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Introduction

In this chapter we put the spectral sequence machinery to work and compute the first nontrivial stable homotopy group of spheres:

\begin{equation*} \pi_1^s \;:=\; \mathop{\mathrm{colim}}_n \pi_{1+n}(S^n) \;\cong\; \mathbb{Z}/2. \end{equation*}

By Freudenthal’s suspension theorem the colimit stabilises at $\pi_4(S^3)\text{,}$ so the claim is $\pi_4(S^3) \cong \mathbb{Z}/2$ (which in turn forces $\pi_4(S^2) \cong \mathbb{Z}/2$ via the Hopf fibration). The route we take is to compute the integral cohomology of $K(\mathbb{Z}, 3)$ through degree $\sim 9$ using the path-space fibration, and then convert that to a statement about $\pi_4(S^3)\text{.}$

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