The cohomology ring of the sapphires that admit the Sol geometry

Abstract

Let $G$ 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 $\mathbb{Z}$ over $\mathbb{Z}G$ and calculate a partial diagonal approximation for this resolution. We also compute the cohomology rings $H^{\ast }(G;A)$ for $A=\mathbb{Z}$ and $A=\mathbb{Z}/p$ for an odd prime $p$, and indicate how to compute the groups $H^{\ast }(G;A)$ and the multiplicative structure given by the cup product for any system of coefficients $A$.

arXiv comment14 pages