Chain functors with isomorphic homology

Abstract

Every chain functor ${\bf K}_{*}$ determines a homology theory on a given category of topological spaces resp. of spectra $H_{*}(\bf K_{*})(\cdot)$ cf. $\S$ 4. If $\bf K_{*}$, ${\bf L}_{*}$ are chain functors such that $H_{*}({\bf K}_{*})(\cdot) \approx H_{*}({\bf L}_{*})(\cdot)$ then there exists a third chain functor ${\bf C}_{*}$ and transformations of chain functors ${}^{K}\gamma :{\bf K}_{*} \longrightarrow {\bf C}_{*}$, ${}^{L}\gamma:\ {\bf L}_{*} \longrightarrow {\bf C}_{*}$ inducing isomorphisms of the associated homology theories (theorem 1.1.). Moreover the distinction between regular and irregular chain functors is introduced.