Homotopy

English

An animation of one possible homotopy between the dashed lines

Etymology

From Template:Der + Template:M; earliest known use in print in 1922, Template:W, Analysis Situs.

Noun

1. Template:Lb A continuous deformation of one continuous function or map to another.

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   • Template:Quote-book
   • Template:Quote-book
   • 2010, Template:W, Homotopy Quantum Field Theory, Template:W, page xi,

     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.

2. Template:Lb The relationship between two continuous functions where homotopy from one to the other is evident.
3. Template:Lb Template:Ellipsis of Template:Gloss.
4. Template:Lb A theory associating a system of groups with each topological space.
5. Template:Lb A system of groups associated with a topological space.

Usage notes

• Formally, there are two alternative formulations:
  • Given topological spaces $X,Y$ and continuous maps $f,g:X\to Y$
    1. A continuous map $H:[0,1]\times X\to Y$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$ $\mathrm{\forall }x\in X$.
    2. A family of continuous maps $h_t:X\to Y,t\in [0,1]$ such that $h_0=f,$ $h_1=g$ and the map $(x,t)\mapsto h_t$ is continuous from $X$ to $Y$. (Note that it is not sufficient to require that each map $h_t(x)$ be continuous.)
  • Replacing the unit interval $[0,1]$ with the affine line A¹ leads to Template:W.
• The adjective Template:M 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 Template:M.

Hyponyms

• Template:Sense Template:L, Template:L

Related terms

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Translations

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See also

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Further reading

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• Homotopy on Template:W