DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 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.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Relative singular chain complex

Definition

Fix an abelian group $G$ . For a subspace $A \subseteq X$ , regard the finite singular chains $C_{n} (A; G)$ as the subgroup of $C_{n} (X; G)$ induced by inclusion. The relative singular chain group is $C_{n} (X, A; G) : = C_{n} (X; G) / C_{n} (A; G) .$ Equip these quotients with the induced maps ${\overline{\partial}}_{n} [c] : = [\partial_{n} c] .$ Their well-definedness and the identity ${\overline{\partial}}^{2} = 0$ are established in Boundary on relative chains ↗. The resulting chain complex is denoted $C_{∙} (X, A; G)$ . Thus both $A = \emptyset$ and $A = X$ are admitted.

Depends on

The singular chain complex and singular homology

Used by

Relative singular homology Definition
Boundary on relative chains Lemma

Dependency tree · two levels

5 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, §2.1 (standard reference, not scraped)