Homotopy

Introduction

Homotopy is a fundamental concept in the field of topology, which is a branch of mathematics. It is a process that allows one mathematical function or shape to be continuously deformed into another. This concept is used in various areas of mathematics, including algebraic topology, differential geometry, and group theory.

Definition

Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined as a continuous function H : X × [0,1] → Y, from the product of the space X with the unit interval [0,1] to Y, such that, if x ∈ X, then H(x,0) = f(x) and H(x,1) = g(x).

Homotopy in Topology

In topology, homotopy is a key tool used to define and study the topological spaces. Two maps f : X → Y and g : X → Y are said to be homotopic (denoted as f ≃ g) if there exists a homotopy H between them. The importance of the concept of homotopy in topology cannot be overstated. It is used to define the fundamental group, homotopy groups, and cohomology rings, among other things.

A clear, high-resolution image of two shapes being continuously deformed into each other, demonstrating the concept of homotopy.

Homotopy Equivalence

A central notion in the study of topological spaces is the concept of homotopy equivalence. Two topological spaces X and Y are said to be homotopy equivalent if there exist continuous maps f : X → Y and g : Y → X such that g ∘ f is homotopic to the identity map idX on X and f ∘ g is homotopic to the identity map idY on Y. Homotopy equivalence is an equivalence relation on the class of all topological spaces.

Homotopy in Algebraic Topology

In algebraic topology, homotopy is used to define the fundamental group and higher homotopy groups. The fundamental group of a topological space X at a point x is the set of all homotopy classes of loops in X based at x, with the group operation given by concatenation of loops. Higher homotopy groups are defined similarly, but with spheres instead of loops.

Homotopy in Differential Geometry

In differential geometry, homotopy is used to study the properties of smooth manifolds. For instance, the de Rham cohomology, which is a fundamental tool in differential geometry, can be defined using homotopy.

Homotopy in Group Theory

In group theory, homotopy is used to study the structure of groups. For instance, the concept of a free group can be understood in terms of homotopy.