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Proceedings of the American Mathematical Society

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

 

 

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