PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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The sphere prespectrum groups are the classical stable stems

Statement

For every $k \geq 0$ , $π_{k} (S)$ is canonically isomorphic to the eventual value of

$π_{n + k} (S^{n}) \overset{E}{\to} π_{n + k + 1} (S^{n + 1}) \overset{E}{\to} \dots .$

This eventual group is the classical $k$ th stable homotopy group of spheres.

Facts & Assumptions

[F1]

For fixed $k \geq 0$ , every bonding map after any index $N > k + 1$ is an isomorphism (Freudenthal identifies the eventual suspension system for spheres).

[F2]

The stable-stem definition is the colimit of the sphere suspension system and records its finite-tail independence (Stable stems of the sphere).

Proof

Given: A fixed integer $k \geq 0$ .

1.1F1

Choose $N > k + 1$ . By [F1], every map in the tail beginning at $N$ is an isomorphism. Sending $x \in π_{N + k} (S^{N})$ to its colimit class is surjective, since every later representative can be transported back uniquely through the intervening isomorphisms.

1.2F1

If two elements at stage $N$ have the same colimit class, their images agree at a later stage. The composite from stage $N$ to that stage is an isomorphism by [F1], so the original elements agree. The stage- $N$ map is therefore injective.

2.1F1

Changing $N$ replaces this isomorphism by transport through canonical bonding isomorphisms; [F2] shows that all choices identify the same colimit. By the definition of the sphere prespectrum and the stable stem, that colimit is $π_{k} (S) = π_{k}^{s}$ . $□$

Depends on

Used by

The stable Pontryagin-Thom theorem identifies framed bordism with stable stems Theorem

Dependency tree · two levels

7 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)