Definition:Homotopy Group

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

Let $* : \pi_n(X,x_0) \times \pi_n(X,x_0) \to \pi_n(X,x_0)$ denote the concatenation of homotopy classes of paths.

That is, if $\overline{c_1},\overline{c_2}$ are two elements of $\pi_n(X,x_0)$, then
 * $\overline{c_1}*\overline{c_2} = \overline{c_1 \cdot c_2}$

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

Then $(\pi_n(X,x_0),*)$ is the $n^\text{th}$ 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 $\pi_1(X,x_0)$ does not depend on $x_0$ and we just write $\pi_1 (X)$.

Also see

 * Homotopy Groups are Groups


 * Fundamental Group is Independent of Base Point for Path-Connected Space


 * List of Fundamental Groups for 2-Manifolds