Let M4k+2 K be the Kervaire manifold: a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product S2k+1 S2k+1 of spheres. We show that a nite group of odd order acts freely on M4k+2 K if and only if it acts freely on S2k+1 S2k+1. If MK is smoothable, then each smooth structure on MK admits a free smooth involution. If k 6= 2j 􀀀 1, then M4k+2 K does not admit any free TOP involutions. Free \exotic" (PL) involutions are constructed on M30 K , M62 K , and M126 K . Each smooth structure on M30 K admits a free Z=2 Z=2 action.
Bibliographical noteDate of Acceptance: 09/06/2015
We would like to thank Bruce Williams, Jim Davis, Martin Olbermann, John Klein,
Mark Behrens and Wolfgang Steimle for useful information. We would also like to thank the referee for helpful comments and suggestions.
- Finite group actions
- Kerviare manifold
- Piecewise linear topology
- Surgery theory
- Smoothing theory