View source for Homotopy

{{short description|Continuous deformation between two continuous functions}}
{{About|topology|chemistry|Homotopic groups}}
[[File:HomotopySmall.gif|thumb|The two dashed [[Path (topology)|paths]] shown above are homotopic relative to their endpoints. The animation represents one possible homotopy.]]
In [[topology]], two [[continuous function (topology)|continuous function]]s from one [[topological space]] to another are called '''homotopic''' (from {{langx|grc|ὁμός}} {{transl|grc|homós}} "same, similar" and {{lang|grc|τόπος}} {{transl|grc|tópos}} "place") if one can be "continuously deformed" into the other, such a deformation being called a '''homotopy''' ({{IPAc-en|h|ə|ˈ|m|ɒ|t|ə|p|iː}},<ref>{{cite web |url = https://www.dictionary.com/browse/homotopy |title = Homotopy Definition & Meaning |access-date = 22 April 2022 }}</ref> {{Respell|hə|MO|tə|pee}}; {{IPAc-en|ˈ|h|oʊ|m|oʊ|ˌ|t|oʊ|p|iː}},<ref>{{cite web |url = https://www.youtube.com/watch?v=Ft8R3-kPDdk |title = Homotopy Type Theory Discussed - Computerphile |website = [[YouTube]] |date = 13 October 2017 |access-date = 22 April 2022 }}</ref> {{Respell|HOH|moh|toh|pee}}) between the two functions. A notable use of homotopy is the definition of [[homotopy groups]] and [[cohomotopy groups]], important [[invariant (mathematics)|invariant]]s in [[algebraic topology]].<ref>{{Cite web|url=https://www.britannica.com/science/homotopy|title=Homotopy {{!}} mathematics|website=Encyclopedia Britannica|language=en|access-date=2019-08-17}}</ref>

In practice, there are technical difficulties in using homotopies with certain spaces.  Algebraic topologists work with [[compactly generated space]]s, [[CW complex]]es, or [[Spectrum (homotopy theory)|spectra]].

==Formal definition==
[[File:Mug and Torus morph.gif|thumb|right|250px|A homotopy and its inverse, between two [[embedding]]s of the [[torus]] into <math>\mathbb{R}^3</math>: as "the surface of a doughnut" and as "the surface of a coffee mug".  This is also an example of an [[#Isotopy|isotopy]].]]
Formally, a homotopy between two [[continuous function (topology)|continuous function]]s ''f'' and ''g'' from a 
topological space ''X'' to a topological space ''Y'' is defined to be a [[Function_of_several_real_variables#Continuity_and_limit|continuous function]] <math>H: X \times  [0,1] \to Y</math> from the [[product topology|product]] of the space ''X'' with the [[unit interval]] [0,&thinsp;1] to ''Y'' such that <math>H(x,0) = f(x)</math> and <math>H(x,1) = g(x)</math> for all <math>x \in X</math>. 

If we think of the second [[parameter]] of ''H'' as time then ''H'' describes a ''continuous deformation'' of ''f'' into ''g'': at time 0 we have the function ''f'' and at time 1 we have the function ''g''. We can also think of the second parameter as a "slider control" that allows us to smoothly transition from ''f'' to ''g'' as the slider moves from 0 to 1, and vice versa.

An alternative notation is to say that a homotopy between two continuous functions <math>f, g: X \to Y</math> is a family of continuous functions <math>h_t: X \to Y</math> for <math>t \in [0,1]</math> such that <math>h_0 = f</math> and <math>h_1 = g</math>, and the [[Map (mathematics)|map]] <math>(x, t) \mapsto h_t(x)</math> is continuous from <math>X \times [0,1]</math> to <math>Y</math>. The two versions coincide by setting <math>h_t(x) = H(x,t)</math>. It is not sufficient to require each map <math>h_t(x)</math> to be continuous.<ref>{{Cite web|url=https://math.stackexchange.com/q/104515 |title=algebraic topology - Path homotopy and separately continuous functions|website=Mathematics Stack Exchange}}</ref>

The animation that is looped above right provides an example of a homotopy between two [[embedding]]s, ''f'' and ''g'', of the torus into {{nowrap|1=''R''<sup>3</sup>}}.  ''X'' is the torus, ''Y'' is {{nowrap|1=''R''<sup>3</sup>}}, ''f'' is some continuous function from the torus to ''R''<sup>3</sup> 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 of ''h''<sub>''t''</sub>(X) as a function of the parameter ''t'', where ''t'' varies with time from 0 to 1 over each cycle of the animation loop.  It pauses, then shows the image as ''t'' varies back from 1 to 0, pauses, and repeats this cycle.

