On cohomology of crystallographic groups with cyclic holonomy of split type. (English) Zbl 1267.20070

Petrosyan, Nansen; Putrycz, Bartosz

J. Algebra 367, 237-246 (2012).

Summary: We disprove a conjecture stating that the integral cohomology of any \(n\)-dimensional crystallographic group \(\mathbb Z^n\rtimes\mathbb Z_m\) admits a decomposition: \[ H^*(\mathbb Z^n\rtimes\mathbb Z_m)\cong\bigoplus_{i+j=*}H^i(\mathbb Z_m,H^j(\mathbb Z^n)) \] 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.

MSC:

20H15	Other geometric groups, including crystallographic groups
20J06	Cohomology of groups
53C29	Issues of holonomy in differential geometry
55T10	Serre spectral sequences
57N16	Geometric structures on manifolds of high or arbitrary dimension

Keywords:

crystallographic groups; group cohomology; spectral sequences; integral cohomology; holonomy

Software:

GAP; ccgch; CARAT

References:

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