Homotopy

This article is about topology. For chemistry, seeHomotopic groups.

Intopology,twocontinuous functionsfrom onetopological spaceto 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/,hə-MO-tə-pee;/ˈhoʊmoʊˌtoʊpiː/,HOH-moh-toh-pee) between the two functions. A notable use of homotopy is the definition ofhomotopy groupsandcohomotopy groups,importantinvariantsinalgebraic topology.

The two dashedpathsshown above are homotopic relative to their endpoints. The animation represents one possible homotopy.

In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work withcompactly generated spaces,CW complexes,orspectra.

Formal definition

A homotopy and its inverse, between twoembeddingsof thetorusinto$\mathbb {R} ^{3}$:as "the surface of a doughnut" and as "the surface of a coffee mug". This is also an example of anisotopy.

Formally, a homotopy between twocontinuous functionsfandgfrom a topological spaceXto a topological spaceYis defined to be acontinuous function$H:X\times [0,1]\to Y$from theproductof the spaceXwith theunit interval[0, 1] toYsuch that$H(x,0)=f(x)$and$H(x,1)=g(x)$for all$x\in X$.

If we think of the secondparameterofHas time thenHdescribes acontinuous deformationoffintog:at time 0 we have the functionfand 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 fromftogas the slider moves from 0 to 1, and vice versa.

An alternative notation is to say that a homotopy between two continuous functions$f,g:X\to Y$is a family of continuous functions$h_{t}:X\to Y$for$t\in [0,1]$such that$h_{0}=f$and$h_{1}=g$,and themap$(x,t)\mapsto h_{t}(x)$is continuous from$X\times [0,1]$to$Y$.The two versions coincide by setting$h_{t}(x)=H(x,t)$.It is not sufficient to require each map$h_{t}(x)$to be continuous.

The animation that is looped above right provides an example of a homotopy between twoembeddings,fandg,of the torus intoR³.Xis the torus,YisR³,fis some continuous function from the torus toR³that takes the torus to the embedded surface-of-a-doughnut shape with which the animation starts;gis some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape. The animation shows the image ofh_t(X) as a function of the parametert,wheretvaries with time from 0 to 1 over each cycle of the animation loop. It pauses, then shows the image astvaries back from 1 to 0, pauses, and repeats this cycle.

Properties

Continuous functionsfandgare said to be homotopic if and only if there is a homotopyHtakingftogas described above. Being homotopic is anequivalence relationon the set of all continuous functions fromXtoY. This homotopy relation is compatible withfunction compositionin the following sense: iff₁,g₁:X→Yare homotopic, andf₂,g₂:Y→Zare homotopic, then their compositionsf₂ ∘ f₁andg₂ ∘ g₁:X→Zare also homotopic.

Examples

• If$f,g:\mathbb {R} \to \mathbb {R} ^{2}$are given by$f(x):=\left(x,x^{3}\right)$and$g(x)=\left(x,e^{x}\right)$,then the map$H:\mathbb {R} \times [0,1]\to \mathbb {R} ^{2}$given by$H(x,t)=\left(x,(1-t)x^{3}+te^{x}\right)$is a homotopy between them.
• More generally, if$C\subseteq \mathbb {R} ^{n}$is aconvexsubset ofEuclidean spaceand$f,g:[0,1]\to C$arepathswith the same endpoints, then there is alinear homotopy(orstraight-line homotopy) given by

  ${\begin{aligned}H:[0,1]\times [0,1]&\longrightarrow C\\(s,t)&\longmapsto (1-t)f(s)+tg(s).\end{aligned}}$

• Let$\operatorname {id} _{B^{n}}:B^{n}\to B^{n}$be theidentity functionon the unitn-disk;i.e. the set$B^{n}:=\left\{x\in \mathbb {R} ^{n}:\|x\|\leq 1\right\}$.Let$c_{\vec {0}}:B^{n}\to B^{n}$be theconstant function$c_{\vec {0}}(x):={\vec {0}}$which sends every point to theorigin.Then the following is a homotopy between them:

  ${\begin{aligned}H:B^{n}\times [0,1]&\longrightarrow B^{n}\\(x,t)&\longmapsto (1-t)x.\end{aligned}}$

Homotopy equivalence

Given two topological spacesXandY,ahomotopy equivalencebetweenXandYis a pair of continuousmapsf:X→Yandg:Y→X,such thatg ∘ fis homotopic to theidentity mapid_Xandf ∘ gis homotopic to id_Y.If such a pair exists, thenXandYare said to behomotopy equivalent,or of the samehomotopy type.This relation of homotopy equivalence is often denoted$\simeq$.Intuitively, two spacesXandYare 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.