===Properties===

Continuous functions ''f'' and ''g'' are said to be homotopic if and only if there is a homotopy ''H'' taking ''f'' to ''g'' as described above. Being homotopic is an [[equivalence relation]] on the set of all continuous functions from ''X'' to ''Y''. 
This homotopy relation is compatible with [[function composition]] in the following sense: if {{nowrap|1=''f''<sub>1</sub>, ''g''<sub>1</sub> : ''X'' → ''Y''}} are homotopic, and {{nowrap|1=''f''<sub>2</sub>, ''g''<sub>2</sub> : ''Y'' → ''Z''}} are homotopic, then their compositions {{nowrap|1=''f''<sub>2</sub>&thinsp;∘&thinsp;''f''<sub>1</sub>}} and {{nowrap|1=''g''<sub>2</sub>&thinsp;∘&thinsp;''g''<sub>1</sub> : ''X'' → ''Z''}} are also homotopic.

== Examples ==
* If <math>f, g: \R \to \R^2</math> are given by <math>f(x) := \left(x, x^3\right)</math> and <math>g(x) = \left(x, e^x\right)</math>, then the map <math>H: \mathbb{R} \times [0, 1] \to \mathbb{R}^2</math> given by <math>H(x, t) = \left(x, (1 - t)x^3 + te^x\right)</math> is a homotopy between them.
* More generally, if <math>C \subseteq \mathbb{R}^n</math> is a [[convex set|convex]] subset of [[Euclidean space]] and <math>f, g: [0, 1] \to C</math> are [[path (topology)|paths]] with the same endpoints, then there is a '''linear homotopy'''<ref>{{Cite book|title=Algebraic topology|last=Allen.|first=Hatcher|date=2002|publisher=Cambridge University Press|isbn=9780521795401|location=Cambridge|pages=185|oclc=45420394}}</ref> (or '''straight-line homotopy''') given by 
*: <math>\begin{align}
  H: [0, 1] \times [0, 1] &\longrightarrow C \\
                   (s, t) &\longmapsto (1 - t)f(s) + tg(s).
\end{align}</math>
* Let <math>\operatorname{id}_{B^n}:B^n\to B^n</math> be the [[identity function]] on the unit ''n''-[[ball (mathematics)|disk]]; i.e. the set <math>B^n := \left\{x\in\mathbb{R}^n: \|x\| \leq 1\right\}</math>. Let <math>c_{\vec{0}}: B^n \to B^n</math> be the [[constant function]] <math>c_\vec{0}(x) := \vec{0}</math> which sends every point to the [[Origin (mathematics)|origin]]. Then the following is a homotopy between them:
*: <math>\begin{align}
  H: B^n \times [0, 1] &\longrightarrow B^n \\
                (x, t) &\longmapsto (1 - t)x.
\end{align}</math>

==Homotopy equivalence==
Given two topological spaces ''X'' and ''Y'', a '''homotopy equivalence''' between ''X'' and ''Y'' is a pair of continuous [[map (mathematics)|map]]s {{nowrap|1=''f'' : ''X'' → ''Y''}} and {{nowrap|1=''g'' : ''Y'' → ''X''}}, such that {{nowrap|1=''g''&thinsp;∘&thinsp;''f''}} is homotopic to the [[identity function|identity map]] id<sub>''X''</sub> and {{nowrap|1=''f''&thinsp;∘&thinsp;''g''}} is homotopic to id<sub>''Y''</sub>. If such a pair exists, then ''X'' and ''Y'' are said to be '''homotopy equivalent''', or of the same '''homotopy type'''. This relation of homotopy equivalence is often denoted <math>\simeq</math>.<ref>{{cite book
 | last = Singh | first = Tej Bahadur
 | doi = 10.1007/978-981-13-6954-4
 | isbn = 9789811369544
 | page = 317
 | publisher = Springer Singapore
 | title = Introduction to Topology
 | year = 2019}} This is the misnamed [[unicode]] symbol {{unichar|2243}}.</ref> Intuitively, two spaces ''X'' and ''Y'' 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 called [[contractible]].

