Weakly additive cohomology
A family \(\{A_ j\}_{j\in J}\) of closed subsets of a space X is called discrete if every point of X has a neighbourhood which meets at most one member of the family. A cohomology theory H,\(\delta\) is called weakly additive if the natural monomorphism \[ i:\oplus_{j\in J}H(A_ j)\to H(\cup_{j\in J}A_ j) \] is an isomorphism for all discrete families \(\{A_ j\}\) in X. The main result of this paper is the following theorem. Theorem. Let \(\phi\) : H,\(\delta\to H',\delta '\) be a homomorphism between cohomology theories both of which are weakly additive and having para- compact supports on the same space X. Suppose there is n such that \(\phi_ x: H(x)\to H'(x)\) is an n-equivalence for all \(x\in X\). If H,\(\delta\) and \(H',\delta '\) are both nonnegative or if X is locally finite dimensional, then \(\phi_ A: H(A)\to H'(A)\) is an n-equivalence for all closed \(A\subset X\).
homomorphism between cohomology theories
scientific article; zbMATH DE number 4161675