On cohomology of crystallographic groups with cyclic holonomy of split type

TL;DR

This paper disproves a conjecture about the cohomology of crystallographic groups with cyclic holonomy, providing counterexamples up to dimension 5 and exploring cohomology in specific 6-dimensional cases.

Contribution

It offers a complete list of counterexamples to the conjecture and computes cohomology for certain 6-dimensional crystallographic groups.

Findings

Counterexamples to the conjecture up to dimension 5

Counterexample with holonomy m=9 in dimension 8

Cohomology computations for 6-dimensional groups from Calabi-Yau orbifolds

Abstract

We disprove a conjecture stating that the integral cohomology of any crystallographic group Z^n \rtimes Z_m is given by the cohomology of Z_m with coefficients in the cohomology of the group Z^n, by providing a complete list of counterexamples up to dimension 5. We also find a counterexample with odd order holonomy, m=9, in dimension 8 and finish the computations of the cohomology of 6-dimensional crystallographic groups arising as orbifold fundamental groups of certain Calabi-Yau toroidal orbifolds.