False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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.

FALSE: every fundamental group is abelian

Statement

False claim: for every pointed topological space $(X, x_{0})$ , the group $π_{1} (X, x_{0})$ is abelian.

Facts & Assumptions

Given: The two-circle wedge $W = S^{1} \lor S^{1}$ with standard loop classes $a$ and $b$ .

[L1]

The group $π_{1} (W, w)$ is the free group on $a$ and $b$ ( $π_{1} (S^{1} \lor S^{1})$ is the free group on two generators).

[F1]

The reduced words on a basis and its formal inverses form the free group on that basis (Reduced words form the free group on an alphabet).

Refutation

technique · direct

1.1

Under [L1], the products $a b$ and $b a$ are represented by the two reduced words with syllable sequences $(a, b)$ and $(b, a)$ .

2.1

These reduced words are distinct by [F1], so $a b \neq b a$ in $π_{1} (W, w)$ .

step 1.1F1

3.1

Thus the fundamental group of the two-circle wedge is not abelian, providing a counterexample to the universal claim.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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, Example 1.21 (standard reference, not scraped)