Topological consequences of classifying maps to flat manifolds

Ascertain whether the existence of a classifying map from a closed cohomologically symplectic manifold M to a closed flat manifold N = K(pi_1(M), 1), obtainable when the fundamental group pi_1(M) is torsion-free (hence a Bieberbach group), imposes any additional topological constraints on M beyond those already established, particularly in the context of non-negative Ricci curvature considered in the paper.

Background

In Section 5, after proving Theorem newnewnew1, the authors observe that for a closed c-symplectic manifold with non-negative Ricci curvature and torsion-free fundamental group, the fundamental group is a Bieberbach group. Consequently, the classifying map of the universal cover can be realized as a map to a closed flat manifold with the same fundamental group.

They then raise the issue of whether the mere existence of such a classifying map to a flat manifold yields further topological implications for M, beyond those already deduced via category and curvature arguments.