresponds to the definition of isotopy. Anambient isotopy, studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. KnotsK1 andK2 are considered equivalent when there is an ambient isotopy which movesK1 toK2. This is the appropriate definition in the topological category.
Similar language is used for the equivalent concept in contexts where one has a stronger notion of equivalence. For example, a path between two smooth embeddings is asmooth isotopy.
On aLorentzian manifold, certain curves are distinguished astimelike (representing something that only goes forwards, not backwards, in time, in every local frame). Atimelike homotopy between twotimelike curves is a homotopy such that the curve remains timelike during the continuous transformation from one curve to another. Noclosed timelike curve (CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to bemultiply connected by timelike curves. A manifold such as the3-sphere can besimply connected (by any type of curve), and yet betimelike multiply connected.[10]
If we have a homotopyH :X × [0,1] →Y and a coverp :Y →Y and we are given a maph0 :X →Y such thatH0 =p ○h0 (h0 is called alift ofh0), then we can lift allH to a mapH :X × [0, 1] →Y such thatp ○H =H. The homotopy lifting property is used to characterizefibrations.
Another useful property involving homotopy is thehomotopy extension property,which characterizes the extension of a homotopy between two functions from a subset of some set to the set itself. It is useful when dealing withcofibrations.
Since the relation of two functions being homotopic relative to a subspace is an equivalence relation, we can look at theequivalence classes of maps between a fixedX andY. If we fix, the unit interval [0, 1]crossed with itselfn times, and we take itsboundary as a subspace, then the equivalence classes form a group, denoted, where is in the image of the subspace.
We can define the action of one equivalence class on another, and so we get a group. These groups are called thehomotopy groups. In the case, it is also called thefundamental group.
The idea of homotopy can be turned into a formal category ofcategory theory. Thehomotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spacesX andY are isomorphic in this category if and only if they are homotopy-equivalent. Then afunctor on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category.
For example, homology groups are afunctorial homotopy invariant: this means that iff andg fromX toY are homotopic, then thegroup homomorphisms induced byf andg on the level ofhomology groups are the same: Hn(f) = Hn(g) : Hn(X) → Hn(Y) for alln. Likewise, ifX andY are in additionpath connected, and the homotopy betweenf andg is pointed, then the group homomorphisms induced byf andg on the level ofhomotopy groups are also the same: πn(f) = πn(g) : πn(X) → πn(Y).
Based on the concept of the homotopy,computation methods foralgebraic anddifferential equations have been developed. The methods for algebraic equations include thehomotopy continuation method[11] and the continuation method (seenumerical continuation). The methods for differential equations include thehomotopy analysis method.
Homotopy theory can be used as a foundation forhomology theory: one canrepresent a cohomology functor on a spaceX by mappings ofX into an appropriate fixed space, up to homotopy equivalence. For example, for any abelian groupG, and any based CW-complexX, the set of based homotopy classes of based maps fromX to the Eilenberg–MacLane space is in natural bijection with then-thsingular cohomology group of the spaceX. One says that theomega-spectrum of Eilenberg-MacLane spaces arerepresenting spaces for singular cohomology with coefficients inG. Using this fact, homotopy classes between a CW complex and a multiply connected space can be calculated using cohomology as described by theHopf–Whitney theorem.
Recently, homotopy theory is used to develop deep learning based generative models likediffusion models andflow-based generative models. Perturbing the complex non-Gaussian states is a tough task. Using deep learning and homotopy, such complex states can be transformed to Gaussian state and mildly perturbed to get transformed back to perturbed complex states.[12]
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