ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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Canonical zero extension of an overlap cochain

Example

Take $U = (- 2, 1)$ and $V = (- 1, 2)$ in $X = (- 2, 2)$ , with overlap vertex $p = 0$ and $U$ -only vertex $q = - 3 / 2$ . The overlap zero-cochain $η$ with value two at $p$ and zero at every other vertex has canonical extension $E_{U} η$ satisfying $(E_{U} η) (3 [p] - [q]) = 6$ .

Facts & Assumptions

Given: The intervals, vertices and basis values above, extended linearly on finite overlap chains.

[F1]

Canonical zero extension retains overlap basis values and vanishes on every other simplex in $U$ (Canonical extension by zero of a singular cochain on a simplex basis).

Proof

1.1

We have $U \cap V = (- 1, 1)$ , so $p$ lies in the overlap and $q$ lies in $U ∖ V$ . By [F1], $(E_{U} η) ([p]) = 2$ and $(E_{U} η) ([q]) = 0$ . Linearity therefore gives $(E_{U} η) (3 [p] - [q]) = 3 \cdot 2 - 0 = 6$ .

givenF1algebra

2.1

Restriction back to the overlap equals $η$ on every vertex: it has value two at $p$ and zero elsewhere, hence agrees on every finite chain. More generally, for any specified overlap cochain and finite chain $\sum a_{s} [s]$ in $U$ , the value is exactly $\sum_{s in the overlap} a_{s} η (s)$ . An empty retained index set gives zero and one retained term gives its coefficient times its value. This is a degreewise extension, not an assertion that extension commutes with coboundary. No basis or representatives are chosen.

F1step 1.1algebra∎

Depends on

Canonical extension by zero of a singular cochain on a simplex basis

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

DG-16 design; Hatcher/Park control (standard reference, not scraped)