LemmaStatement: AI-generatedProof: AI-generated

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.

Cofinal tails and level invariance of stable groups

PDF p. 3: Filtered S1 spectra: stable enhancement prerequisites

Statement

For a sequential prespectrum $E$ and integer $k$ , write $G_{k} (E)$ for the sequential colimit of the groups $π_{n + k} (E_{n})$ , starting at any $N$ with $n + k \geq 2$ for $n \geq N$ and using the sphere-first bonding maps of the published definition. Increasing $N$ gives a canonical isomorphism. A compatible level map inducing an isomorphism on every stage group induces an isomorphism on $G_{k}$ . The same assertions hold for pointed simplicial prespectra, with simplicial spheres and suspension in the same order.

Facts & Assumptions

Given: $k$ , the specified sequential system and, for the second assertion, compatible stage isomorphisms.

Goal: Prove cutoff independence and level invariance.

[F1]

Stable homotopy groups of a sequential prespectrum defines the groups, bonding maps and their sequential colimit by representatives and eventual equality.

Proof

1.1F1given

For $N^{'} \geq N$ , send a representative at stage $n$ to its image at any stage at least $\max (n, N^{'})$ . Any two choices agree after further advancement; eventual equality is preserved. This defines a homomorphism from the original colimit to the tail colimit. Composing in either order with the map induced by inclusion of the tail fixes every representative after advancement to a common stage. Thus the maps are inverse.

2.1F1step 1.1given∎

A compatible family $f_{n}$ of stage isomorphisms induces a map on representatives. Its inverses are compatible because $f_{n + 1} b_{n} = b_{n}^{'} f_{n}$ , so they induce its inverse on colimits. This argument uses only sequential systems of groups and applies equally to the simplicial systems. Both assertions follow.

Depends on

Stable homotopy groups of a sequential prespectrum

Used by · 3

Dependency tree · 3 levels

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