Homotopy: Difference between revisions
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== Definition == | == Definition == | ||
If <math>f,g | If <math>f,g\colonX\to Y</math> are continuous functions between topological spaces <math>X</math> and <math>Y</math>, a homotopy <math>H</math> from <math>f</math> to <math>g</math> is a continuous map | ||
<math display="block">H | <math display="block">H\colon[0,1] \times X \to Y</math> | ||
such that for all <math>x \in X</math>: | such that for all <math>x \in X</math>: | ||
| Line 19: | Line 19: | ||
Two topological spaces <math>X</math> and <math>Y</math> are '''homotopy equivalent''' if there exist continuous maps | Two topological spaces <math>X</math> and <math>Y</math> are '''homotopy equivalent''' if there exist continuous maps | ||
<math display="block">f | <math display="block">f\colonX \to Y \quad \text{and} \quad g\colonY \to X</math> | ||
such that | such that | ||
Revision as of 15:15, 11 April 2026
A homotopy is a continuous deformation between two continuous functions from one topological space to another. Specifically, a homotopy between two functions is a continuous map that, for each point in the domain, provides a path from its image under the first function to its image under the second. If such a function exists between two functions, they are said to be homotopic.
Intuitively, a homotopy is the continuous transformation of paths that varies over time. It shows how one function can be smoothly bent, stretched, or deformed into the other without tearing or folding.
Definition
If
are continuous functions between topological spaces 
and 
, a homotopy 
from 
to 
is a continuous map
such that for all 
:
,
.
Two continuous functions and are called homotopic if there exists a homotopy between them, denoted 
[Note 1].
Homotopy equivalence
Two topological spaces and are homotopy equivalent if there exist continuous maps
such that
,
.
In this case, and have the same "essential shape" from the perspective of homotopy. This concept is distinct from homeomorphism, which is a stricter condition requiring the maps to be inverses of each other. For instance, a solid sphere is homotopy equivalent to a single point, but they are not homeomorphic.
Note
- ↑ Not to be confused with "
" for homeomorphism.
See also
Terminology
| en | fr | de | zh | ja | |
|---|---|---|---|---|---|
| homotopy | homotopie | Homotopie | 同伦 | 同倫 | ホモトピー |
| homotopic | homotope | homotop | 同伦的 | 同倫的 | ホモトピック |
| homotopy equivalence | équivalence d'homotopie | Homotopieäquivalenz | 同伦等价 | 同倫等價 | ホモトピー同値 |
| homotopy equivalent | homotopiquement équivalent | homotopieäquivalent | 同伦等价的 | 同倫等價的 | ホモトピー同値な |