| Tue, Apr 23, 2024
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Topology Seminar
4:00 PM
MSCS 514 | | Classification of closed 4-manifolds with Sol × $\mathbb{R}$-geometry
Scott Thuong, Pittsburg State
| | | Abstract: Let $G$ denote the solvable Lie group $\mathrm{Sol} \times \mathbb{R}$, equipped with left invariant metric.
A crystallographic group of $G$ is a discrete subgroup
$\varPi$ of the isometry group of $G$,
$\varPi \subset \mathrm{Isom}(G)$, so that the quotient space
$\varPi \backslash G$
is compact. When $\varPi$ is torsion free, it acts freely on $G$, and the quotient
$M=\varPi \backslash G$ is a closed 4-dimensional manifold modeled on $G$, a so-called infra-solvmanifold of $G$, and $\pi_1(M) \cong \varPi$. In fact, such $M$ are determined up to
affine diffeomorphism by $\varPi$. Therefore, to classify the infra-solvmanifolds of $\mathrm{Sol} \times \mathbb{R}$, it is sufficient to find all the crystallographic groups of $\mathrm{Sol} \times \mathbb{R}$, and then determine which are torsion free.
Here we present a different approach. Every such infra-solvmanifold is Seifert fibered over a $1$-dimensional base with $3$-dimensional fiber. The fibers are $3$-dimensional flat manifolds. This insight leads to simple presentations for the fundamental groups of infra-solvmanifolds of $\mathrm{Sol} \times \mathbb{R}$. |
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