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Section 21.1 Strategy: What Do We Know So Far?

Before lecture 21 we have:

\begin{equation*} \renewcommand{\arraystretch}{1.2} \begin{array}{c|ccccc} \pi_i(S^n) & i = 1 & 2 & 3 & 4 & 5 \\ \hline n = 1 & \mathbb{Z} & 0 & 0 & 0 & 0 \\ n = 2 & 0 & \mathbb{Z} & \mathbb{Z} & ? & ? \\ n = 3 & 0 & 0 & \mathbb{Z} & ? & ? \\ n = 4 & 0 & 0 & 0 & \mathbb{Z} & ? \\ n = 5 & 0 & 0 & 0 & 0 & \mathbb{Z} \end{array} \end{equation*}

The diagonal entries $\pi_n(S^n) \cong \mathbb{Z}$ are the Hurewicz computation; $\pi_3(S^2) \cong \mathbb{Z}$ uses the Hopf fibration $S^1 \to S^3 \to S^2\text{;}$ the $\pi_i(S^n) = 0$ for $i < n$ is from cellular approximation. We compute the first "off-diagonal" entry $\pi_4(S^3)\text{.}$

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