=== Homotopy equivalence vs. homeomorphism ===
A [[homeomorphism]] is a special case of a homotopy equivalence, in which {{nowrap|1=''g''&thinsp;∘&thinsp;''f''}} is equal to the identity map id<sub>''X''</sub> (not only homotopic to it), and {{nowrap|1=''f''&thinsp;∘&thinsp;''g''}} is equal to id<sub>''Y''</sub>.<ref>Archived at [https://ghostarchive.org/varchive/youtube/20211211/XxFGokyYo6g Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20200829013025/https://www.youtube.com/watch?v=XxFGokyYo6g&gl=US&hl=en Wayback Machine]{{cbignore}}: {{Cite web|last=Albin|first=Pierre|date=2019|title=History of algebraic topology|website=[[YouTube]] |url=https://www.youtube.com/watch?v=XxFGokyYo6g}}{{cbignore}}</ref>{{Rp|0:53:00}} 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 no [[bijection]] between them (since one is an infinite set, while the other is finite).
* The [[Möbius strip]] and 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 <math>\mathbb{R}^n</math> with a point, denoted <math>\mathbb{R}^n \simeq \{ 0\}</math>. The part that needs to be checked is the existence of a homotopy <math>H: I \times \mathbb{R}^n \to \mathbb{R}^n</math> between <math>\operatorname{id}_{\mathbb{R}^n}</math> and <math>p_0</math>, the projection of <math>\mathbb{R}^n</math> onto the origin. This can be described as <math>H(t,\cdot) = t\cdot p_0 + (1-t)\cdot\operatorname{id}_{\mathbb{R}^n}</math>.
* There is a homotopy equivalence between <math>S^1</math> (the [[n-sphere|1-sphere]]) and <math>\mathbb{R}^2-\{0\}</math>.
** More generally, <math>\mathbb{R}^n-\{ 0\} \simeq S^{n-1}</math>.
* Any [[fiber bundle]] <math>\pi: E \to B</math> with fibers <math>F_b</math> homotopy equivalent to a point has homotopy equivalent total and base spaces. This generalizes the previous two examples since <math>\pi:\mathbb{R}^n - \{0\} \to S^{n-1}</math> is a fiber bundle with fiber <math>\mathbb{R}_{>0}</math>.
* Every [[vector bundle]] is a fiber bundle with a fiber homotopy equivalent to a point.
* <math>\mathbb{R}^n - \mathbb{R}^k \simeq S^{n-k-1}</math> for any <math>0 \le k < n</math>, by writing <math>\mathbb{R}^n - \mathbb{R}^k</math> as the total space of the fiber bundle <math>\mathbb{R}^k \times (\mathbb{R}^{n-k}-\{0\})\to (\mathbb{R}^{n-k}-\{0\})</math>, then applying the homotopy equivalences above.
* If a subcomplex <math>A</math> of a [[CW complex]] <math>X</math> is contractible, then the [[quotient space (topology)|quotient space]] <math>X/A</math> is homotopy equivalent to <math>X</math>.<ref>{{Cite book|title=Algebraic topology|last=Allen.|first=Hatcher|date=2002|publisher=Cambridge University Press|isbn=9780521795401|location=Cambridge|pages=11|oclc=45420394}}</ref>
* A [[deformation retraction]] is a homotopy equivalence.

===Null-homotopy===
A function <math>f</math> is said to be '''null-homotopic''' {{anchor|null homotopic}} if it is homotopic to a constant function.  (The homotopy from <math>f</math> to a constant function is then sometimes called a '''null-homotopy'''.)  For example, a map <math>f</math> from the [[unit circle]] <math>S^1</math> to any space <math>X</math> is null-homotopic precisely when it can be continuously extended to a map from the [[unit disk]] <math>D^2</math> to <math>X</math> that agrees with <math>f</math> on the boundary.

