Borsuk--Sieklucki theorem in cohomological dimension theory (Q2773367)

Investigating dimension properties of ANR-compacta, \textit{K.~Borsuk} [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 9, 685--687 (1961; Zbl 0108.35606)] and \textit{K.~Sieklucki} [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 10, 433--436 (1962); correction, ibid. 12, 695 (1964; Zbl 0119.38301)] proved that for every uncountable family \(\{X_{\alpha}\}_{\alpha\in J}\) of \(n\)-dimensional closed subsets of an \(n\)-dimensional ANR-compactum, there exist \(\alpha\neq\beta\) such that \(\dim(X_{\alpha}\cap X_{\beta})=n\).NEWLINENEWLINEIn the present paper the authors prove several cohomological versions of this theorem for certain compacta. Note that for a given abelian group \(G\), a compactum \(X\) is said to be \(\text{clc}^n_G\), if it is cohomology locally \(k\)-connected with respect to \(G\) for all \(k\leq n\). The last property means that for each point \(x\in X\) and neighborhood \(N\) of \(x\), there is a neighborhood \(M\subset N\) of \(x\) such that the homomorphism \(i^{k}_{M,N}:H^k(N;G)\to H^k(M;G)\), induced on the \(k\)-dimensional reduced Čech cohomology groups by the inclusion \(i_{M,N}:M\to N\), is trivial. Note also that an abelian group \(G\) satisfies the descending chain condition if every decreasing chain of subgroups of \(G\) stabilizes.NEWLINENEWLINEThe main version of the Borsuk--Sieklucki theorem proved in the paper says that for every abelian group \(G\) satisfying the descending chain condition and every uncountable family \(\{X_{\alpha}\}_{\alpha\in J}\) of closed subsets of a \(\text{clc}^n_G\)-compactum \(X\), where \(n\geq 1\), if \(\dim_GX=\dim_GX_{\alpha}=n\) for all \(\alpha \in J\), then \(\dim_G(X_{\alpha}\cap X_{\beta})=n\) for some \(\alpha \neq \beta\). For \(G\) being a countable principal ideal domain the above result was proved by \textit{J.~S.~Choi} and \textit{G.~Kozlowski} [Topol. Proc. 23 (Spring), 135--142 (1998; Zbl 0951.55003)]. Independently, \textit{J.~Dydak} and \textit{A.~Koyama} [Topology Appl. 113, No. 1--3, 39--50 (2001; Zbl 1094.55002)] proved it for \(G\) being an arbitrary principal ideal domain.NEWLINENEWLINEFurthermore, the authors derive the following version of the Borsuk--Sieklucki theorem. Namely, for every abelian group \(G\) and every uncountable family \(\{X_{\alpha}\}_{\alpha\in J}\) of closed subsets of a \(\text{clc}^{n+1}_{\mathbb Z}\)-compactum \(X\), where \(n\geq 1\), if \(\dim_GX=\dim_GX_{\alpha}=n\) for all \(\alpha \in J\), then \(\dim_G(X_{\alpha}\cap X_{\beta})=n\) for some \(\alpha \neq \beta\). Finally, this theorem is applied to investigation of some properties of strong cohomological dimension extending previous results of Dydak and Koyama [op. cit.].