Nielsen numbers of iterates and Nielsen type periodic numbers of periodic maps on tori and nilmanifolds (Q1704384)

The classical Nielsen number \(N(f)\) of a selfmap \(f: M\to M\) on a compact manifold \(M\) is a sharp lower bound for the number of fixed points of maps in the homotopy class of \(f\), provided \(M\) is not a \(2\)-manifold with \(\chi(M)<0\). The computation of \(N(f)\) is notoriously difficult in general. For \(M\) a torus or a nilmanifold, \(N(f)=|L(f)|\) where \(L(f)\) is the Lefschetz number of \(f\). Since \(L(f)\) is a homological invariant, the computation of \(N(f)\) in these cases amounts to computations involving matrices and their determinants. By fibering a nilmanifold by tori over a lower dimensional nilmanifold, the Nielsen number is the product of the Nielsen numbers of the corresponding maps on the fiber and on the base. Such fiberwise techniques furnish the inductive step in the computation. Generalizing to periodic points, i.e., fixed points of the iterates, \(N(f)\) can also be generalized to similar Nielsen type invariants \(NP_n(f)\) and \(N\Phi_n(f)\), which are homotopy lower bounds for the minimal number of periodic points of least period \(n\) and of periodic points of period \(n\), respectively. \textit{J. Jezierski} [Topology 42, No. 5, 1101--1124 (2003; Zbl 1026.55001)] showed that for PL-manifolds of dimension \(\dim M\geq 3\), \(NP_n(f)\) and \(N\Phi_n(f)\) are indeed sharp lower bounds. In this paper, formulas for these invariants are given when \(M\) is a torus or a nilmanifold. Many of these formulas involve or are derived from Möbius inversion and properties of prime decomposition of integers.