DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-05

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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The singular boundary operator

Definition

Let $X$ be a topological space and let $n \geq 1$ . For a singular $n$ -simplex $σ : Δ^{n} \to X$ , its singular boundary is $\partial_{n} σ : = \sum_{i = 0}^{n} (- 1)^{i} σ \circ δ_{i} \in C_{n - 1} (X; Z),$ where the $δ_{i}$ are the affine face maps from Singular simplices and singular chain groups with coefficients.

Extend this formula $Z$ -linearly to a homomorphism $\partial_{n} : C_{n} (X; Z) \to C_{n - 1} (X; Z) .$ For coefficients in an abelian group $G$ , the boundary on $C_{n} (X; G)$ is the tensor extension $\partial_{n} \otimes {id}_{G} : C_{n} (X; G) \to C_{n - 1} (X; G),$ again denoted $\partial_{n}$ .

The degree-zero boundary is the zero map $\partial_{0} : C_{0} (X; G) \to 0,$ and for negative indices the chain groups are taken to be zero.

Depends on

Singular simplices and singular chain groups with coefficients

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 (standard reference, not scraped)