Involutions on sapphire Sol 3-manifolds and the Borsuk-Ulam theorem for maps into $R^n$

TL;DR

This paper classifies free involutions on sapphire Sol 3-manifolds and determines the Borsuk-Ulam property for maps into R^n, focusing on dimensions 2 and 3, with implications for topological symmetry and mapping properties.

Contribution

It provides a complete classification of free involutions on sapphire Sol 3-manifolds and characterizes the Borsuk-Ulam property for these manifolds in low dimensions.

Findings

Classified all free involutions on sapphire Sol 3-manifolds.

Determined the Borsuk-Ulam property for maps into R^2 and R^3.

Showed the property does not hold for n>3 regardless of involution.

Abstract

For each sapphire Sol $3$-manifold, we classify the free involutions. For each triple $(M, \tau; R^n)$ where $M$ is a sapphire Sol $3$-manifold and $\tau$ is a free involution, we show if $(M, \tau; R^n)$ has the Borsuk-Ulam property or not. It is known that for $n>3$ the Borsuk-Ulam property does not hold independent of the involution, so we provide a classification when $n=2$ and $3$.