TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-04‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The Euler-Poincare formula for a finite simplicial complex with free homology

Statement

Let $K$ be a finite simplicial complex. Assume that each simplicial homology group $H_{n}^{s i m p} (K)$ is free of finite rank. Then $χ (K) = \sum_{n \geq 0} (- 1)^{n} rank H_{n}^{s i m p} (K) .$

Proof

Given: A finite simplicial complex $K$ whose simplicial homology groups are free of finite rank.

1.1

For each $n$ , the chain group $C_{n} (K)$ is free abelian on the oriented $n$ -simplices of $K$ , so $rank C_{n} (K) = f_{n} (K)$ . Since $K$ is finite, only finitely many of these groups are nonzero.

given

2.1

Therefore the simplicial chain complex of $K$ is a bounded chain complex of finite-rank free abelian groups, and its homology groups are free of finite rank by hypothesis. The finite-free Euler-Poincare theorem applies and gives $\sum_{n \geq 0} (- 1)^{n} rank C_{n} (K) = \sum_{n \geq 0} (- 1)^{n} rank H_{n}^{s i m p} (K)$ .

step 1.1

3.1

Replace $rank C_{n} (K)$ by $f_{n} (K)$ using step 1.1, and replace the left-hand side by $χ (K)$ by definition. This yields the stated formula.

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

20 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)