Homotopy group

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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.