From en.wiktionary.org:
** English
[wikipedia] right
*** Etymology
From [en] + [grc]; earliest known use in print in 1922, [Oswald Veblen],
_Analysis Situs_.[1]
*** Pronunciation
[en] <!--* [en]-->
*** Noun
[~]
1. [en] A continuous deformation of one continuous function or map to another.
2. * {{ quote-book | en | year=1998 | author=Paul Sellick | chapter=Space Exponents for Loop Spaces of Spheres | editor= [William Gerard Dwyer] | title=Stable and Unstable HOMOTOPY | publisher=w:American Mathematical Society | pageurl=https://books.google.com.au/books?id=nM1ugknmGtwC&pg=PA279&dq=%22homotopy%22%7C%22homotopies%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiNt43ShIbtAhXDYcAKHVcgCgYQ6AEwDXoECGMQAg#v=onepage&q=%22homotopy%22%7C%22homotopies%22&f=false | page=279
|passage=An integer _M_ is called an exponent for the torsion of an abelian
group _G_ if _M_ * (torsion of _G_) = 0. We say that _M_ is a HOMOTOPY
exponent for a space _X_ if _M_ is an exponent for π<sub>_k_</sub> (_X_) for
all _k_.}}
1. * {{ quote-book | en | year=2001 | author=F. R. Cohen; S. Gitler | chapter=Loop-spaces of configuration spaces, braid-like groups, and knots | editors=Jaume Aguadé; Carles Broto; Carles Casacuberta | title=Cohomological Methods in HOMOTOPY Theory | publisher=Springer (Birkhäuser) | pageurl=https://books.google.com.au/books?id=uiLyBwAAQBAJ&printsec=frontcover&dq=%22homotopy%22%7C%22homotopies%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiNt43ShIbtAhXDYcAKHVcgCgYQ6AEwDHoECF0QAg#v=onepage&q=%22homotopy%22%7C%22homotopies%22&f=false | page=63
|passage=A graded Lie algebra arises from these maps via the Samelson product
in HOMOTOPY, the so-called [Homotopy Lie algebra] which is discussed below.}}
1. * 2010 , [Vladimir Turaev] , _HOMOTOPY Quantum Field Theory_ , [European Mathematical Society] , page xi (see https://books.google.com.au/books?id=v4hty8W-guAC&pg=PR11&dq=%22homotopy%22%7C%22homotopies%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiNt43ShIbtAhXDYcAKHVcgCgYQ6AEwDnoECGIQAg#v=onepage&q=%22homotopy%22%7C%22homotopies%22&f=false) ,
2. *: In this monograph we apply the idea of a TQFT to maps from manifolds to topological spaces. This leads us to a notion of a ( _d_ + 1)-dimensional HOMOTOPY quantum field theory (HQFT) which may be described as a TQFT for closed oriented _d_ -dimensional manifolds and compact oriented ( _d_ + 1)-dimensional cobordisms endowed with maps to a given space _X_ .
3. [en] The relationship between two continuous functions where homotopy from one to the other is evident.
4. [en] [en] .
5. [en] A theory associating a system of group s with each topological space .
6. [en] A system of groups associated with a topological space.
**** Usage notes
- Formally, there are two alternative formulations: [2]
- Given topological space s <math> X, Y </math> and continuous maps <math> f,g: X\rightarrow Y </math>
- # A continuous map <math> H: [ 0,1 ] \times X\rightarrow Y </math> such that <math> H(x,0)=f(x) </math> and <math> H(x,1)=g(x) </math> <math> \forall x\in X </math> .
- # A family of continuous maps <math> h_t: X\rightarrow Y, t\in [ 0,1 ] </math> such that <math> h_0=f, </math> <math> h_1=g </math> and the map <math> (x,t)\mapsto h_t </math> is continuous from <math> X </math> to <math> Y </math> . (Note that it is not sufficient to require that each map <math> h_t(x) </math> be continuous.)
- Replacing the unit interval <math> [ 0,1 ] </math> with the affine line A¹ leads to [A¹ homotopy theory] .
- The adjective [en] is used specifically in the sense, with respect to two functions, of "having the relationship of being in homotopy".
- Being homotopic is an equivalence relation on the class of all continuous functions between given topological spaces. An equivalence class of such a relation is called a [en] .
**** Hyponyms
- [continuous deformation] [en] , [en]
**** Derived terms
{{col|en |cohomotopy |homotopic |homotopy category |homotopy class |homotopy
equivalence |homotopy equivalent |homotopy fiber,homotopy fibre |homotopy
group |homotopy invariant |homotopy relative to a subspace |homotopy theory
|homotopy type |linear homotopy |regular homotopy |straight-line homotopy }}
**** Related terms
- [en]
**** Translations
[(topology) continuous deformation of one continuous function to another]
- Danish: [da]
- French: [fr]
- Greek: [el]
- Hungarian: [hu]
- Icelandic: [is]
- Japanese: [ja]
- Korean: [ko] , [ko]
- Persian: [fa]
- Portuguese: [pt]
- Spanish: [es]
- Ukrainian: [uk]
- Vietnamese: [vi]
[trans-bottom]
[(topology) theory associating a system of groups with each topological space]
- French: [fr]
- Hungarian: [hu]
- Spanish: [es]
[trans-bottom]
[(topology) system of groups associated with a topological space]
- French: [fr]
- Hungarian: [hu]
- Spanish: [es]
[trans-bottom]
*** See also
- [en]
- [en]
*** References
References:
[1]. Earliest Known Uses of Some of the Words of Mathematics (H) (see http://jeff560.tripod.com/h.html)
[2]. [Homotopy#Formal definition]
*** Further reading
- [Homotopy group]
- [Fundamental group]
- [Homotopy theory]
- [A¹ homotopy theory]
- [Homeotopy]
- [Fiber-homotopy equivalence]
- [Poincaré conjecture]
- Homotopy (see https://encyclopediaofmath.org/wiki/Homotopy) on [Encyclopedia of Mathematics]