TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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A chain homotopy equivalence is a quasi-isomorphism

Statement

Every chain homotopy equivalence is a quasi-isomorphism.

Facts & Assumptions

Given: A chain homotopy equivalence $f : C_{∙} \to D_{∙}$ with homotopy inverse $g : D_{∙} \to C_{∙}$ .

[L1]

A chain homotopy equivalence has maps $g$ with $g f ≃ 1_{C}, f g ≃ 1_{D}$ (A chain homotopy equivalence).

[L2]

Chain-homotopic maps induce the same map on homology (Chain-homotopic maps induce the same map on homology).

[L3]

Homology respects identities and composition (Homology respects identities and composition).

[L4]

A quasi-isomorphism is a chain map inducing isomorphisms on all homology objects (Quasi-isomorphism).

Proof

technique · direct

1.1

From [L1] and [L2], the homotopies $g f ≃ 1_{C}$ and $f g ≃ 1_{D}$ imply $H_{n} (g) H_{n} (f) = 1_{H_{n} (C)}, H_{n} (f) H_{n} (g) = 1_{H_{n} (D)}$ for every $n \in Z$ .

L1L2givenalgebra

2.1

By [L3], the equalities in step 1.1 say exactly that each $H_{n} (f)$ has inverse $H_{n} (g)$ . Therefore every $H_{n} (f)$ is an isomorphism, and [L4] makes $f$ a quasi-isomorphism.

L3L4step 1.1algebra∎

Depends on

Used by

A contractible complex is acyclic Corollary
A quasi-isomorphism with no homotopy inverse Counterexample
FALSE: every quasi-isomorphism is a chain homotopy equivalence False statement
Shift preserves homotopy equivalences, contractibility, and quasi-isomorphisms Proposition

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

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
Joseph J. Rotman, An Introduction to Homological Algebra, 2nd ed. (standard reference, not scraped)