Homotopy equivalence vs. homeomorphism

Ahomeomorphismis a special case of a homotopy equivalence, in whichg ∘ fis equal to the identity map id_X(not only homotopic to it), andf ∘ gis equal to id_Y.Therefore, if X and Y are homeomorphic then they are homotopy-equivalent, but the opposite is not true. Some examples:

• A solid disk is homotopy-equivalent to a single point, since you can deform the disk along radial lines continuously to a single point. However, they are not homeomorphic, since there is nobijectionbetween them (since one is an infinite set, while the other is finite).
• TheMöbius stripand an untwisted (closed) strip are homotopy equivalent, since you can deform both strips continuously to a circle. But they are not homeomorphic.

Examples

• The first example of a homotopy equivalence is$\mathbb {R} ^{n}$with a point, denoted$\mathbb {R} ^{n}\simeq \{0\}$.The part that needs to be checked is the existence of a homotopy$H:I\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}$between$\operatorname {id} _{\mathbb {R} ^{n}}$and$p_{0}$,the projection of$\mathbb {R} ^{n}$onto the origin. This can be described as$H(t,\cdot )=t\cdot p_{0}+(1-t)\cdot \operatorname {id} _{\mathbb {R} ^{n}}$.
• There is a homotopy equivalence between$S^{1}$(the1-sphere) and$\mathbb {R} ^{2}-\{0\}$.
  • More generally,$\mathbb {R} ^{n}-\{0\}\simeq S^{n-1}$.
• Anyfiber bundle$\pi:E\to B$with fibers$F_{b}$homotopy equivalent to a point has homotopy equivalent total and base spaces. This generalizes the previous two examples since$\pi:\mathbb {R} ^{n}-\{0\}\to S^{n-1}$is a fiber bundle with fiber$\mathbb {R} _{>0}$.
• Everyvector bundleis a fiber bundle with a fiber homotopy equivalent to a point.
• $\mathbb {R} ^{n}-\mathbb {R} ^{k}\simeq S^{n-k-1}$for any$0\leq k<n$,by writing$\mathbb {R} ^{n}-\mathbb {R} ^{k}$as the total space of the fiber bundle$\mathbb {R} ^{k}\times (\mathbb {R} ^{n-k}-\{0\})\to (\mathbb {R} ^{n-k}-\{0\})$,then applying the homotopy equivalences above.
• If a subcomplex$A$of aCW complex$X$is contractible, then thequotient space$X/A$is homotopy equivalent to$X$.
• Adeformation retractionis a homotopy equivalence.

Null-homotopy

A function$f$is said to benull-homotopicif it is homotopic to a constant function. (The homotopy from$f$to a constant function is then sometimes called anull-homotopy.) For example, a map$f$from theunit circle$S^{1}$to any space$X$is null-homotopic precisely when it can be continuously extended to a map from theunit disk$D^{2}$to$X$that agrees with$f$on the boundary.

It follows from these definitions that a space$X$is contractible if and only if the identity map from$X$to itself—which is always a homotopy equivalence—is null-homotopic.

Invariance

Homotopy equivalence is important because inalgebraic topologymany concepts arehomotopy invariant,that is, they respect the relation of homotopy equivalence. For example, ifXandYare homotopy equivalent spaces, then:

• Xispath-connectedif and only ifYis.
• Xissimply connectedif and only ifYis.
• The (singular)homologyandcohomology groupsofXandYareisomorphic.
• IfXandYare path-connected, then thefundamental groupsofXandYare isomorphic, and so are the higherhomotopy groups.(Without the path-connectedness assumption, one has π₁(X, x₀) isomorphic to π₁(Y,f(x₀)) wheref:X→Yis a homotopy equivalence andx₀∈X.)

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).

Variants

Relative homotopy

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: iffandgare continuous maps fromXtoYandKis asubsetofX,then we say thatfandgare homotopic relative toKif there exists a homotopyH:X× [0, 1] →Ybetweenfandgsuch thatH(k, t) =f(k) =g(k)for allk∈Kandt∈ [0, 1].Also, ifgis aretractionfromXtoKandfis the identity map, this is known as a strongdeformation retractofXtoK. WhenKis a point, the termpointed homotopyis used.

