# Homotopy group

## Algebraic construct classifying topological spaces / From Wikipedia, the free encyclopedia

#### Dear Wikiwand AI, let's keep it short by simply answering these key questions:

Can you list the top facts and stats about Homotopy group?

Summarize this article for a 10 year old

In mathematics, **homotopy groups** are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted $\pi _{1}(X),$ 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**, $\pi _{n}(X),$ 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]

Oops something went wrong: