Rossetti, J. P.; Tirao, P. A. Compact flat manifolds with holonomy group \(\mathbb Z_2\oplus\mathbb Z_2\). II. (English) Zbl 0970.53029 Rend. Semin. Mat. Univ. Padova 101, 99-136 (1999). The classification of compact flat manifolds (up to affine equivalence) is equivalent, after Bieberbach’s results, to the classification of their fundamental groups (up to isomorphism). In this paper the authors give a complete classification of compact flat manifolds with holonomy group \({\mathbb Z}_2\oplus{\mathbb Z}_2\), with the property that the holonomy representation is a direct sum of \({\mathbb Z}_2\oplus{\mathbb Z}_2\)-indecomposable representations of \({\mathbb Z}\)-rank equal to 1 or 2. The compact flat manifolds in this class which have first Betti number zero are exactly those considered in [the authors, Proc. Am. Math. Soc. 124, No. 8, 2491–2499 (1996; Zbl 0864.53027)]. Reviewer: Pascual Lucas Saorín (Murcia) Cited in 2 Documents MSC: 53C29 Issues of holonomy in differential geometry 57S30 Discontinuous groups of transformations 53C20 Global Riemannian geometry, including pinching 57M05 Fundamental group, presentations, free differential calculus Keywords:fundamental group; compact flat manifold; holonomy group; Betti number Citations:Zbl 0864.53027 PDFBibTeX XMLCite \textit{J. P. Rossetti} and \textit{P. A. Tirao}, Rend. Semin. Mat. Univ. Padova 101, 99--136 (1999; Zbl 0970.53029) Full Text: Numdam EuDML References: [1] H. Brown - R. Bülow - J. Neub - H. Wondratschok - H. Zassenhaus , Crystallografic Groups of Four-Dimensional Space , Wiley , New York ( 1978 ). · Zbl 0381.20002 [2] L. Charlap , Bieberbach Groups and Flat Manifolds , Springer-Verlag , New York ( 1986 ). MR 862114 | Zbl 0608.53001 · Zbl 0608.53001 [3] L. Charlap , Compact Flat Manifolds: I , Ann. of Math. , 81 ( 1965 ), pp. 15 - 30 . MR 170305 | Zbl 0132.16506 · Zbl 0132.16506 · doi:10.2307/1970379 [4] P. Cobb , Manifolds with holonomy group Z2 \oplus Z2 and first Betti number zero , J. Differential Geometry , 10 ( 1975 ), pp. 221 - 224 . Zbl 0349.53027 · Zbl 0349.53027 [5] I. Dotti Miatello - R. Miatello , Isospectral compact flat manifolds , Duke Mathematical Journal , 68 ( 1992 ), pp. 489 - 498 . Article | MR 1194952 | Zbl 0781.53032 · Zbl 0781.53032 · doi:10.1215/S0012-7094-92-06820-7 [6] D. Heisler , On the cohomology of modules over the Klein group, Ph. D. Thesis, State University of New York at Stony Brook ( 1973 ). [7] H. Hiller - C. Sah , Holonomy offlat manifolds with \beta 1= 0 , Quart. J. Math. Oxford ( 2 ), 37 ( 1986 ), pp. 177 - 187 . Zbl 0598.57014 · Zbl 0598.57014 · doi:10.1093/qmath/37.2.177 [8] P. Hindman - L. Klingler - C.J. Odenthal , On the Krull-Schmidt-Azumaya theorem for integral group rings , Comm. in Algebra , to appear. MR 1647074 | Zbl 0914.20003 · Zbl 0914.20003 · doi:10.1080/00927879808826371 [9] L. Nazarova , Integral representations of the Klein’s four group , Dokl. Akad. Nauk. SSSR , 140 ( 1961 ), p. 1011 ; English transl. in Soviet Math. Dokl. , 2 ( 1961 ) p. 1304 . MR 130916 | Zbl 0106.02602 · Zbl 0106.02602 [10] I. Reiner , Integral representations of cyclic groups of prime order , Proc. Amer. Math. Soc. , 8 ( 1957 ), pp. 142 - 146 . MR 83493 | Zbl 0077.25103 · Zbl 0077.25103 · doi:10.2307/2032829 [11] I. Reiner , Failure of the Krull-Schmidt theorem for integral representations , Michigan Math. Jour. , 9 ( 1962 ), pp. 225 - 231 . Article | MR 144942 | Zbl 0106.02504 · Zbl 0106.02504 · doi:10.1307/mmj/1028998721 [12] J. Rossetti - P. Tirao , Compact Flat Manifolds with Holonomy Group Z2\oplus Z2 , Proc. Amer. Math Soc. , 124 ( 1996 ), pp. 2491 - 2499 . Zbl 0864.53027 · Zbl 0864.53027 · doi:10.1090/S0002-9939-96-03633-7 [13] H. Zassenhaus , Uber einen Algorithmus zur Bestimmung der Raumgruppen , Comment. Math. Helvetici , 21 ( 1948 ), pp. 117 - 141 . MR 24424 | Zbl 0030.00902 · Zbl 0030.00902 · doi:10.1007/BF02568029 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.