DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Two open set de rham mayer vietoris cochain maps

Definition

For an open cover $M = U \cup V$ , put $W = U \cap V$ . The two-open-set de Rham maps are $r : Ω^{∙} (M) \to Ω^{∙} (U) \oplus Ω^{∙} (V)$ , $r ω = (ω ∣_{U}, ω ∣_{V})$ , and $s : Ω^{∙} (U) \oplus Ω^{∙} (V) \to Ω^{∙} (W)$ , $s (α, β) = β ∣_{W} - α ∣_{W}$ .

The complexes are De rham cochain complex. Restrictions are pullbacks along open inclusions, so Pullback is a morphism of de rham complexes gives $d r = r d$ and $d s = s d$ . Both maps are real linear. The middle differential acts componentwise. Empty opens have zero form spaces. The order second minus first fixes the sign of every connecting map below.

Source locator

Lee, Theorem 17.20, pp.449–450, and its full proof pp.462–463. This page reverses Lee’s difference convention consistently: $s (α, β) = β - α$ .

Depends on

Dependency tree · 2 levels

Sources

John M. Lee, Introduction to Smooth Manifolds, second edition