It follows from these definitions that a space <math>X</math> is contractible if and only if the identity map from <math>X</math> to itself&mdash;which is always a homotopy equivalence&mdash;is null-homotopic.

==Invariance==
Homotopy equivalence is important because in [[algebraic topology]] many concepts are '''homotopy invariant''', that is, they respect the relation of homotopy equivalence.  For example, if ''X'' and ''Y'' are homotopy equivalent spaces, then:
* ''X'' is [[connected space|path-connected]] if and only if ''Y'' is.
* ''X'' is [[simply connected]] if and only if ''Y'' is.
* The (singular) [[homology (mathematics)|homology]] and [[cohomology group]]s of ''X'' and ''Y'' are [[group isomorphism|isomorphic]].
* If ''X'' and ''Y'' are path-connected, then the [[fundamental group]]s of ''X'' and ''Y'' are isomorphic, and so are the higher [[homotopy group]]s. (Without the path-connectedness assumption, one has π<sub>1</sub>(''X'',&thinsp;''x''<sub>0</sub>) isomorphic to π<sub>1</sub>(''Y'', ''f''(''x''<sub>0</sub>)) where {{nowrap|1=''f'' : ''X'' → ''Y''}} is a homotopy equivalence and {{nowrap|1=''x''<sub>0</sub> &isin; ''X''.)}}

An example of an algebraic invariant of topological spaces which is not homotopy-invariant is [[compactly supported homology]] (which is, roughly speaking, the homology of the [[compactification (mathematics)|compactification]], and compactification is not homotopy-invariant).

== Variants ==

===Relative homotopy===

In order to define the [[fundamental group]], one needs the notion of '''homotopy relative to a subspace'''. These are homotopies which keep the elements of the subspace fixed. Formally: if ''f'' and ''g'' are continuous maps from ''X'' to ''Y'' and ''K'' is a [[subset]] of ''X'', then we say that ''f'' and ''g'' are homotopic relative to ''K'' if there exists a homotopy {{nowrap|1=''H'' : ''X'' &times; [0,&thinsp;1] → ''Y''}} between ''f'' and ''g'' such that {{nowrap|1=''H''(''k'',&thinsp;''t'') = ''f''(''k'') = ''g''(''k'')}} for all {{nowrap|1=''k'' ∈ ''K''}} and {{nowrap|1=''t'' ∈ [0,&thinsp;1].}} Also, if ''g'' is a [[retraction (topology)|retraction]] from ''X'' to ''K'' and ''f'' is the identity map, this is known as a strong [[deformation retract]] of ''X'' to ''K''.
When ''K'' is a point, the term '''pointed homotopy''' is used.

=== Isotopy ===
{{multiple image
| total_width = 320
| image1 = Blue Unknot.png
| image2 = Blue Trefoil Knot.png
| footer = The [[unknot]] is not equivalent to the [[trefoil knot]] since 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 functions ''f'' and ''g'' from the topological space ''X'' to the topological space ''Y'' are [[embedding]]s, one can ask whether they can be connected 'through embeddings'. This gives rise to the concept of '''isotopy''', which is a homotopy, ''H'', in the notation used before, such that for each fixed ''t'', ''H''(''x'',&thinsp;''t'') gives an embedding.<ref>{{MathWorld|Isotopy|Isotopy}}</ref>

A related, but different, concept is that of [[ambient isotopy]].

Requiring that two embeddings be isotopic is a stronger requirement than that they be homotopic. For example, the map from the interval [−1,&thinsp;1] into the real numbers defined by ''f''(''x'') = &minus;''x'' is ''not'' isotopic to the identity ''g''(''x'') = ''x''.  Any homotopy from ''f'' 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 and ''g'' has not, which is impossible under an isotopy. However, the maps are homotopic; one homotopy from ''f'' to the identity is ''H'':&nbsp;[−1,&thinsp;1]&nbsp;&times;&nbsp;[0,&thinsp;1]&nbsp;→&nbsp;[−1,&thinsp;1] given by ''H''(''x'',&thinsp;''y'')&nbsp;=&nbsp;2''yx''&nbsp;−&nbsp;''x''.

