Acyclic covers

Abstract

The following question, which is directly related to the Whitehead problem of subcomplexes of acyclic 2-complexes, is studied: If $\mathfrak{P}$ is a class of groups, X is a 2-dimensional CW-complex and X' is an acyclic, infinite cyclic cover of X with ${\pi }_1(X^{\prime })$ in $\mathfrak{P}$ , must X' be contractible? A positive answer is given if X is finite and $\mathfrak{P}$ is the class of amenable groups.