Cotorsion and wild homology
An abelian group \(G\) is called cotorsion if it is a direct summand in every extension by a torsion-free group, a group \(G\) is called algebraically compact if it is a direct summand of every extension where it embeds as a pure subgroup. There is an equivalence of the notion of algebraically compact with the property of solving equations, namely \(G\) is algebraically compact if every system of equations \(x_{i}=w_{i}(f_{i},x_{i+1})\) is solvable for any sequence \(f_{1},f_{2},\ldots\) in \(G\). This last characterization generalizes to nonabelian groups: a group \(G\) is called Higman-complete if such a system of equations is solvable in \(G\). In the present paper, the authors relate these notions to a special class of topological spaces with these kinds of fundamental groups. One of the main theorems is as follows: {Theorem 4.} Let \(X\) be a space such that every sequence of elements \(g_{i}\) in the fundamental group \(G=\pi_{1}(X,x)\) has representatives \(\gamma_{i}\) of \(g_{i}\) that converge pointwise to the trivial path at \(x\). Then \(G\) is Higman-complete. As a corollary of this result, the authors describe the first homology of such spaces. A space is called an archipelago if it is homotopy equivalent to a mapping cone of a weak wedge of circles to itself. As applications of their methods, the authors prove the following: {Theorem 8.} The abelianization of any archipelago group \(G=\circledast_{i\geq 1}G_{i}/\ast_{i\geq 1}G_{i}\) is isomorphic to \(\prod_{i\geq 1}\mathbb{Z}/\bigoplus_{i\geq 1}\mathbb{Z}\) for arbitrary nontrivial groups \(G_{1},G_{2},\ldots \) of cardinality at most the continuum.
scientific article; zbMATH DE number 6808572
scientific article