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

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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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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.

1.1F1F2F3

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.

2.1F1F2F3step 1.1∎

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.

Depends on

Dependency tree · 2 levels

Sources

Hatcher, Algebraic Topology, Chapter 4