$\mathbb{S}ol^3\times\mathbb{E}^1$-manifolds

TL;DR

This paper classifies 4-manifolds modeled on the product of Sol^3 and Euclidean 1-space, showing they are Seifert fibered with torus fibers over specific flat 2-orbifolds.

Contribution

It provides a classification of Sol^3×E^1-manifolds, demonstrating their Seifert fibered structure and identifying possible base orbifolds.

Findings

Sol^3×E^1-manifolds are Seifert fibered with torus fibers.

Base orbifolds are among seven specific flat 2-orbifolds.

Outline of a classification scheme for these 4-manifolds.

Abstract

We show that $\mathbb{S}ol^3\times\mathbb{E}^1$-manifolds are Seifert fibred, with general fibre the torus, and base one of the seven flat 2-orbifolds $T, Kb, \mathbb{A}, \mathbb{M}b, S(2,2,2,2), P(2,2)$ or $\mathbb{D}(2,2)$, and outline a classification of such 4-manifolds.