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Fusion systems and group actions with abelian isotropy subgroups

Part of: Representation theory of groups

Published online by Cambridge University Press:  10 July 2013

Özgün Ünlü  and
Ergün Yalçin
Özgün Ünlü
Affiliation:
Department of Mathematics, Bilkent University, Ankara 06800, Turkey (unluo@fen.bilkent.edu.tr; yalcine@fen.bilkent.edu.tr)
Ergün Yalçin
Affiliation:
Department of Mathematics, Bilkent University, Ankara 06800, Turkey (unluo@fen.bilkent.edu.tr; yalcine@fen.bilkent.edu.tr)
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Abstract

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We prove that if a finite group G acts smoothly on a manifold M such that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly on M × × … × for some positive integers n1, …, nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres, with trivial action on homology.

Keywords

fusion systemsfree actions on manifoldsproducts of spheres

MSC classification

Secondary: 57S25: Groups acting on specific manifolds 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 3 , October 2013 , pp. 873 - 886
Copyright
Copyright © Edinburgh Mathematical Society 2013 

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