Localization of group rings and applications to 2-complexes
\textit{S. Rosset} [Math. Z. 185, 211-215 (1984; Zbl 0549.57010)] has shown that for a group G which has a nontrivial normal abelian torsion free subgroup (``G is an R-group'') any finite aspherical complex X with \(\pi_ 1(X)\approx G\) fulfills \(\chi (X)=0\). Rosset's technique concerns localization of group rings, and in the present paper it is applied to problems of 2-complexes. The main result generalizes Rosset's theorem to an exact sequence \(L\hookrightarrow G\twoheadrightarrow H\) with H an R-group, G finitely presented, \(H_ 1(L)\) finitely generated as an abelian group and the minimal number of elements, whose normal closure in G is L, being finite: If \(X^ 2\) is a 2-complex with \(\pi_ 1(G)\approx G\), then \(\chi (X)=0\), and \(\chi (X)=0\) holds if and only if (a) \(H_ 2(L)=0\) and (b) the image of \(\pi_ 2(X)\) in \(H_ 2\) of the covering of X corresponding to L is trivial. After the description of Rosset's method the paper gives an extension of Kaplansky's invariant to localized group rings, and then several beautiful applications to Whitehead's asphericity problem and the deficiency of groups.
scientific article; zbMATH DE number 4017913