Definition:Homotopy Group

Definition
The $n^{th}$ homotopy group of a topological space $X$ at a base point $x_0$, written $\pi_n(X,x_0)$, is the group whose members are the homotopy classes of continuous mappings $c:[0,1]^n \to X$ satisfying $c(\partial([0,1]^n))=x_0$, and whose operation is concatenation on homotopy class members.

The group $\pi_1(X,x_0)$ is called the fundamental group.

For a path-connected manifold, the base point is irrelevant and we just write $\pi_n (X)$.