Editing Homotopy (section)

==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.