On cohomology of crystallographic groups with cyclic holonomy of split type
Abstract
We disprove a conjecture stating that the integral cohomology of any crystallographic group Zn Zm is given by the cohomology of Zm with coefficients in the cohomology of the group Zn, 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.
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