A chain homotopy

Depends on

• A graded morphism of chain complexes
• Chain map
• The Hom complex of chain complexes

Used by

• Homotopic maps induce the same map on singular homology Corollary
• A chain homotopy equivalence Definition
• A null-homotopic chain map Definition
• Complexes, homotopies and contractibility in an additive category Definition
• Shifted chain maps and shifted chain homotopies Definition
• Signed totalization of graded Aₘ-bimodule actions Definition
• A contractible simplicial cone Example
• A contracting homotopy for the two-term identity complex Example
• Two homotopic maps with different components Example
• FALSE: chain-homotopic maps are equal as chain maps False statement
• FALSE: mapping cone is a functor on the homotopy category with no extra data False statement
• A chain contraction makes the odd-to-even parity map invertible Lemma
• A lifted finite CW equivalence has a contractible group-ring mapping cone Lemma
• Bimodule tensor totalization respects differentials and homotopies Lemma
• Chain homotopy is compatible with addition and composition Lemma
• Extending a partial comparison homotopy by one degree Lemma
• Homotopic maps have chain-isomorphic mapping cones Lemma
• Koszul Generator Contraction Homotopy Lemma
• Lifting a morphism from an exact complex into an injective resolution Lemma
• Pushforward along a closed immersion preserves sheaf cohomology Lemma
• Simplicial chain maps carried by specified cones are chain homotopic Lemma
• The bounded projective comparison for the derived category Lemma
• The induced cohomology map is independent of the chosen injective comparison extension Lemma
• The induced homology map is independent of the chosen comparison lift Lemma
• Chain homotopy is an equivalence relation Proposition
• Cones preserve chain-homotopy equivalences of arrows Proposition
• The cone triangle of a null-homotopic map splits in the homotopy category Proposition
• The target is a strong deformation retract of the mapping cylinder Proposition
• A chain map is a homotopy equivalence exactly when its cone is contractible Theorem
• Chain-homotopic maps induce the same map on homology Theorem
• Contiguous simplicial maps induce the same map on simplicial homology Theorem
• Homotopic continuation data give chain homotopic maps Theorem
• Projective comparison maps are unique up to chain homotopy Theorem
• Refinement choices induce the same Čech map Theorem
• Termwise Hochschild homology respects bimodule chain homotopies 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)