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The second homotopy group $\pi_2(X)$ is particularly important for the study of surfaces and two-dimensional 'holes'. A non-trivial second homotopy group indicates the presence of such 'holes' in a space.

In algebraic topology, fibration sequences can be used to calculate homotopy groups. They consist of a sequence of spaces and fibrations where one can often deduce the homotopy groups of one space from the known groups of the others.

Unlike the fundamental group, which can be non-Abelian, all higher homotopy groups $\pi_n$ for $n > 1$ are Abelian groups. This property simplifies computations and means that the group operation is commutative.

Relative homotopy groups $\pi_n(X, A, x_0)$ generalize usual homotopy groups by considering maps from the $n$-dimensional disk $D^n$ to $X$ which restrict to a pointed subspace $A$. These groups can detect 'holes' relative to the subspace $A$.

Suspension is an operation in algebraic topology that helps compute homotopy groups. It involves building a new space by extending a given space 'upwards' and 'downwards' and can change the homotopy groups of that space.

Higher homotopy groups are generalizations of the fundamental group, which capture information about 'holes' in spaces of higher dimensions. The $n$-th homotopy group $\pi_n(X, x_0)$ measures the equivalence classes of maps from the $n$-dimensional sphere $S^n$ to a space $X$ based at a point $x_0$.

The Freudenthal suspension theorem describes how suspension affects homotopy groups. It asserts that suspension induces an isomorphism between $\pi_n(X)$ and $\pi_{n+1}(SX)$ for large enough $n$, where $SX$ is the suspension of $X$.

The Whitehead product is a way of combining elements in different homotopy groups of a space to get a new element in a higher homotopy group. It reflects the fact that homotopy groups have richer algebraic structures beyond just being groups.

Homotopy groups of spheres are a central object of study in algebraic topology. For the $n$-sphere $S^n$, the group $\pi_n(S^n)$ is always isomorphic to the integers $\mathbb{Z}$, representing the degree of maps from the sphere to itself.