Nielsen type numbers and homotopy minimal periods for maps on $3$–solvmanifolds

Abstract

For all continuous maps on $3$–solvmanifolds, we give explicit formulas for a complete computation of the Nielsen type numbers $N{P}_n\left(f\right)$ and $N{\Phi }_n\left(f\right)$. The most general cases were explored by Heath and Keppelmann [Topology Appl. 76 (1997) 217–247] and the complementary part is studied in this paper. While studying the homotopy minimal periods of all maps on $3$–solvmanifolds, we give a complete description of the sets of homotopy minimal periods of all such maps, including a correction to Jezierski, Kȩedra and Marzantowicz’s results in [Topology Appl. 144 (2004) 29–49].