DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Stable stems of the sphere

Definition

For $k \in Z$ , the $k$ th stable stem of the sphere is

$π_{k}^{s} : = π_{k} (S) = {colim}_{n \geq n_{0}} π_{n + k} (S^{n}),$

where $n_{0}$ is any index for which $n + k \geq 1$ thereafter. The bonding map is the suspension homomorphism determined by the sphere-prespectrum structure homeomorphism $S^{1} \land S^{n} ≅ S^{n + 1}$ . Tail independence makes the notation independent of $n_{0}$ .

Depends on

Used by

Stabilizing a map between spheres Example
The zero stable stem is the integers Example
Freudenthal identifies the eventual suspension system for spheres Lemma
The sphere prespectrum groups are the classical stable stems Proposition

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. P. May, A Concise Course in Algebraic Topology (standard reference, not scraped)