Abstract.
If a finite group acts freely on a homology 3-sphere, then it has periodic cohomology. To say that a finite group F has periodic cohomology is equivalent to say that any Sylow subgroup of F of odd order is cyclic and a Sylow 2-subgroup of F is either cyclic or a quaternion group. In this paper we consider more generally smooth actions of finite groups G on homology 3-spheres which may have fixed points. We prove that any Sylow subgroup of G of odd order is either cyclic or the direct sum of two cyclic groups. Moreover, we show that if G has odd order, then it splits as a semidirect product of a subgroup A and a normal subgroup B such that B acts freely and there exist some simple closed curves in the homology 3-sphere which are fixed pointwise by some non-trivial element of A. We discuss the relation between these algebraic results and some classical constructions of the theory of 3-manifolds.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received 25 September 1997; in revised form 2 June 1998
Rights and permissions
About this article
Cite this article
Reni, M. Finite Group Actions with Fixed Points on Homology 3-Spheres. Mh Math 128, 23–33 (1999). https://doi.org/10.1007/PL00010081
Issue Date:
DOI: https://doi.org/10.1007/PL00010081