Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted ${\displaystyle \pi _{1}(X),}$ which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.

To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the n-th homotopy group, ${\displaystyle \pi _{n}(X),}$ of the given space X with base point. Topological spaces with differing homotopy groups are never homeomorphic, but topological spaces that are not homeomorphic can have the same homotopy groups.

The notion of homotopy of paths was introduced by Camille Jordan.^[1]