Compact flat manifolds with holonomy group 𝐙₂⊕𝐙₂

Rossetti, J.; Tirao, P.

doi:10.1090/S0002-9939-96-03633-7

Compact flat manifolds with holonomy group $\mathbf {Z}_2 \oplus \mathbf {Z}_2$
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by J. P. Rossetti and P. A. Tirao PDF

Proc. Amer. Math. Soc. 124 (1996), 2491-2499 Request permission

Abstract:

In this paper we construct a family of compact flat manifolds, for all dimensions $n\ge 3$, with holonomy group isomorphic to $\mathbf {Z}_2^2$ and first Betti number zero.

References

George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
Calabi E., Closed, locally euclidean, 4-dimensional manifolds, Bull. Amer. Math. Soc. 63 (1957), 135 (abstract).
Leonard S. Charlap, Bieberbach groups and flat manifolds, Universitext, Springer-Verlag, New York, 1986. MR 862114, DOI 10.1007/978-1-4613-8687-2
Peter V. Z. Cobb, Manifolds with holonomy group $Z_{2}\oplus Z_{2}$ and first Betti number zero, J. Differential Geometry 10 (1975), 221–224. MR 375147
Isabel Dotti Miatello and Roberto J. Miatello, Isospectral compact flat manifolds, Duke Math. J. 68 (1992), no. 3, 489–498. MR 1194952, DOI 10.1215/S0012-7094-92-06820-7
Howard Hiller, Cohomology of Bieberbach groups, Mathematika 32 (1985), no. 1, 55–59. MR 817108, DOI 10.1112/S002557930001086X
Howard Hiller and Chih-Han Sah, Holonomy of flat manifolds with $b_1=0$, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 146, 177–187. MR 841425, DOI 10.1093/qmath/37.2.177
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740