### Abstract

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.

Original language | English |
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Pages (from-to) | 88-129 |

Number of pages | 42 |

Journal | Advances in Mathematics |

Volume | 283 |

Early online date | 25 Jul 2015 |

DOIs | |

Publication status | Published - 1 Oct 2015 |

### Keywords

- Finite group actions
- Kerviare manifold
- Piecewise linear topology
- Surgery theory
- Smoothing theory

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## Cite this

Crowley, D., & Hambleton, I. (2015). Finite Group Actions on Kervaire Manifolds.

*Advances in Mathematics*,*283*, 88-129. https://doi.org/10.1016/j.aim.2015.06.010