TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Short exact chain Mayer–Vietoris sequence

Statement

For $X = U \cup V$ open, there is a short exact sequence $0 \to C_{∙} (U \cap V; G) \overset{i}{\to} (C_{∙} (U; G) \oplus C_{∙} (V; G)) \overset{j}{\to} C_{∙}^{{U, V}} (X; G) \to 0,$ where $i (c) = (c, - c)$ and $j (u, v) = u + v$ .

Facts & Assumptions

Given: An open cover $X = U \cup V$ .

Proof

technique · direct

1.1

Both $i$ and $j$ commute with boundaries; $i$ is injective and $j$ is surjective by the sum description of cover-small chains.

givenconstruct

2.1

If $u + v = 0$ , then $u = - v$ is a chain in both $U$ and $V$ , so $(u, v) = i (u)$ . Thus the kernel of $j$ is the image of $i$ .

step 1.1algebra∎

Depends on

Cover-small chains for a two-open cover

Used by

Well-definedness of the Mayer–Vietoris connector Lemma
Mayer–Vietoris sequence in singular homology Theorem
Naturality of singular Mayer–Vietoris Theorem

Dependency tree · two levels

2 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, §2.2 (standard reference, not scraped)