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 $$\pi_n (X) \ $$.