For questions related to higher homotopy groups. A higher homotopy group, $\pi_n(X,x_0)$, is the set of based homotopy classes of based maps $\gamma:(S^n,s_0)\rightarrow(X,x_0).$
The fundamental group, $\pi_1(X,x_0)$, is a powerful tool of algebraic topology used to study topological spaces by studying how loops behave within them. Higher homotopy groups, which will be our main object of study, are the natural generalization of the fundamental group.
While $\pi_1(X,x_0)$ considers how loops can live in a space up to based homotopy, higher homotopy groups more generally consider how closed surfaces (maps from the $n$-sphere) can be mapped into spaces.
A higher homotopy group, $\pi_n(X,x_0)$, is the set of based homotopy classes of based maps $\gamma:(S^n,s_0)\rightarrow(X,x_0).$
As in the case of the fundamental group, higher homotopy groups are also groups.