PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Functoriality of relative homology

Statement

A continuous map of pairs $f : (X, A) \to (Y, B)$ , meaning $f (A) \subseteq B$ , induces $f_{*} : H_{n} (X, A; G) \to H_{n} (Y, B; G)$ for every $n$ . Identity maps and composites induce identity maps and composites.

Facts & Assumptions

Given: A continuous $f : X \to Y$ with $f (A) \subseteq B$ .

Proof

technique · direct

1.1givenconstruct

The induced singular-chain map sends $C_{n} (A; G)$ into $C_{n} (B; G)$ , hence $[c] \mapsto [f_{#} c]$ defines a quotient-chain map.

2.1step 1.1algebra∎

It commutes with the quotient boundaries, so it maps cycles to cycles and boundaries to boundaries and hence descends to homology. The chain-level identity and composition laws persist after quotienting.

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Algebraic Topology, §2.1