Local cohomological properties of homogeneous ANR compacta (Q2804297)

This work is motivated by the Bing-Borsuk conjecture [\textit{R. H. Bing} and \textit{K. Borsuk}, Ann. Math. (2) 81, 100--111 (1965; Zbl 0127.13302)]. This asserts that each homogeneous Euclidean neighborhood retract is a topological manifold.NEWLINENEWLINEThe notion of the \(G\)-cohomological dimension of a space arises. If \(X\) is a nonempty space and \(G\) is an abelian group, then this dimension, \(\mathrm{dim}_G X\), is an element of \(\{0,1,\dots,\infty\}\). The term \((n,G)\)-bubble comes up, so let us give its definition. A closed subset \(A\subset X\) of a space \(X\) is called a cohomological carrier of a non-zero element \(\alpha\in H^n(A;G)\) if \(j_{A,B}^n(\alpha)=0\) for every proper closed subset \(B\subset A\). Here, the homomorphism \(j_{A,B}^n\) is that induced by the inclusion \(B\hookrightarrow A\), and the cohomology is reduced Čech. If \(H^n(A;G)\neq0\), but for every proper closed subset \(B\subset A\), \(H^n(B;G)=0\), then \(A\) is called an \((n,G)\)-bubble. So in this case \(A\) is by default a cohomological carrier for every non-zero element of \(H^n(A;G)\). We will not define certain other terms that appear in the next result; these can be found in the paper.NEWLINENEWLINE\textbf{Theorem 1.1.} Let \(X\) be a homogeneous metrizable ANR continuum with \(\mathrm{dim}_G X=n\), where \(G\) is a countable PID with unity and \(n\geq2\). Then every point \(x\) of \(X\) has a basis \(\mathcal{B}_x\) of open sets \(U\subset X\) satisfying the following conditions:NEWLINENEWLINE(1) \(\mathrm{int}\,\overline U=U\) and the complement of \(\mathrm{bd}\,U\) has two components one of which is \(U\);NEWLINENEWLINE(2) \(H^{n-1}(\overline U;G)=0\) and \(\overline U\) is an \((n-1)\)-cohomology membrane spanned on \(\mathrm{bd}\,U\) for any non-zero \(\gamma\in H^{n-1} (\mathrm{bd}\,U;G)\);NEWLINENEWLINE(3) \(\mathrm{bd\, U}\) is an \((n-1,G)\)-bubble and \(H^{n-1}(\mathrm{bd}\,U;G)\) is a finitely generated \(G\)-module.NEWLINENEWLINE\textbf{Theorem 1.2.} Let \(X\) be as in Theorem 1.1 and \(G\) be a countable group. If a closed subset \(K\subset X\) is an \((n-1)\)-cohomology membrane spanned on \(A\) for some closed set \(A\subset K\) and some \(\gamma\in H^{n-1}(A;G)\), then \((K\setminus A)\cap \overline{X\setminus K}=\emptyset\).NEWLINENEWLINE\textbf{Corollary 1.3.} In the setting of Theorem 1.2, if \(U\subset X\) is open and \(f:U\to X\) is an injective map, then \(f(U)\) is open in \(X\).NEWLINENEWLINE\textbf{Theorem 1.4.} The following conditions are equivalent for any metrizable ANR compactum \(X\) of dimension \(\mathrm{dim}\,X=n\):NEWLINENEWLINE(1) \(X\) is dimensionally full-valued;NEWLINENEWLINE(2) there is a point \(x\in X\) with \(H_n(X,X\setminus x;\mathbb{Z}) \neq0\);NEWLINENEWLINE(3) \(\mathrm{dim}_{S^1} X=n\).NEWLINENEWLINE\textbf{Corollary 1.5.} Every homogeneous metrizable ANR compactum \(X\) with \(\mathrm{dim}\,X=3\) is dimensionally full-valued.NEWLINENEWLINEAll of these are proved in Sections 3 and 4. Section 2 contains other, preliminary results.