A contractible complex

Depends on

• A chain homotopy equivalence
• A null-homotopic chain map

Used by

• A contractible complex is acyclic Corollary
• An acyclic noncontractible cone Counterexample
• Complexes, homotopies and contractibility in an additive category Definition
• A contracting homotopy for the two-term identity complex Example
• FALSE: an acyclic mapping cone is contractible False statement
• FALSE: every acyclic complex is contractible False statement
• FALSE: every acyclic complex of projective objects is contractible False statement
• fs-a-derived-functor-is-canonical-without-supplied-replacement-data.md False statement
• A chain contraction makes the odd-to-even parity map invertible Lemma
• A degreewise split exact complex with compatible splittings is contractible Lemma
• A lifted finite CW equivalence has a contractible group-ring mapping cone Lemma
• An augmented simplicial cone has an explicit chain contraction Lemma
• The augmented simplicial chain complex of a simplex is contractible Lemma
• Shift preserves homotopy equivalences, contractibility, and quasi-isomorphisms Proposition
• Zero homology does not make an object zero in the homotopy category Proposition
• A chain map is a homotopy equivalence exactly when its cone is contractible Theorem
• The cone of an identity map is contractible Theorem

Dependency tree · two levels

5 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)