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

Used by

Equal Euler characteristic without homeomorphism Counterexample
Degree of a self map of an oriented sphere Definition
Orientation local system and orientation cover Definition
Orientation-preserving homeomorphisms and the geometric definition of quasiconformality Definition
The unreduced Burau relative homology module Definition
Nonzero degree forces surjectivity on closed oriented manifolds Example
Gauss-Bonnet alone does not classify surfaces False statement
Braid mapping classes lift equivariantly to the Burau cover Lemma
Coordinate-ball classes identify local homology stalks Lemma
Properties of bounded-domain degree Lemma
Smooth Beltrami coefficients admit quasiconformal solutions Lemma
Smooth orientation sign is the local integral homology multiplier Lemma
The geometric half twist acts on the lifted-edge basis by the Burau block Lemma
The inverse of a quasiconformal map is quasiconformal with the same dilatation Lemma
Excision for singular homology Theorem
Local homology detects manifold dimension, interior, and boundary Theorem
Naturality of the pair long exact sequence Theorem
Regular-value formula for degree Theorem
The geometric and analytic definitions of quasiconformality agree Theorem
The topological and matrix Burau representations agree Theorem
Top homology of a connected manifold Theorem

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, §2.1 (standard reference, not scraped)