Consider the Whitehead tower of the $n$-sphere $X = S^n$: $$... \rightarrow X' \stackrel{p}{\rightarrow} X,$$ where $X'$ is $n$-connected. Explicitly, what is $X'$ and what is the fibration $p$? How do I compute them?
Thoughts: Intuitively, I am looking for the first nontrivial "delooping" of the $n$-sphere. For example, for $n=1$, I know that $X' = \mathbb{R}$ and $p$ is the universal covering of $S^1$. However for $n \geq 2$, I have no idea how to compute $X'$ or $p$. Perhaps in the case $n=2$, $p$ is the Hopf fibration.