DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Simplicial chain groups and the boundary operator

Definition

For an abstract simplicial complex $K$ and an integer $n \geq 0$ , the simplicial chain group $C_{n} (K)$ is the free abelian group generated by the oriented $n$ -simplices of $K$ , subject to the relation $[v_{π (0)}, \dots, v_{π (n)}] = sgn (π) [v_{0}, \dots, v_{n}]$ for every permutation $π$ of the vertices of a simplex. For $n < 0$ , set $C_{n} (K) = 0$ .

The boundary operator $\partial_{n} : C_{n} (K) \to C_{n - 1} (K)$ is $\partial_{0} = 0$ in degree $0$ . For $n \geq 1$ , it is defined on an oriented simplex by $\partial_{n} [v_{0}, \dots, v_{n}] = \sum_{i = 0}^{n} (- 1)^{i} [v_{0}, \dots, \hat{v_{i}}, \dots, v_{n}] .$

The well-definedness of this formula with respect to the chosen oriented representative is recorded in The simplicial boundary is independent of the chosen oriented representative ↗ through justified_by.

Depends on

An orientation of a simplex

Used by

Augmentation and reduced simplicial homology Definition
Simplicial cycles, boundaries, and homology Definition
The induced graded homomorphism of a simplicial map Definition
The simplicial homology of a point and an edge Example
The simplicial augmentation is a chain map Lemma
The simplicial boundary is independent of the chosen oriented representative Lemma
Simplicial homology of a disjoint union is the direct sum Proposition
The simplicial boundary squares to zero 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 (standard reference, not scraped)
Vidit Nanda, Computational Algebraic Topology, Lecture 03: Homology (standard reference, not scraped)