Nielsen type numbers and homotopy minimal periods for maps on solvmanifolds with $\operatorname{Sol}_{1}^{4}$ -geometry

Table 1 Homotopy minimal periods

HPer( f )	Linearization matrix F is conjugate to
∅	or (γδ = ±1)
{1}	(γδ = 0) or
$\mathbb{N}-\{2\}$
$\mathbb{N}$	(ζ ≠ 0,±1,−2)
$\mathbb{N}-2\mathbb{N}$	(γδ ≠ 0,±1)

HPer( f )	Linearization matrix F is conjugate to
∅	or (γδ = ±1)
{1}	(γδ = 0) or
\(\mathbb{N}-\{2\}\)
\(\mathbb{N}\)	(ζ ≠ 0,±1,−2)
\(\mathbb{N}-2\mathbb{N}\)	(γδ ≠ 0,±1)