Homotopy

An introduction to the concept of homotopy in topology

Homotopy

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

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

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

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

• Homotopy group
• Fundamental group
• Homology
• Cohomology
• Fiber bundle

Related pages

• Topology
• Algebraic topology
• Continuous function
• Topological space