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Introduction

The first two stable stems came out to $\pi_1^s \cong \mathbb{Z}/2$ and $\pi_2^s \cong \mathbb{Z}/2\text{.}$ In general, stable stems are infamously hard to compute, but Serre proved a striking qualitative result: every positive-degree stable stem is a finite abelian group. In fact:

\begin{equation*} \pi_q^s \;=\; \mathop{\mathrm{colim}}_n \pi_{q+n}(S^n) \;\cong\; \mathbb{Z}/m_1 \oplus \cdots \oplus \mathbb{Z}/m_s, \qquad m_i \geq 2, \end{equation*}

for each $q > 0\text{.}$ In this chapter we develop the finite-type technology required for the proof and prove the key statement: for odd $n\text{,}$ $\pi_i(S^n)$ is finite for all $i > n\text{.}$

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