The cohomology ring of the sapphires that admit the Sol geometry

Abstract

Let [Formula: see text] be the fundamental group of a sapphire that admits the Sol geometry and is not a torus bundle. We determine a finite free resolution of [Formula: see text] over [Formula: see text] and calculate a partial diagonal approximation for this resolution. We also compute the cohomology rings [Formula: see text] for [Formula: see text] and [Formula: see text] for an odd prime [Formula: see text], and indicate how to compute the groups [Formula: see text] and the multiplicative structure given by the cup product for any system of coefficients [Formula: see text].