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Geometric realization of a finite subgroup of $ \pi \sb{0}\varepsilon (M)$. II

Author: Kyung Bai Lee
Journal: Proc. Amer. Math. Soc. 87 (1983), 175-178
MSC: Primary 57S17
MathSciNet review: 677256
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Abstract: Let $ M$ be a closed aspherical manifold with a virtually $ 2$-step nilpotent fundamental group. Then any finite group $ G$ of homotopy classes of self-homotopy equivalences of $ M$ can be realized as an effective group of self-homeomorphisms of $ M$ if and only if there exists a group extension $ E$ of $ \pi $ by $ G$ realizing $ G \to {\operatorname{Out }}{\pi _1}M$ so that $ {C_E}(\pi )$, the centralizer of $ \pi $ in $ E$, is torsion-free. If this is the case, the action $ (G,M)$ is equivalent to an affine action $ (G,M')$ on a complete affinely flat manifold homeomorphic to $ M$. This generalizes the same result for flat Riemannian manifolds.

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  • [CR] P. E. Conner and F. Raymond, Deforming homotopy equivalences to homeomorphisms in aspherical manifolds, Bull. Amer. Math. Soc. 83 (1977), 36-87. MR 0467777 (57:7629)
  • [FH] F. T. Farrell and W. C. Hsiang, Topological characterization of flat and almost flat Riemannian manifolds $ {M^n}(n \ne 3,4)$ (to appear). MR 704219 (84k:57017)
  • [L1] K. B. Lee, Geometric realization of $ {\pi _0}\varepsilon (M)$, Proc. Amer. Math. Soc. 86 (1982), 353-357. MR 667306 (84m:57026)
  • [L2] -, Seifert relatives of flat Riemannian manifolds, Ph. D. Thesis, University of Michigan, 1981.
  • [L3] -, Aspherical manifolds with virtually $ 3$-step nilpotent fundamental group, Amer. J. Math. (to appear).
  • [LR1] K. B. Lee and F. Raymond, Topological, affine and isometric actions on flat Riemannian manifolds, J. Differential Geometry 16 (1981), 255-269. MR 638791 (84k:57027)
  • [R] F. Raymond, The Nielsen theorem for Seifert fibered space over locally symmetric spaces, J. Korean Math. Soc. 16 (1979), 87-93. MR 543085 (81h:57029)
  • [Zi] B. Zimmermann, Über Gruppen von Homöomorphismen Seifertscher Faserräume und flacher Mannigfaltigkeiten, Manuscripta Math. 30 (1980), 361-373. MR 567213 (82c:57024)
  • [ZZ] H. Zieschang and B. Zimmermann, Endliche Gruppen von Abbildungsklassen gefaserter $ 3$-Mannigfaltigkeiten, Math. Ann. 240 (1979), 41-52. MR 524001 (80h:57025)

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Keywords: Geometric realization, infranilmanifold, crystallographic group, virtually nilpotent group, homotopy class of self-homotopy equivalences, affine diffeomorphism, complete affinely flat manifold
Article copyright: © Copyright 1983 American Mathematical Society

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