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

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.


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


GAP; CARAT; ccgch
Full Text: DOI arXiv


[1] Adem, A.; Pan, J., Toroidal orbifolds, Gerbes and group cohomology, Trans. Amer. Math. Soc., 358, 9, 3969-3983 (2006) · Zbl 1166.20043
[2] Adem, A.; Ge, J.; Pan, J.; Petrosyan, N., Compatible actions and cohomology of crystallographic groups, J. Algebra, 320, 1, 341-353 (2008) · Zbl 1163.20032
[3] Cartan, H.; Eilenberg, S., Homological Algebra (1956), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0075.24305
[4] Charlap, L. S., Bieberbach Groups and Flat Manifolds, Universitext (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0608.53001
[5] Degrijse, D.; Petrosyan, N., Characteristic classes for cohomology of split Hopf algebra extensions, J. Algebra, 322, 366-385 (2011) · Zbl 1242.18015
[6] Ellis, G., Homological algebra programming, (Computational Group Theory and the Theory of Groups. Computational Group Theory and the Theory of Groups, Contemp. Math., vol. 470 (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 63-74 · Zbl 1155.20051
[8] Langer, M.; Lück, W., On the group cohomology of the semi-direct product \(Z^n \rtimes_\rho Z / m\) and a conjecture of Adem-Ge-Pan-Petrosyan, J. Pure Appl. Algebra, 216, 6, 1318-1339 (2012) · Zbl 1278.20072
[9] Petrosyan, N., Cohomology of split group extensions and characteristic classes, J. Algebra, 321, 2916-2925 (2009) · Zbl 1188.20062
[10] Plesken, W.; Schulz, T., Counting crystallographic groups in low dimensions, Experiment. Math., 9, 3, 407-411 (2000) · Zbl 0965.20031
[11] Putrycz, B., Package ccgch - Implementation of twisted tensor product for crystallographic groups of split type and cyclic holonomy and implementation of checks of Conjecture 1.1 up to dimension 6 (2012)
[12] Sah, C.-H., Cohomology of split group extensions, J. Algebra, 255-302 (1974) · Zbl 0277.20071
[13] Wall, C. T.C., Resolutions for extensions of groups, Proc. Cambridge Philos. Soc., 57, 251-255 (1961) · Zbl 0106.24903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.