Elsevier

Advances in Mathematics

Volume 283, 1 October 2015, Pages 88-129
Advances in Mathematics

Finite group actions on Kervaire manifolds

https://doi.org/10.1016/j.aim.2015.06.010Get rights and content
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Abstract

Let MK4k+2 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 finite group of odd order acts freely on MK4k+2 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 k2j1, then MK4k+2 does not admit any free TOP involutions. Free “exotic” (PL) involutions are constructed on MK30, MK62, and MK126. Each smooth structure on MK30 admits a free Z/2×Z/2 action.

MSC

57S25
57Q91
57R67
57R10

Keywords

Finite group actions
Kervaire manifold
Piecewise linear topology
Surgery theory
Smoothing theory

Cited by (0)

Research partially supported by NSERC Discovery Grant A4000. The authors would like to thank the Max Planck Institut für Mathematik and the Hausdorff Research Institute for Mathematics in Bonn for hospitality and support.