DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-09-06

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.

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Subdivision prism homotopy

Definition

Let $ι_{n} : Δ^{n} \to Δ^{n}$ be the identity affine simplex. Put $P_{0} = 0$ . Inductively, after $P_{m}$ has been defined for $m < n$ , set $P (λ) = λ_{#} P_{m}$ for every affine $m$ -simplex $λ : Δ^{m} \to Δ^{n}$ . Then define the linear $(n + 1)$ -chain $P_{n} = b_{n} (ι_{n} - P (\partial ι_{n})),$ where $b_{n} (-)$ is affine coning inside the convex simplex $Δ^{n}$ to its barycenter. For a singular simplex $σ : Δ^{n} \to X$ , set $T (σ) = σ_{#} P_{n}$ and extend $G$ -linearly. This degree-one operator is the subdivision prism homotopy.

Depends on

Used by

Dependency tree · two levels

4 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

Allen Hatcher, Algebraic Topology, Proposition 2.21 (standard reference, not scraped)