Algebraic Criteria to Decide if a Finite Group Acts Effectively on a Model Aspherical Manifold
✍ Scribed by Wim Malfait
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 204 KB
- Volume
- 206
- Category
- Article
- ISSN
- 0021-8693
No coin nor oath required. For personal study only.
✦ Synopsis
For an aspherical manifold M, arising from a Seifert fiber space construction, it is known that, under some additional conditions on M, a finite abstract kernel : Ž Ž .. Ž . FªOut M can be effectively geometrically realized by a group of fiber 1 preserving homeomorphisms of M if and only if can be realized by an Ž . Ž . admissible group extension 1 ª M ª E ª F ª 1. Hence, the study of the 1
Ž
. symmetry of such a manifold in terms of finite effective actions on M can be Ž . converted into a group-theoretical study of realizing algebraically finite abstract Ž Ž .. kernels F ª Out M . This question, conceptually, is well understood: there 1 exists an extension realizing a given abstract kernel if and only if the corresponding third cohomology class, called the obstruction, vanishes. Unfortunately, a straightforward computation of this obstruction can be extremely hard. In this paper, we present some criteria which solve this problem for certain finite abstract kernels. Instead of using cohomological arguments, a completely different, rather technical and computational approach, based on the Reidemeister᎐Schreier method for presenting subgroups of finite index in a given finitely presented group, is followed. Not only the actual results but also this approach is of interest since it certainly allows one to produce similar criteria for other finite groups.