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

1.1L1L2givenalgebra

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 $n \in Z$ .

2.1L3L4step 1.1algebra∎

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.

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
Euler class of a bounded projective complex is derived invariant and triangle additive Lemma
Quasi isomorphisms admit the roof calculus in the homotopy category Lemma
Relative homology is invariant under homotopy equivalence of arrows Proposition
Shift preserves homotopy equivalences, contractibility, and quasi-isomorphisms Proposition
The long exact sequence of relative homology for a composable pair Theorem

Dependency tree · two levels

12 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)