On a question of Atiyah (Q2704622)

Atiyah raised the question as to whether the \(L^2\) Betti numbers of the universal cover of a closed manifold are always rational numbers. Subsequent revisions of this question relate the possible denominators to the orders of finite subgroups of the fundamental group, and this has been confirmed in many important cases. This paper gives an example of a 7-manifold with fundamental group an extension of \(Z^2\) by the normal subgroup \((Z/2Z)^\infty\), for which the third \(L^2\) Betti number is \(1/3\), thus providing a counterexample to a strong version of Atiyah's question. (The original question remains open, as does the question as to whether the \(L^2\) Betti numbers are integers if the fundamental group is torsion free).