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.

Augmentation and reduced simplicial homology

Definition

For a simplicial complex $K$ , the augmentation $ε : C_{0} (K) \to Z$ is the homomorphism determined by $ε ([v]) = 1$ for every vertex $v$ .

The augmented simplicial chain complex is $\dots \to C_{2} (K) \overset{\partial_{2}}{\to} C_{1} (K) \overset{\partial_{1}}{\to} C_{0} (K) \overset{ε}{\to} Z \to 0.$ Its homology groups are the reduced simplicial homology groups ${\tilde{H}}_{n}^{s i m p} (K)$ .

If $K$ has no vertices, then $C_{n} (K) = 0$ for all $n \geq 0$ , so ${\tilde{H}}_{- 1}^{s i m p} (K) ≅ Z$ and ${\tilde{H}}_{n}^{s i m p} (K) = 0$ for $n \neq - 1$ .

Depends on

Used by

A simplex has zero reduced simplicial homology Corollary
Oriented simplicial subdivision operator Definition
A contractible simplicial cone Example
The simplicial homology of a point and an edge Example
An augmented simplicial cone has an explicit chain contraction Lemma
The augmented simplicial chain complex of a simplex is contractible Lemma
The simplicial augmentation is a chain map Lemma

Dependency tree · two levels

6 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)