The classification of flat solvmanifolds
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- by Alexander Morgan PDF
- Trans. Amer. Math. Soc. 239 (1978), 321-351 Request permission
Abstract:
This paper contains a complete algebraic characterization of the fundamental groups of flat solvmanifolds. This characterization is in terms of finite integral representations of free abelian groups and the associated cohomology. A classification of compact flat solvmanifolds follows, and a list of all compact flat solvmanifolds of dimensions 3, 4, and 5 (except the 5-dimensional with first betti number 1) is given. Some theorems on the classification of noncompact flat solvmanifolds have also been obtained. These give full results in some cases, partial results in others. For example, the odd order holonomy group case is completely settled.References
- Louis Auslander, Some compact solvmanifolds and locally affine spaces, J. Math. Mech. 7 (1958), 963–975. MR 0101280, DOI 10.1512/iumj.1958.7.57056
- Louis Auslander, Fundamental groups of compact solvmanifolds, Amer. J. Math. 82 (1960), 689–697. MR 122910, DOI 10.2307/2372935
- Louis Auslander, Bieberbach’s theorems on space groups and discrete uniform subgroups of Lie groups, Ann. of Math. (2) 71 (1960), 579–590. MR 121423, DOI 10.2307/1969945
- Louis Auslander, An exposition of the structure of solvmanifolds. I. Algebraic theory, Bull. Amer. Math. Soc. 79 (1973), no. 2, 227–261. MR 486307, DOI 10.1090/S0002-9904-1973-13134-9
- L. Auslander and M. Auslander, Solvable Lie groups and locally euclidean Riemann spaces, Proc. Amer. Math. Soc. 9 (1958), 933–941. MR 103240, DOI 10.1090/S0002-9939-1958-0103240-X
- L. Auslander and M. Kuranishi, On the holonomy group of locally Euclidean spaces, Ann. of Math. (2) 65 (1957), 411–415. MR 86341, DOI 10.2307/1970053
- L. Auslander and R. H. Szczarba, Characteristic classes of compact solvmanifolds, Ann. of Math. (2) 76 (1962), 1–8. MR 141136, DOI 10.2307/1970261
- L. Auslander and R. H. Szczarba, Vector bundles over tori and noncompact solvmanifolds, Amer. J. Math. 97 (1975), 260–281. MR 383443, DOI 10.2307/2373671
- L. Auslander and R. H. Szczarba, On free nilmanifolds and their associated non compact solvmanifolds, Duke Math. J. 42 (1975), 357–369. MR 379751, DOI 10.1215/S0012-7094-75-04234-9
- L. Auslander and R. Tolimieri, Splitting theorems and the structure of solvmanifolds, Ann. of Math. (2) 92 (1970), 164–173. MR 276995, DOI 10.2307/1970700
- Ludwig Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume, Math. Ann. 70 (1911), no. 3, 297–336 (German). MR 1511623, DOI 10.1007/BF01564500
- Ludwig Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich, Math. Ann. 72 (1912), no. 3, 400–412 (German). MR 1511704, DOI 10.1007/BF01456724
- Leonard S. Charlap, Compact flat riemannian manifolds. I, Ann. of Math. (2) 81 (1965), 15–30. MR 170305, DOI 10.2307/1970379
- L. S. Charlap and A. T. Vasquez, Compact flat Riemannian manifolds. II. The cohomology of $Z_{p}$-manifolds, Amer. J. Math. 87 (1965), 551–563. MR 189064, DOI 10.2307/2373062
- L. S. Charlap and A. T. Vasquez, Compact flat riemannian manifolds. III. The group of affinities, Amer. J. Math. 95 (1973), 471–494. MR 336658, DOI 10.2307/2373726
- Claude Chevalley, On the topological structure of solvable groups, Ann. of Math. (2) 42 (1941), 668–675. MR 4632, DOI 10.2307/1969255
- Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
- Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
- A. I. Mal′cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949), 9–32 (Russian). MR 0028842 A Morgan, The classification of flat solvmanifolds, Ph.D. Dissertation, Yale Univ., 1975.
- G. D. Mostow, Factor spaces of solvable groups, Ann. of Math. (2) 60 (1954), 1–27. MR 61611, DOI 10.2307/1969700
- Morris Newman, Integral matrices, Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972. MR 0340283
- George B. Seligman, On two-dimensional algebraic groups, Scripta Math. 29 (1973), no. 3-4, 453–465. MR 409489
- Ken ichi Tahara, On the finite subgroups of $\textrm {GL}(3,\,Z)$, Nagoya Math. J. 41 (1971), 169–209. MR 272910, DOI 10.1017/S002776300001415X
- Joseph A. Wolf, Spaces of constant curvature, 5th ed., Publish or Perish, Inc., Houston, TX, 1984. MR 928600
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 239 (1978), 321-351
- MSC: Primary 22E25
- DOI: https://doi.org/10.1090/S0002-9947-1978-0470138-4
- MathSciNet review: 0470138