Two homeomorphisms (which are special cases of embeddings) of the unit ball which agree on the boundary can be shown to be isotopic using [[Alexander's trick]]. For this reason, the map of the [[unit disc]] in <math>\mathbb{R}^2</math> defined by ''f''(''x'',&thinsp;''y'') = (&minus;''x'',&nbsp;&minus;''y'') is isotopic to a 180-degree [[rotation]] around the origin, and so the identity map and ''f'' are isotopic because they can be connected by rotations.

In [[geometric topology]]&mdash;for example in [[knot theory]]&mdash;the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same?  We take two knots, ''K''<sub>1</sub> and ''K''<sub>2</sub>, in three-[[dimension]]al space. A knot is an [[embedding]] 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 can ''deform'' one embedding to another through a path of embeddings: a continuous function starting at ''t''&nbsp;=&nbsp;0 giving the ''K''<sub>1</sub> embedding, ending at ''t''&nbsp;=&thinsp;1 giving the ''K''<sub>2</sub> embedding, with all intermediate values corresponding to embeddings. This corresponds to the definition of isotopy. An [[ambient isotopy]], studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. Knots ''K''<sub>1</sub> and ''K''<sub>2</sub> are considered equivalent when there is an ambient isotopy which moves ''K''<sub>1</sub> to ''K''<sub>2</sub>. 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 a '''smooth isotopy'''.

=== Timelike homotopy ===
On a [[Lorentzian manifold]], certain curves are distinguished as [[timelike]] (representing something that only goes forwards, not backwards, in time, in every local frame). A [[timelike homotopy]] between two [[timelike curve]]s is a homotopy such that the curve remains timelike during the continuous transformation from one curve to another. No [[closed 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 be [[multiply connected]] by timelike curves. A manifold such as the [[3-sphere]] can be [[simply connected]] (by any type of curve), and yet be [[timelike multiply connected]].<ref>{{Cite journal|last=Monroe|first=Hunter|date=2008-11-01|title=Are Causality Violations Undesirable?|journal=Foundations of Physics|language=en|volume=38|issue=11|pages=1065–1069|doi=10.1007/s10701-008-9254-9|issn=0015-9018|arxiv=gr-qc/0609054|bibcode=2008FoPh...38.1065M|s2cid=119707350 }}</ref>

== Properties ==

=== Lifting and extension properties ===
{{main|Homotopy lifting property}}
If we have a homotopy {{nowrap|1=''H'' : ''X'' &times; [0,1] &rarr; ''Y''}} and a cover {{nowrap|1=''p'' : <span style="text-decoration: overline;">''Y''</span> &rarr; ''Y''}} and we are given a map {{nowrap|1=<span style="text-decoration: overline;">''h''</span><sub>0</sub> : ''X'' &rarr; <span style="text-decoration: overline;">''Y''</span>}} such that {{nowrap|1=''H''<sub>0</sub> = ''p'' ○ <span style="text-decoration: overline;">''h''</span><sub>0</sub>}}   (<span style="text-decoration: overline;">''h''</span><sub>0</sub> is called a [[Lift (mathematics)|lift]] of ''h''<sub>0</sub>), then we can lift all ''H'' to a map {{nowrap|1=<span style="text-decoration: overline;">''H''</span> : ''X'' &times; [0,&thinsp;1] &rarr; <span style="text-decoration: overline;">''Y''</span>}} such that {{nowrap|1=''p'' ○ <span style="text-decoration: overline;">''H''</span> = ''H''.}} The homotopy lifting property is used to characterize [[fibration]]s.

Another useful property involving homotopy is the [[homotopy 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 with [[cofibration]]s.

===Groups===

{{main|Homotopy group}}
Since the relation of two functions <math>f, g\colon X\to Y</math> being homotopic relative to a subspace is an equivalence relation, we can look at the [[equivalence class]]es of maps between a fixed ''X'' and ''Y''. If we fix <math>X = [0,1]^n</math>, the unit interval [0,&thinsp;1] [[cartesian product|crossed]] with itself ''n'' times, and we take its [[Boundary (topology)|boundary]] <math>\partial([0,1]^n)</math> as a subspace, then the equivalence classes form a group, denoted <math>\pi_n(Y,y_0)</math>, where <math>y_0</math> is in the image of the subspace <math>\partial([0,1]^n)</math>.

