DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

The induced singular chain map of a continuous map

Definition

Let $f : X \to Y$ be a continuous map of topological spaces. For each $n \geq 0$ , the induced singular chain map in degree $n$ is the homomorphism $f_{#, n} : C_{n} (X; Z) \to C_{n} (Y; Z)$ defined on a singular simplex $σ : Δ^{n} \to X$ by $f_{#, n} (σ) : = f \circ σ,$ and extended $Z$ -linearly.

For coefficients in an abelian group $G$ , tensor with ${id}_{G}$ to obtain $f_{#, n} : C_{n} (X; G) \to C_{n} (Y; G) .$ The family $(f_{#, n})_{n \geq 0}$ is denoted $f_{#} : C_{∙} (X; G) \to C_{∙} (Y; G)$ .

Depends on

Used by

Alexander–Whitney map and diagonal approximation Definition
The reduced Burau homology module Definition
Braid mapping classes lift equivariantly to the Burau cover Lemma
Induced singular chain maps commute with boundaries Lemma
Universal-cover boundaries, maps and homotopies respect the right group-ring action Lemma
Functoriality of relative homology Proposition
Singular chain cross products are natural Proposition
Singular chains and singular homology are covariantly functorial Proposition
Singular cohomology is contravariantly functorial Proposition
The singular chain homotopy formula Theorem

Dependency tree · two levels

7 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)