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

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Stable homotopy colimits are independent of a cofinal tail

Statement

Let $n_{0} \geq 0$ , and let $A_{n_{0}} \to A_{n_{0} + 1} \to A_{n_{0} + 2} \to \dots$ be the sequential group system used to define a stable homotopy group. For every $N \geq n_{0}$ , inclusion of the tail induces a canonical group isomorphism

${colim}_{n \geq N} A_{n} \overset{≅}{\to} {colim}_{n \geq n_{0}} A_{n} .$

In particular, deleting finitely many legal terms from the system defining $π_{k} (E)$ does not change the resulting group.

Facts & Assumptions

[F1]

In the disjoint-union quotient model, $[n, x] = [m, y]$ exactly when the two elements have equal images at some common later stage (Stable homotopy groups of a sequential prespectrum).

[F2]

Every finite set of natural-number indices has a common later index.

Proof

Given: The sequential group system, legal initial cutoff $n_{0}$ , and tail index $N \geq n_{0}$ in the statement.

1.1

Surjectivity. A class $[n, x]$ in the colimit beginning at $n_{0}$ has the same value as $[N, b_{N - 1} \dots b_{n} x]$ if $n < N$ , and already has a representative in the tail if $n \geq N$ . Hence every class comes from the tail.

F1F2

1.2

Injectivity. Suppose two tail representatives have the same image in the colimit beginning at $n_{0}$ . By [F1], their images agree at a common stage $r$ . Replace $r$ by $\max (r, N)$ if needed. This witnesses their equality using only the tail relation, so the tail map is injective.

1.3

Group structure and canonicity. The inclusion respects bonding maps, hence respects addition on common-stage representatives. Its inverse is forced by advancing a representative into the tail; [F1] shows that this is independent of the chosen later stage. Thus the isomorphism is canonical.

$□$

Depends on

Used by

Shift and suspension of sequential prespectra Definition
Strict prespectrum maps act functorially on stable homotopy groups Proposition

Dependency tree · two levels

8 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)