PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

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Singular homology is invariant under deformation retracts

Statement

If $A$ is a deformation retract of a topological space $X$ , then for every $n \geq 0$ and every abelian group $G$ the inclusion $i : A ↪ X$ induces an isomorphism $H_{n}^{s i n g} (i_{#}) : H_{n}^{s i n g} (A; G) \to H_{n}^{s i n g} (X; G) .$

Facts & Assumptions

Given: A deformation retract inclusion $i : A ↪ X$ , an abelian group $G$ , and an integer $n \geq 0$ .

[L1]

A deformation retract inclusion is a homotopy equivalence (The inclusion of a deformation retract is a homotopy equivalence with the retraction as homotopy inverse).

[L2]

Homotopy equivalences induce isomorphisms on singular homology (Homotopy equivalences induce isomorphisms on singular homology).

Proof

technique · direct

1.1

By [L1], the inclusion $i$ is a homotopy equivalence.

L1given

2.1

Applying [L2] to $i$ gives the displayed isomorphism on singular homology in every degree.

L2step 1.1∎

Depends on

Used by

The homology of punctured Euclidean space by deformation retraction Example

Dependency tree · two levels

6 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)