Homotopy - Revision history

InfernalAtom683 at 07:15, 11 April 2026

2026-04-11T07:15:46Z

← Older revision		Revision as of 15:15, 11 April 2026
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	== Definition ==		== Definition ==
	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		If <math>f,g\colon X\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\colon[0,1] \times X \to Y</math>		<math display="block">H\colon[0,1] \times X \to Y</math>
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	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\~~colonX~~ \to Y \quad \text{and} \quad g\~~colonY~~ \to X</math>		<math display="block">f\colon X \to Y \quad \text{and} \quad g\colon Y \to X</math>

	such that		such that

InfernalAtom683 at 07:15, 11 April 2026

2026-04-11T07:15:12Z

← Older revision		Revision as of 15:15, 11 April 2026
Line 4:		Line 4:

	== Definition ==		== Definition ==
	If <math>f,g:X\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		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:[0,1] \times X \to Y</math>		<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>:
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	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:X \to Y \quad \text{and} \quad g:Y \to X</math>		<math display="block">f\colonX \to Y \quad \text{and} \quad g\colonY \to X</math>

	such that		such that

InfernalAtom683 at 07:48, 9 April 2026

2026-04-09T07:48:37Z

← Older revision		Revision as of 15:48, 9 April 2026
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	* <math>H_1(x) = g(x)</math>.		* <math>H_1(x) = g(x)</math>.

	Two continuous functions <math>f</math> and <math>g</math> are called homotopic if there exists a homotopy between them, denoted <math>f\simeq g</math>.		Two continuous functions <math>f</math> and <math>g</math> are called homotopic if there exists a homotopy between them, denoted <math>f\simeq g</math><ref group="Note">Not to be confused with "<math>\cong</math>" for homeomorphism.</ref>.

	== Homotopy equivalence ==		== Homotopy equivalence ==
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	In this case, <math>X</math> and <math>Y</math> 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.		In this case, <math>X</math> and <math>Y</math> 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 ==
			<references group="Note" />

	== See also ==		== See also ==

InfernalAtom683 at 07:30, 9 April 2026

2026-04-09T07:30:19Z

← Older revision		Revision as of 15:30, 9 April 2026
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	If <math>f,g:X\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		If <math>f,g:X\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:~~X \times~~ [0,1] \to Y</math>		<math display="block">H:[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>:

	* <math>H(x,0) = f(x)</math>,		* <math>H_0(x) = f(x)</math>,

	* <math>H(x,1) = g(x)</math>.		* <math>H_1(x) = g(x)</math>.

	Two continuous functions <math>f</math> and <math>g</math> are called homotopic if there exists a homotopy between them, denoted <math>f\simeq g</math>.		Two continuous functions <math>f</math> and <math>g</math> are called homotopic if there exists a homotopy between them, denoted <math>f\simeq g</math>.

InfernalAtom683 at 03:28, 6 April 2026

2026-04-06T03:28:38Z

← Older revision		Revision as of 11:28, 6 April 2026
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	In this case, <math>X</math> and <math>Y</math> 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.		In this case, <math>X</math> and <math>Y</math> 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.

			== See also ==

			* [[Homeomorphism]]
			* [[Continuous function]]
			* [[Homeotopy]]
			* [[Homotopy type theory]]

	== Terminology ==		== Terminology ==

InfernalAtom683 at 11:30, 4 April 2026

2026-04-04T11:30:27Z

← Older revision		Revision as of 19:30, 4 April 2026
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	In this case, <math>X</math> and <math>Y</math> 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.		In this case, <math>X</math> and <math>Y</math> 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.

			== Terminology ==
			{{Terminology_table\|
			{{Terminology_table/row \| homotopy \| homotopie \| Homotopie \| 同伦 \| 同倫 \| ホモトピー }}
			{{Terminology_table/row \| homotopic \| homotope \| homotop \| 同伦的 \| 同倫的 \| ホモトピック }}
			{{Terminology_table/row \| homotopy equivalence \| équivalence d'homotopie \| Homotopieäquivalenz \| 同伦等价 \| 同倫等價 \| ホモトピー同値 }}
			{{Terminology_table/row \| homotopy equivalent \| homotopiquement équivalent \| homotopieäquivalent \| 同伦等价的 \| 同倫等價的 \| ホモトピー同値な }}
			}}

InfernalAtom683: Created page with "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..."

2026-03-22T13:21:31Z

Created page with "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..."

New page

A '''homotopy''' is a continuous deformation between two [[Continuous function|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 <math>f,g:X\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:X \times [0,1] \to Y</math>

such that for all <math>x \in X</math>:

* <math>H(x,0) = f(x)</math>,

* <math>H(x,1) = g(x)</math>.

Two continuous functions <math>f</math> and <math>g</math> are called homotopic if there exists a homotopy between them, denoted <math>f\simeq g</math>.

== Homotopy equivalence ==
Two topological spaces <math>X</math> and <math>Y</math> are '''homotopy equivalent''' if there exist continuous maps

<math display="block">f:X \to Y \quad \text{and} \quad g:Y \to X</math>

such that

* <math>g \circ f \simeq \operatorname{id}_X</math>,

* <math>f \circ g \simeq \operatorname{id}_Y</math>.

In this case, <math>X</math> and <math>Y</math> 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.