$Sol_1^4$-geometry

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group $Sol_1^4$. The maximal compact subgroup of $Isom(Sol_1^4)$ is $D_4=\mathbb Z_4\rtimes\mathbb Z_2$. We shall exhibit an infra-solvmanifold with $Sol_1^4$-geometry whose holonomy is $D_4$. This implies that all possible holonomy groups do occur; $\{1\}$, $\mathbb Z_2$ (5 families), $\mathbb Z_4$, $\mathbb Z_2\times\mathbb Z_2$ (5 families),and $\mathbb Z_4\rtimes\mathbb Z_2$ (2 families). Of course, this includes the classification of 3-dimensional infra-$Sol$ manifolds. We also show that all infra-$Sol_1^4$-manifolds are un-oriented boundaries.