# Definition:Homotopy Group

This article needs to be linked to other articles.In particular: in particular, concatenation of classes must be definedYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

## Definition

Let $X$ be a topological space, and $x_0 \in X$.

Let $\mathbb S^n \subseteq \R^{n+1}$ be the $n$-sphere, and $a \in \mathbb S^n$.

Let $\map {\pi_n} {X, x_0}$ be the set of homotopy classes relative to $a$ of continuous mappings $c: \mathbb S^n \to X$ such that $\map c a = x_0$.

Let $* : \map {\pi_n} {X, x_0} \times \map {\pi_n} {X, x_0} \to \map {\pi_n} {X, x_0}$ denote the concatenation of homotopy classes of paths.

That is, if $\overline {c_1}, \overline {c_2}$ are two elements of $\map {\pi_n} {X, x_0}$, then:

- $\overline {c_1} * \overline {c_2} = \overline {c_1 \cdot c_2}$

where $\cdot$ denotes the usual concatenation of paths.

Then $\struct {\map {\pi_n} {X, x_0}, *}$ is the **$n$th fundamental group** of $X$.

The first homotopy group is usually called the **fundamental group** when higher homotopy groups are not in sight.

For a path-connected manifold, by Fundamental Group is Independent of Base Point for Path-Connected Space, the isomorphism class of $\map {\pi_1} {X, x_0}$ does not depend on $x_0$ and we just write $\map {\pi_1} X$.