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Local cohomological properties of homogeneous ANR compacta. (English) Zbl 1347.55002

This work is motivated by the Bing-Borsuk conjecture [R. H. Bing and K. Borsuk, Ann. Math. (2) 81, 100–111 (1965; Zbl 0127.13302)]. This asserts that each homogeneous Euclidean neighborhood retract is a topological manifold.
The 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.
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:
(1) \(\mathrm{int}\,\overline U=U\) and the complement of \(\mathrm{bd}\,U\) has two components one of which is \(U\);
(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 \(\gamma\in H^{n-1} (\mathrm{bd}\,U;G)\);
(3) \(\mathrm{bd\, U}\) is an \((n-1,G)\)-bubble and \(H^{n-1}(\mathrm{bd}\,U;G)\) is a finitely generated \(G\)-module.
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 \((K\setminus A)\cap \overline{X\setminus K}=\emptyset\).
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 \(X\).
Theorem 1.4. The following conditions are equivalent for any metrizable ANR compactum \(X\) of dimension \(\mathrm{dim}\,X=n\):
(1) \(X\) is dimensionally full-valued;
(2) there is a point \(x\in X\) with \(H_n(X,X\setminus x;\mathbb{Z}) \neq0\);
(3) \(\mathrm{dim}_{S^1} X=n\).
Corollary 1.5. Every homogeneous metrizable ANR compactum \(X\) with \(\mathrm{dim}\,X=3\) is dimensionally full-valued.
All of these are proved in Sections 3 and 4. Section 2 contains other, preliminary results.

MSC:

55M10 Dimension theory in algebraic topology
55M15 Absolute neighborhood retracts
54F45 Dimension theory in general topology
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)

Citations:

Zbl 0127.13302
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References:

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