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$\pi_2(S^2)$ is isomorphic to $\mathbb{Z}$, the group of integers. This is a fundamental result in algebraic topology demonstrating that maps from the 2-sphere to itself can wrap around an integer number of times.

$\pi_3(S^2)$ is also isomorphic to $\mathbb{Z}$. The Hopf fibration is an example of a map from $S^3$ to $S^2$ representing a generator of this group.

The Hopf invariant is an integer that classifies maps from $S^3$ to $S^2$ based on the degree of linking of the preimages of points. The homotopy group $\pi_3(S^2)$ being isomorphic to $\mathbb{Z}$ is directly related to this classification, where each integer value corresponds to a different homotopy class of maps, with the Hopf fibration being a representative of the class with invariant 1.

$\pi_1(S^2)$ is trivial, meaning it is the trivial group consisting of a single element. This indicates that the 2-sphere is simply connected.

$\pi_4(S^3)$ is trivial. This result can be understood using the Freudenthal suspension theorem and the fact that four-dimensional obstructions in a 3-sphere can be 'smoothed out' homotopically.

Suspension increases the dimension of a sphere by 1 and corresponds to the Freudenthal suspension theorem, which states that the homotopy group $\pi_{n+k}(S^n)$ is isomorphic to $\pi_{n+k+1}(S^{n+1})$ for $k < 2n - 1$. This theorem is a key tool in studying homotopy groups of spheres.

$\pi_7(S^3)$ is not trivial; it is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}_{12}$, where $\mathbb{Z}_{12}$ is the cyclic group of order 12. This reflects the nontrivial structure that arises due to the higher-dimensional analogs of the Hopf fibration.

$\pi_4(S^2)$ is trivial. Although it might not be intuitively obvious, this result can be derived using tools such as the long exact sequence of a fibration and the Freudenthal suspension theorem.

The Bott periodicity theorem reveals that there is a periodicity in the stable homotopy groups of spheres. Specifically, it asserts that $\pi_{n+k}(S^n)$ is isomorphic to $\pi_{n+k+8}(S^{n+8})$ for $k$ sufficiently large, indicating an 8-fold periodicity in the stable range.