On compacta of cohomological dimension one over nonabelian groups (Q2716451)

Let \(G\) be a (not necessarily abelian) group. The cohomological dimension of a compactum \(X\) with respect to \(G\) is of \(\dim_GX\leq 1\) provided that every map of a closed subset of \(X\) to the Eilenberg-MacLane complex \(K(G,1)\) admits a continuous extension over the whole space \(X\). \textit{A. N. Dranishnikov} and the second author [Topology Appl. 74, No. 1-3, 123-140 (1996; Zbl 0873.55001)] started an investigation of this kind of cohomological dimension in order to attack the cell-like mapping problem of 4-dimensional manifolds. They introduced several kinds of interesting classes of compacta. Let \(T=(S^1\times S^1)\smallsetminus \text{Int} B\) be a torus with a hole and we denote the boundary by \(\partial T\). A compactum \(X\) is said to be nonabelian if for every closed subset \(A\) of \(X\) and every map \(f:A\to \partial T\) there exists a map \(\overline f:X\to T\) such that \(\overline f|_A=f\). A compactum \(Y\) is Cainian if \(\dim_\Pi Y\leq 1\) for every perfect group \(\Pi\). Let \(M^*\) be the minimal grope [see \textit{R. J. Daverman}'s book ``Decomposition of manifolds'', Pure Appl. Math. Academic Press 124 (1986; Zbl 0608.57002), Chapter 7]. A compactum \(X\) is a Cannon-Shtan'ko compactum if \(\dim_{\pi_1 (M^*)}Z\leq 1\). It was shown that every Cainian compactum is at most 2-dimensional and every 2-dimensional nonabelian compactum is Cainian and an \(n\)-dimensional Cannon-Shtan'ko compactum for every \(n\geq 1\) was constructed. In this paper the authors construct a 2-dimensional Cannon-Shtan'ko compactum which is not nonabelian. However we do not know whether for \(n\geq 3\) there exists an \(n\)-dimensional Cannon-Shtan'ko compactum which is not nonabelian.