ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-14

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.

First nontrivial Postnikov stage of a simply connected space

Claim

Let $X$ be a simply connected CW complex, and let $n \geq 2$ be least such that $π_{n} (X) \neq 0$ . Then

$P_{n} X ≃ K (π_{n} X, n) .$

Facts & Assumptions

[F1]

A Postnikov section $X \to P_{n} X$ is an isomorphism on $π_{i}$ for $i \leq n$ and has $π_{i} (P_{n} X) = 0$ for $i > n$ (Postnikov towers exist for connected CW complexes).

[F2]

Higher homotopy groups are abelian (Higher homotopy classes form groups and are abelian above degree one).

Verification

Given: $X$ and the least index $n$ in the claim.

1.1

Simple connectedness gives $π_{1} (X) = 0$ , and minimality gives $π_{i} (X) = 0$ for $1 < i < n$ . By [F1], the same is true for $P_{n} X$ , while $π_{n} (P_{n} X) ≅ π_{n} (X)$ and every group above $n$ vanishes.

2.1

The surviving group is abelian by [F2]. Since the Postnikov construction supplies a connected CW model, Step 1.1 is exactly the defining homotopy-group condition for $K (π_{n} X, n)$ . The hypothesis that a least nonzero group exists excludes the weakly contractible case. $□$

F2step 1.1

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Haynes Miller, MIT 18.906 Algebraic Topology II lecture notes (standard reference, not scraped)
James Davis and Paul Kirk, Lecture Notes in Algebraic Topology (standard reference, not scraped)