Homotopy: Difference between revisions

Latest revision as of 03:42, 13 February 2025

An introduction to the concept of homotopy in topology

Homotopy[edit]

A visual representation of a homotopy between two functions.

In the field of topology, homotopy is a fundamental concept that describes when two continuous functions from one topological space to another can be continuously deformed into each other. This concept is central to the study of algebraic topology, where it is used to classify topological spaces up to homotopy equivalence.

Definition[edit]

A homotopy between two continuous functions \( f, g : X \to Y \) is a continuous map \( H : X \times [0, 1] \to Y \) such that:

\( H(x, 0) = f(x) \) for all \( x \in X \)
\( H(x, 1) = g(x) \) for all \( x \in X \)

The parameter \( t \in [0, 1] \) can be thought of as "time," and the map \( H \) describes a continuous deformation of the function \( f \) into the function \( g \).

Homotopy Equivalence[edit]

Two topological spaces \( X \) and \( Y \) are said to be homotopy equivalent if there exist continuous maps \( f : X \to Y \) and \( g : Y \to X \) such that \( g \circ f \) is homotopic to the identity map on \( X \) and \( f \circ g \) is homotopic to the identity map on \( Y \). In this case, \( X \) and \( Y \) are said to have the same "homotopy type."

Applications[edit]

Homotopy theory is used in many areas of mathematics, including the study of homotopy groups, which are algebraic invariants that classify spaces up to homotopy equivalence. It also plays a crucial role in the development of homological algebra and the study of fiber bundles.

Related Concepts[edit]

Related pages[edit]

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-'''Homotopy''' is a fundamental concept in the field of [[topology]], a major area of [[mathematics]] that studies the properties of space that are preserved under continuous transformations. Homotopy provides a way to classify [[topological spaces]] through the concept of continuous deformation between functions. This article aims to elucidate the concept of homotopy, its significance in mathematics, and its applications in various fields.
+{{Short description|An introduction to the concept of homotopy in topology}}
-==Definition==
+== Homotopy ==
-A homotopy between two continuous functions \(f, g: X \rightarrow Y\), where \(X\) and \(Y\) are topological spaces, is a continuous function \(H: X \times [0,1] \rightarrow Y\) such that \(H(x,0) = f(x)\) and \(H(x,1) = g(x)\) for all \(x \in X\). The parameter \(t\) in the interval \([0,1]\) allows for a continuous transformation from \(f\) to \(g\), where each intermediate function \(H(x,t)\) is also continuous. Two functions \(f\) and \(g\) are said to be homotopic if there exists a homotopy between them.
+[[File:HomotopySmall.gif|thumb|right|A visual representation of a homotopy between two functions.]]
+In the field of [[topology]], '''homotopy''' is a fundamental concept that describes when two continuous functions from one topological space to another can be continuously deformed into each other. This concept is central to the study of [[algebraic topology]], where it is used to classify topological spaces up to homotopy equivalence.
-==Importance==
+== Definition ==
-Homotopy is a central concept in topology because it provides a way to classify spaces and maps in a flexible manner. Unlike homeomorphisms, which require a strict equivalence between spaces, homotopy allows for a more relaxed similarity that can capture the essence of a space's shape or structure without being confined to exact details. This makes homotopy particularly useful in the study of [[topological invariants]], properties of spaces that remain unchanged under continuous deformations.
+A homotopy between two continuous functions \( f, g : X \to Y \) is a continuous map \( H : X \times [0, 1] \to Y \) such that:
+* \( H(x, 0) = f(x) \) for all \( x \in X \)
+* \( H(x, 1) = g(x) \) for all \( x \in X \)
-==Types of Homotopy==
+The parameter \( t \in [0, 1] \) can be thought of as "time," and the map \( H \) describes a continuous deformation of the function \( f \) into the function \( g \).
-There are several specialized forms of homotopy, including:
-- *Path homotopy*, which deals with the continuous deformation of paths in a space.
-- *Homotopy equivalence*, a stronger condition where two spaces are considered equivalent if there exist continuous maps between them that are inverses up to homotopy.
-- *Relative homotopy*, which considers homotopies between maps that keep a certain subset of the domain fixed.
-==Applications==
+== Homotopy Equivalence ==
-Homotopy theory has applications across various fields of mathematics and science. In [[algebraic topology]], it is used to define and study complex invariants such as [[fundamental groups]], [[homology]], and [[cohomology]]. These invariants play crucial roles in understanding the global properties of spaces. Homotopy also finds applications in [[differential geometry]], [[algebraic geometry]], and even in theoretical physics, particularly in the study of [[quantum field theory]] and [[string theory]], where the concept of homotopy groups provides insights into the possible configurations and symmetries of physical systems.
+Two topological spaces \( X \) and \( Y \) are said to be homotopy equivalent if there exist continuous maps \( f : X \to Y \) and \( g : Y \to X \) such that \( g \circ f \) is homotopic to the identity map on \( X \) and \( f \circ g \) is homotopic to the identity map on \( Y \). In this case, \( X \) and \( Y \) are said to have the same "homotopy type."
-==See Also==
+== Applications ==
-* [[Topology]]
+Homotopy theory is used in many areas of mathematics, including the study of [[homotopy groups]], which are algebraic invariants that classify spaces up to homotopy equivalence. It also plays a crucial role in the development of [[homological algebra]] and the study of [[fiber bundles]].
-* [[Topological space]]
-* [[Continuous function]]
+== Related Concepts ==
-* [[Topological invariant]]
+* [[Homotopy group]]
-* [[Algebraic topology]]
 * [[Fundamental group]]
-* [[Homology (mathematics)|Homology]]
+* [[Homology]]
 * [[Cohomology]]
+* [[Fiber bundle]]
-==References==
+== Related pages ==
-<references/>
+* [[Topology]]
+* [[Algebraic topology]]
+* [[Continuous function]]
+* [[Topological space]]
 [[Category:Topology]]
-[[Category:Mathematical concepts]]
+[[Category:Algebraic topology]]
-[[Category:Homotopy theory]]
-{{math-stub}}