CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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.

Hopf formula is presentation-independent

Statement

Assume dependent choice and supplied projective-resolution data for group homology. Then Hopf's quotient is independent of the free presentation through $M (G)$ .

Facts & Assumptions

Given: Dependent choice, the supplied group-homology resolution data, and two free presentations $G = F / R = F^{'} / R^{'}$ .

Proof

technique · direct

1.1givenalgebra

For each presentation, the conditional five-term sequence injects $H_{2} (G; Z)$ into $R / [F, R]$ with image the kernel of $R / [F, R] \to F_{a b}$ . That kernel is $(R \cap [F, F]) / [F, R]$ , since the map sends $r [F, R]$ to $r [F, F]$ . By the definition of the Schur multiplier this gives $(R \cap [F, F]) / [F, R] ≅ M (G)$ , and the same argument gives $(R^{'} \cap [F^{'}, F^{'}]) / [F^{'}, R^{'}] ≅ M (G)$ .

2.1step 1.1algebra∎

Composing the first isomorphism with the inverse of the second identifies the two presentation quotients. Thus their isomorphism type depends only on $G$ , through $M (G)$ .

Depends on

Used by

FALSE: the numerator in Hopf's formula is presentation-independent False statement

Dependency tree · two levels

19 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

Clara Löh, Group Cohomology (standard reference, not scraped)