LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Cubical concatenation is well defined on higher homotopy classes

Statement

For $n \geq 1$ , the coordinate-1 concatenation in the cubical definition defines a representative-independent product on $π_{n} (X, x_{0})$ . The same construction works in each coordinate whose two opposite faces are fixed at $x_{0}$ .

Facts & Assumptions

[F1]

Representatives and their homotopies fix every boundary face. Higher homotopy group by based cubes

[F2]

Continuous pieces agreeing on a finite closed cover paste continuously. Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous

[F3]

Continuity into a product is coordinatewise. A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice

Proof

Given: The spaces, maps, and hypotheses in the statement above.

1.1

The two affine maps $(s, u) \mapsto (2 s, u)$ and $(s, u) \mapsto (2 s - 1, u)$ are continuous on the closed half-cubes. On their common face the values of $a, b$ are both $x_{0}$ . Thus the concatenation is continuous by closed pasting. Its outer boundary maps to $x_{0}$ , since either s is an endpoint or a coordinate of u is an endpoint.

F1F2F3

2.1

If $A, B : I^{n} \times I \to X$ are boundary-fixed homotopies between the two respective pairs of representatives, paste $A (2 s, u, t)$ and $B (2 s - 1, u, t)$ . The seam values are $x_{0}$ for every t; the same boundary calculation applies. This is a continuous boundary-fixed homotopy between the concatenations. Permuting the selected coordinate with coordinate 1 gives the identical proof whenever its two faces are fixed.

F1F2F3step 1.1∎

Depends on

Used by

Dependency tree · two levels

18 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

Hatcher, Algebraic Topology, Chapter 4 (standard reference, not scraped)