Functorial dependence between homology and cohomology groups (Q1081884)

Given two functors \(F_ 1,F_ 2: {\mathcal A}\to {\mathcal B}\) between categories \({\mathcal A}\) and \({\mathcal B}\), we say that \(F_ 1\) depends functorially on \(F_ 2\), \(F_ 1\Rightarrow F_ 2\), if for every morphism f for which \(F_ 2(f)\) is an isomorphism, \(F_ 1(f)\) is also an isomorphism. Let \({\mathcal L}\) denote the category of locally compact spaces satisfying the second axiom of countability, and continuous mappings among them. Let \(H^*(-;G)\) (resp. \(H^ c_*(-;G))\) denote the ordinary Alexander-Spanier cohomology (resp. the Steenrod-Sitnikov homology with compact supports). The main result of the paper is that for every module G over a countable principal ideal ring R, the following functorial dependencies are in effect in the category \({\mathcal L}:\) (1) \(H^*(-;R)\Rightarrow H^ c_*(-;G)\); (2) \(H^ c_*(-;R)\Rightarrow H^*(-;R)\); and (3) \(H^*(-;R)\Rightarrow H^*(-;G)\). The proof is based on the algebraic construction of the cone of a chain map.