Abstract
Let M be a compact manifold and A a finite subgroup of the outer automorphism group Out π1(M) of π1(M). A necessary condition for realising A by an isomorphic group of homeomorphisms of M is the existence of an extension to the abstract kernel (A, π1(M), Ω: A→Out π1(M)). If the center of π1(M) is nontrivial this condition need not be fulfilled. We show however, that we can find a finite group B with a normal abelian subgroup C with B/C≅A, and such that there exists an extension to the abstract kernel (B, π1(M), Ω′: B→A→Out π1(M)). In the case of Seifert fiber spaces or flat Riemannian manifolds B can be ralized by an isomorphic group of homeomorphisms of M.
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Zimmermann, B. Über Gruppen von Homöomorphismen Seifertscher Faserräume und Flacher Mannigfaltigkeiten. Manuscripta Math 30, 361–373 (1979). https://doi.org/10.1007/BF01301256
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DOI: https://doi.org/10.1007/BF01301256