Nielsen numbers for roots of maps of aspherical manifolds (Q1919873)

Let \(f:X \to Y\) be a map of closed orientable manifolds of the same dimension, and let \(a \in Y\). The topological degree of \(f\) is an algebraic count of the number of solutions to \(f(x) = a\), but not an actual count. The Nielsen number \(N(f,a)\) of roots is an actual lower bound for the number of solutions. We investigate conditions under which \(N(f,a) = |\text{degree }f|\). Our question is analogous to the question in fixed point theory: when is the Lefschetz number equal to the fixed point Nielsen number? We find equality when \(X = Y\) is an aspherical manifold whose fundamental group satisfies the ascending chain condition on normal subgroups, or if \(X\) and \(Y\) are aspherical manifolds with virtually polycyclic fundamental groups. This includes infrasolvmanifolds. Similar results are obtained for nonorientable manifolds by considering their orientable double covers.