Long exact sequence of homotopy groups

Remarks

The key step is that $p_* \colon \pi_n(E, F, x_0) \to \pi_n(B, b_0)$ is an isomorphism for $n \ge 1$, proved by lifting maps of disks with the homotopy lifting property. The sequence then comes from the long exact sequence of the pair $(E, F)$ in relative homotopy groups. The map $\partial$ lifts a based map of a disk into $B$ to $E$ and restricts it to the boundary sphere, which lands in $F$. For the Hopf fibration $S^1 \to S^3 \to S^2$, the sequence gives $\pi_2(S^2) \cong \pi_1(S^1) \cong \Z$ and $\pi_n(S^3) \cong \pi_n(S^2)$ for $n \ge 3$. For the covering $\R \to S^1$ the fibre $\Z$ is discrete, so $\pi_n(S^1) \cong \pi_n(\R) = 0$ for $n \ge 2$.

Going deeper

For a student

This is the most used tool for computing homotopy groups. It gives $\pi_3(S^2) \cong \Z$ from the Hopf fibration, and computes homotopy groups of Lie groups and homogeneous spaces from fibrations such as $SO(n) \to SO(n+1) \to S^n$. For the path space fibration $\Omega B \to PB \to B$ with contractible total space, it gives $\pi_n(\Omega B) \cong \pi_{n+1}(B)$. The analogous sequence of a pair $A \subseteq X$ has the form $\cdots \to \pi_n(A) \to \pi_n(X) \to \pi_n(X, A) \to \pi_{n-1}(A) \to \cdots$.