We can define the action of one equivalence class on another, and so we get a group. These groups are called the [[homotopy group]]s. In the case <math>n = 1</math>, it is also called the [[fundamental group]].

===Homotopy category===
{{Main articles|Homotopy category}}
The idea of homotopy can be turned into a formal category of [[category theory]]. The '''[[homotopy category]]''' is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces ''X'' and ''Y'' are isomorphic in this category if and only if they are homotopy-equivalent.  Then a [[functor]] 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 a ''functorial'' homotopy invariant: this means that if ''f'' and ''g'' from ''X'' to ''Y'' are homotopic, then the [[group homomorphism]]s induced by ''f'' and ''g'' on the level of [[homology group]]s are the same: H<sub>''n''</sub>(''f'') = H<sub>''n''</sub>(''g'') : H<sub>''n''</sub>(''X'') → H<sub>''n''</sub>(''Y'') for all ''n''.  Likewise, if ''X'' and ''Y'' are in addition [[connectedness|path connected]], and the homotopy between ''f'' and ''g'' is pointed, then the group homomorphisms induced by ''f'' and ''g'' on the level of [[homotopy group]]s are also the same: π<sub>''n''</sub>(''f'') = π<sub>''n''</sub>(''g'') : π<sub>''n''</sub>(''X'') → π<sub>''n''</sub>(''Y'').

==Applications==
Based on the concept of the homotopy, [[Numerical methods|computation methods]] for [[algebraic equations|algebraic]] and [[differential equations]] have been developed. The methods for algebraic equations include the [[homotopy continuation]] method<ref>{{Cite book |last=Allgower |first=E. L. |url=https://www.worldcat.org/oclc/52377653 |title=Introduction to numerical continuation methods |date=2003 |publisher=SIAM |others=Kurt Georg |isbn=0-89871-544-X |location=Philadelphia |oclc=52377653}}</ref> and the continuation method (see [[numerical continuation]]). The methods for differential equations include the [[homotopy analysis method]].

Homotopy theory can be used as a foundation for [[Homology (mathematics)|homology theory]]: one can [[Representable functor|represent]] a cohomology functor on a space ''X'' by mappings of ''X'' into an appropriate fixed space, up to homotopy equivalence. For example, for any abelian group ''G'', and any based CW-complex ''X'', the set <math>[X,K(G,n)]</math> of based homotopy classes of based maps from ''X'' to the [[Eilenberg–MacLane space]] <math>K(G,n)</math> is in natural bijection with the ''n''-th [[singular cohomology]] group <math>H^n(X,G)</math> of the space ''X''. One says that the [[Spectrum (topology)|omega-spectrum]] of Eilenberg-MacLane spaces are [[representing space]]s for singular cohomology with coefficients in ''G''. Using this fact, homotopy classes between a CW complex and a multiply connected space can be calculated using cohomology as described by the [[Hopf–Whitney theorem]]. 

==See also==
*[[Fiber-homotopy equivalence]] (relative version of a homotopy equivalence)
*[[Homeotopy]]
*[[Homotopy type theory]]
*[[Mapping class group]]
*[[Poincaré conjecture]]
*[[Regular homotopy]]

== References ==
{{reflist|30em}}

== Sources ==
*{{cite book |last=Armstrong |first=M.A. |title=Basic Topology |year=1979 |publisher=Springer |isbn=978-0-387-90839-7}}
*{{Springer |title=Homotopy |id=p/h047920}}
*{{Springer |title=Isotopy (in topology) |id=p/i052940}}
*{{cite book |last=Spanier |first=Edwin |title=Algebraic Topology |date=December 1994 |publisher=Springer |isbn=978-0-387-94426-5}}

{{Topology}}

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[[Category:Homotopy theory|*]]
[[Category:Maps of manifolds]]
[[Category:Theory of continuous functions]]