Isotopy

Theunknotis not equivalent to thetrefoil knotsince one cannot be deformed into the other through a continuous path of homeomorphisms of the ambient space. Thus they are not ambient-isotopic.

When two given continuous functionsfandgfrom the topological spaceXto the topological spaceYareembeddings,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.

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) = −xisnotisotopic to the identityg(x) =x.Any homotopy fromfto the identity would have to exchange the endpoints, which would mean that they would have to 'pass through' each other. Moreover,fhas changed the orientation of the interval andghas not, which is impossible under an isotopy. However, the maps are homotopic; one homotopy fromfto 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 discin$\mathbb {R} ^{2}$defined byf(x, y) = (−x,−y) is isotopic to a 180-degreerotationaround the origin, and so the identity map andfare 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,K₁andK₂,in three-dimensionalspace. A knot is anembeddingof 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 candeformone embedding to another through a path of embeddings: a continuous function starting att= 0 giving theK₁embedding, ending att= 1 giving theK₂embedding, 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. KnotsK₁andK₂are considered equivalent when there is an ambient isotopy which movesK₁toK₂.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.

Timelike homotopy

On aLorentzian manifold,certain curves are distinguished astimelike(representing something that only goes forwards, not backwards, in time, in every local frame). Atimelike homotopybetween twotimelike curvesis 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 connectedby timelike curves. A manifold such as the3-spherecan besimply connected(by any type of curve), and yet betimelike multiply connected.

Properties

Lifting and extension properties

Main article:Homotopy lifting property

If we have a homotopyH:X× [0,1] →Yand a coverp:Y→Yand we are given a maph₀:X→Ysuch thatH₀=p○h₀(h₀is called aliftofh₀), then we can lift allHto a mapH:X× [0, 1] →Ysuch 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.

Groups

Main article:Homotopy group

Since the relation of two functions$f,g\colon X\to Y$being homotopic relative to a subspace is an equivalence relation, we can look at theequivalence classesof maps between a fixedXandY.If we fix$X=[0,1]^{n}$,the unit interval [0, 1]crossedwith itselfntimes, and we take itsboundary$\partial ([0,1]^{n})$as a subspace, then the equivalence classes form a group, denoted$\pi _{n}(Y,y_{0})$,where$y_{0}$is in the image of the subspace$\partial ([0,1]^{n})$.

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$n=1$,it is also called thefundamental group.

Homotopy category

Main article:Homotopy category

The idea of homotopy can be turned into a formal category ofcategory theory.Thehomotopy categoryis the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spacesXandYare isomorphic in this category if and only if they are homotopy-equivalent. Then afunctoron 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 afunctorialhomotopy invariant: this means that iffandgfromXtoYare homotopic, then thegroup homomorphismsinduced byfandgon the level ofhomology groupsare the same: H_n(f) = H_n(g): H_n(X) → H_n(Y) for alln.Likewise, ifXandYare in additionpath connected,and the homotopy betweenfandgis pointed, then the group homomorphisms induced byfandgon the level ofhomotopy groupsare also the same: π_n(f) = π_n(g): π_n(X) → π_n(Y).

Applications

Based on the concept of the homotopy,computation methodsforalgebraicanddifferential equationshave been developed. The methods for algebraic equations include thehomotopy continuationmethodand 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 canrepresenta cohomology functor on a spaceXby mappings ofXinto an appropriate fixed space, up to homotopy equivalence. For example, for any abelian groupG,and any based CW-complexX,the set$[X,K(G,n)]$of based homotopy classes of based maps fromXto theEilenberg–MacLane space$K(G,n)$is in natural bijection with then-thsingular cohomologygroup$H^{n}(X,G)$of the spaceX.One says that theomega-spectrumof Eilenberg-MacLane spaces arerepresenting spacesfor 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.

See also

• Fiber-homotopy equivalence(relative version of a homotopy equivalence)
• Homeotopy
• Homotopy type theory
• Mapping class group
• Poincaré conjecture
• Regular homotopy

Sources

• Armstrong, M.A. (1979).Basic Topology.Springer.ISBN978-0-387-90839-7.
• "Homotopy",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• "Isotopy (in topology)",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Spanier, Edwin (December 1994).Algebraic Topology.Springer.ISBN978-0-387-94426-5.