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Section 11.5 Stable Homotopy Groups

Definition 11.14.

For a based space \(X\text{,}\) the \(q\)th stable homotopy group is
\begin{equation*} \pi_q^{s}(X) := \operatorname*{colim}_{n} \pi_{q+n}(\Sigma^n X), \end{equation*}
where the colimit is taken along the suspension homomorphisms.
By Theorem 11.10, the colimit stabilizes. That is, once \(n\) is large enough (depending on \(q\)), the map \(\pi_{q+n}(\Sigma^n X) \to \pi_{q+n+1}(\Sigma^{n+1} X)\) is an isomorphism. Explicitly, let \(X\) be a CW complex. The cells of \(\Sigma^n X\) other than the basepoint have dimension at least \(n\text{,}\) so \(\Sigma^n X\) is \((n-1)\)-connected, and the suspension map above is an isomorphism as soon as \(q + n < 2(n-1) + 1\text{.}\) Hence
\begin{equation*} \pi_q^s(X) \cong \pi_{q+n}(\Sigma^n X) \qquad \text{for any } n \geq q + 2. \end{equation*}
In particular, Corollary 11.12 can be rephrased as
\begin{equation*} \pi_0^{s}(S^0) \cong \mathbb{Z}. \end{equation*}
For \(X = S^0\) we write \(\pi_q^s := \pi_q^s(S^0) = \pi_{q+n}(S^n)\) for \(n \geq q + 2\text{.}\) These groups are the stable stems.

Remark 11.15.

The computation of the stable homotopy groups of spheres \(\pi_q^{s}(S^0)\) — the so-called stable stems — is one of the deepest and most studied problems in algebraic topology. The first few are
\begin{equation*} \pi_0^s, \ldots, \pi_7^s \;=\; \mathbb{Z},\; \mathbb{Z}/2,\; \mathbb{Z}/2,\; \mathbb{Z}/24,\; 0,\; 0,\; \mathbb{Z}/2,\; \mathbb{Z}/240, \end{equation*}
but there is no known pattern in general, and the stable stems are known only through a finite range of degrees.

Remark 11.16.

The failure of excision that opened this chapter disappears after stabilization. Combining Freudenthal’s theorem with a version of homotopy excision for quotients (Lemma 13.6), one shows that every CW pair \((X, A)\) has a long exact sequence
\begin{equation*} \cdots \to \pi_q^s(A) \to \pi_q^s(X) \to \pi_q^s(X/A) \to \pi_{q-1}^s(A) \to \cdots. \end{equation*}
In fact \(X \mapsto \pi_*^s(X)\) is a reduced generalized homology theory on based CW complexes. It is not ordinary homology, since \(\pi_1^s(S^0) \cong \mathbb{Z}/2\) (Corollary 21.2) while \(\widetilde{H}_1(S^0) = 0\text{.}\)
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