Acyclic covers
Whitehead's asphericity question, cf. e.g. [\textit{R. Brown} and \textit{J. Huebschmann}, Identities among relations, in `Low-dimensional topology', Lond. Math. Soc. Lect. Note Ser. 48, 153-202 (1982; Zbl 0485.57001)], asks whether any subcomplex of an aspherical 2-complex is itself aspherical. Given a class \(\mathcal P\) of groups containing the trivial group, the acyclic cover questions asks whether, under the assumption that \(X'\) be acyclic, \(\pi_1(X')\) is in \(\mathcal P\) only if \(X'\) is contractible. In the paper under review, it is proved that, given a class of groups \(\mathcal P\), if there exists a subcomplex \(L\) of an aspherical 2-complex \(K\) which is not aspherical, and if the maximal perfect subgroup of \(\pi_1(L)\) is in \(\mathcal P\), there exists a regular, acyclic, infinite cyclic cover \(Y'\) of some 2-complex \(Y\) having \(\pi_1(Y')\) in \(\mathcal P\). Hence a positive answer to the acyclic cover question would limit the maximal perfect subgroups that could appear in any counterexample to the asphericity question. In the paper, some evidence for a positive answer to the acyclic cover question is given for the class of finite groups. More precisely, it is proved that, given a regular infinite cyclic cover \(X'\) of a finite 2-complex \(X\), when \(X'\) is acyclic and has \(\pi_1(X')\) amenable, it is necessarily aspherical; hence when \(\pi_1(X')\) is finite, it is necessarily trivial and \(X'\) is contractible. (A group is amenable if every action thereof on a compact metric space has an invariant Borel measure.).
scientific article; zbMATH DE number 1333318