Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Flat bundles with solvable holonomy

Authors: William M. Goldman and Morris W. Hirsch
Journal: Proc. Amer. Math. Soc. 82 (1981), 491-494
MSC: Primary 57R15; Secondary 55R15, 57R22
MathSciNet review: 612747
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a solvable linear Lie group. We show that for every flat principal $ G$-bundle $ \xi $ over a CW-complex $ M$, there is a finite-sheeted covering space $ p:\hat M \to M$ such that $ {p^ * }\xi $ is trivial as a principal $ G$-bundle. This result is used to show that every affine manifold with solvable fundamental group has a finite covering which is parallelizable.

References [Enhancements On Off] (What's this?)

  • [1] P. Deligne and D. Sullivan, Fibrés vectoriels á groupe structural discret, C. R. Acad. Sci. Paris 281 (1975), 1081-1083. MR 0397729 (53:1587)
  • [2] M. Hirsch and W. Thurston, Foliated bundles, invariant measures and flat manifolds, Ann. of Math. (2) 101 (1975), 369-390. MR 0370615 (51:6842)
  • [3] G. Hochschild, The structure of Lie groups, Holden-Day, San Francisco, Calif., 1965. MR 0207883 (34:7696)
  • [4] J. Milnor, On the existence of a connection with curvature zero, Comment Math. Helv. 32 (1958), 215-223. MR 0095518 (20:2020)
  • [5] J. Smillie, Flat manifolds with nonzero Euler characteristic, Comment Math. Helv. 52 (1977), 453-456. MR 0461521 (57:1506)
  • [6] D. Sullivan, Conversation with W. Goldman, February, 1980.
  • [7] L Auslander and R. H. Szczarba, Characteristic classes of compact solvmanifolds, Ann. of Math. (2) 76 (1962), 1-8. MR 0141136 (25:4547)
  • [8] B. Kostant and D. Sullivan, The Euler characteristic of an affine space form is zero, Bull. Amer. Math. Soc. 81 (1975), 937-938. MR 0375341 (51:11536)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57R15, 55R15, 57R22

Retrieve articles in all journals with MSC: 57R15, 55R15, 57R22

Additional Information

Keywords: Solvable Lie group, flat bundle, covering space, affine manifold
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society