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

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 \(3\).