---
tag: 5KERJ85BXJ
kind: definition
name: Stable homotopy theory
msc: [55P42, 55Q10]
defines: 545EV0W4CE
uses: [2VD7GR64JJ, T9DG8DDE75, 5W95QFHXPR, P976WYEFMB, 7DM2KSS2DB]
sources:
  - url: https://en.wikipedia.org/wiki/Stable_homotopy_theory
---

## Definition

Stable homotopy theory is the study of homotopy invariants that are unchanged by suspension, such as the following groups.
Let $X$ and $Y$ be [[pointed|T9DG8DDE75]] [[CW complexes|5W95QFHXPR]], with $X$ finite, and let $\Sigma$ be the reduced [[suspension|2VD7GR64JJ]].
Let $[A, B]_*$ be the set of pointed homotopy classes of pointed maps $A \to B$.
The group of *stable maps* from $X$ to $Y$ is the colimit
$$\set{X, Y} = \operatorname{colim}_{k} \, [\Sigma^k X, \Sigma^k Y]_*,$$
taken along the maps $[\Sigma^k X, \Sigma^k Y]_* \to [\Sigma^{k+1} X, \Sigma^{k+1} Y]_*$ that send $f$ to $\Sigma f$.
For $n \ge 0$, the $n$-th *stable homotopy group* of $Y$ is
$$\pi_n^s(Y) = \operatorname{colim}_k \, \pi_{n+k}(\Sigma^k Y) = \set{S^n, Y}.$$

## Remarks

For $k \ge 2$ the set $[\Sigma^k X, \Sigma^k Y]_*$ is an abelian group, so the colimit is an abelian group.
Let $Y$ be $(c-1)$-connected and let $X$ have dimension $d$.
The [[Freudenthal suspension theorem|7DM2KSS2DB]] shows that the maps in the colimit are bijective once $d + k < 2(c + k) - 1$, that is, once $k > d - 2c + 1$.
So the colimit is reached at a finite stage.

Example: $\pi_0^s(S^0) = \Z$, since $\pi_k(S^k) = \Z$ for all $k \ge 1$ and suspension preserves degree.
Also $\pi_1^s(S^0) = \Z/2$, generated by the stable class of the Hopf map $\eta$.

Non-example: the group $\pi_3(S^2) \cong \Z$, generated by the Hopf map, is not a stable group.
Its suspension lands in $\pi_4(S^3) \cong \Z/2$, and the stable value $\pi_1^s(S^0)$ is $\Z/2$.

Today stable homotopy theory is usually set in the stable homotopy category of spectra, where $\set{X, Y}$ becomes a group of morphisms between suspension spectra.