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 cycles, boundaries, and homology

Definition

For a simplicial complex $K$ , define $Z_{n} (K) : = \ker (\partial_{n}) \subseteq C_{n} (K), B_{n} (K) : = im (\partial_{n + 1}) \subseteq C_{n} (K) .$ By $\partial^{2} = 0$ , every boundary is a cycle. The $n$ th simplicial homology group is $H_{n}^{s i m p} (K) : = Z_{n} (K) / B_{n} (K) .$

Depends on

Used by

The boundary of a triangle has first homology Z Example
The simplicial homology of the tetrahedron boundary Example
Simplicial homology of a disjoint union is the direct sum Proposition
Zero-th simplicial homology is free on connected components Proposition
The Euler-Poincare formula for a finite simplicial complex with free homology Theorem

Dependency tree · two levels

10 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)