The cohomology ring of the sapphires that admit the Sol geometry

TL;DR

This paper computes the cohomology ring structure of fundamental groups of certain 3-manifolds with Sol geometry, providing explicit resolutions and methods for calculating cohomology with various coefficients.

Contribution

It determines a finite free resolution for these groups and develops a framework to compute their cohomology rings with different coefficient systems.

Findings

Explicit finite free resolution of $bZ$ over $bZ G$

Calculation of cohomology rings $H^*(G;A)$ for specific coefficients

Methodology for computing cup product structures

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 $H^*(G; A)$ for $A = \mathbb{Z}$ and $A = \mathbb{Z}/p$ for an odd prime $p$, and indicate how to compute the groups $H^*(G; A)$ and the multiplicative structure given by the cup product for any system of coefficients $A$.