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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Dependency tree · 2 levels

Sources

J. P. May, A Concise Course in Algebraic Topology