I will discuss the extension of free finite cyclic  group actions on the Brieskorn integral homology 3-spheres to their canonical smooth 4-manifold resolutions. [joint with Nima Anvari]
Every finite group can act freely and smoothly on some closed, oriented 3-manifold M with the rational homology of the 3-sphere. However, if the action induces the identity on the integral homology of M, then new group theoretic restrictions arise.  [joint  with Alejandro Adem]
  

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