DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Coefficient normalized morphism of ordinary homology theories

Definition

For ordinary theories $h, k$ as in Unreduced homology theory on cw pairs, a morphism $η : h \to k$ consists of homomorphisms $η_{n} (X, A) : h_{n} (X, A) \to k_{n} (X, A)$ , natural for all maps of CW pairs and all $n \in Z$ , satisfying $\partial^{k} η_{n} = η_{n - 1} \partial^{h}$ .

For a specified homomorphism $u : h_{0} (*) \to k_{0} (*)$ , the morphism is coefficient-normalized by $u$ if $η_{0} (*) = u$ . A comparison equivalence has every component invertible and is normalized by a specified coefficient isomorphism. Neither the existence nor uniqueness of such an extension is part of this definition.

Depends on

Unreduced homology theory on cw pairs

Used by

Dependency tree · two levels

2 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

May, A Concise Course in Algebraic Topology, 15§2, pp.119–120, natural comparison and boundary compatibility (standard reference, not scraped)
Hatcher, Algebraic Topology, Axioms for Homology, p.161, uniqueness with coefficients (standard reference, not scraped)