Geometric realization of a finite subgroup of . II

Author:
Kyung Bai Lee

Journal:
Proc. Amer. Math. Soc. **87** (1983), 175-178

MSC:
Primary 57S17

DOI:
https://doi.org/10.1090/S0002-9939-1983-0677256-3

MathSciNet review:
677256

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a closed aspherical manifold with a virtually -step nilpotent fundamental group. Then any finite group of homotopy classes of self-homotopy equivalences of can be realized as an effective group of self-homeomorphisms of if and only if there exists a group extension of by realizing so that , the centralizer of in , is torsion-free. If this is the case, the action is equivalent to an affine action on a complete affinely flat manifold homeomorphic to . This generalizes the same result for flat Riemannian manifolds.

**[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*(to appear). MR**704219 (84k:57017)****[L1]**K. B. Lee,*Geometric realization of*, 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**-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**-Mannigfaltigkeiten*, Math. Ann.**240**(1979), 41-52. MR**524001 (80h:57025)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
57S17

Retrieve articles in all journals with MSC: 57S17

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0677256-3

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