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.

1.1F1F2

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.

1.2F1

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.

1.3F1

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
Stable unoriented Thom homotopy is injectively detected Theorem

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)