Cohomology with supports (Q1083064)

In this paper the study of cohomology theories, on a Hausdorff space X, introduced by the author in two previous papers [Contemp. Math. 12, 315- 329 (1982; Zbl 0518.55003); ''Cohomology theory on spaces'' (to appear)] is continued. If \(\Phi\) is a family of supports on X and H,\(\delta\) is a cohomology theory on X, then H,\(\delta\) has supports in \(\Phi\) if given \(u\in H(A)\) there exist B closed, \(C\in \Phi\) with \(A=B\cup C\) and \(u/B=0.\) If H,\(\delta\) and H',\(\delta\) ' are cohomology theories on the same space X, a homomorphism \(\phi\) from H,\(\delta\) to H',\(\delta\) ' is a natural transformation from H to H' commuting up to sign with \(\delta\) and \(\delta\) '. An ES theory [see the second op. cit. above] has supports in a family \(\Phi\) if the corresponding cohomology theory has supports in \(\Phi\). A cohomology theory H,\(\delta\) on a space X is said to be concentrated on a subset \(Y\subset X\) if \(H(A)=0\) for all closed \(A\subset X-Y\). An ES theory is said to be concentrated on Y if the corresponding cohomology theory is concentrated on Y. The paper is very dense, containing many major results. From these results we quote the following: a uniqueness theorem for two cohomology theories with the same family of supports, a characterization of cohomology with supports in suitable families in terms of limit properties, the construction of cohomology theories on a space X with supports in a given family \(\Phi\) from an ES theory on X, the relation between cohomology theories on X and on open subsets of X, a bijection between cohomology theories on X concentrated on an open set Y with supports in a suitable family \(\Phi\) /Y, a study of the particular cases of compact and paracompact supports, the proof of the fact that cohomology theories on a locally compact (locally paracompact) space X with compact (relatively paracompact) supports correspond to cohomology theories on the one-point compactification (paracompactification) \(X^+\) which are concentrated on X, a uniqueness theorem for additive cohomology theories with paracompact supports on finite dimensional normal spaces. All necessary notions, including those previously introduced by the author, are given in the paper.