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.

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Reading spectrum homotopy from an Omega tower

PDF p. 4: Shift [2j] in equation (2.1)

Statement

Let $Y = (Y_{n})$ be a coherent Omega tower of pointed spaces, meaning that its specified maps $Y_{n} \to Ω Y_{n + 1}$ are equivalences, and let $E$ be its associated spectrum. For every integer $i$ and every sufficiently large $n$ , the natural comparison identifies $π_{i} (E)$ with $π_{n + i} (Y_{n})$ and with the stable group of the tower.

Facts & Assumptions

Given: $Y$ , $E$ , and $i$ as stated.

Goal: Identify the stable group with the enhanced homotopy group.

[F1]

Omega towers and the suspension index gives the comparison when $n + i \geq 2$ , compatible with advancement in the tower.

Proof

1.1F1givenchoose

Choose $n$ with $n + i \geq 2$ . The first isomorphism and its compatibility with advancement are exactly those of [F1].

2.1F1step 1.1∎

Each tower transition induces an isomorphism on the relevant homotopy group, so its stable colimit is identified with any such sufficiently late group. Combining this with step 1.1 proves the claim.

Depends on

Omega towers and the suspension index

Used by · 1

Complex shifts and spectral homotopy indices

Dependency tree · 3 levels

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