The Nielsen type numbers for maps on a $3$-dimensional flat Riemannian manifold

Abstract

Let $f\colon M\to M$ be a self-map on a $3$-dimensional flat Riemannian $M$. We compute the Lefschetz number and the Nielsen number of $f$ by using the infra-nilmanifold structure of $M$ and the averaging formulasfor the Lefschetz numbers and the Nielsen numbers of maps on infra-nilmanifolds. For each positive integer $n$, we provide an explicit algorithm for a complete computation of the Nielsen type numbers $N\mathrm{P}_n(f)$ and $N\Phi_n(f)$ of $f^n$.