Abstract
We disprove a conjecture stating that the integral cohomology of any n-dimensional crystallographic group Zn⋊Zm admits a decomposition:H⁎(Zn⋊Zm)≅⊕i+j=⁎Hi(Zm,Hj(Zn)) by providing a complete list of counterexamples up to dimension 6. This finishes the computations of the cohomology of 6-dimensional crystallographic groups arising as orbifold fundamental groups of certain Calabi–Yau toroidal orbifolds. We also find a counterexample with odd order holonomy, m=9, in dimension 8.
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