Journal article
Finite group actions on Kervaire manifolds
D Crowley, I Hambleton
Advances in Mathematics | ACADEMIC PRESS INC ELSEVIER SCIENCE | Published : 2015
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 k≠2j-1, 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.
Grants
Awarded by Natural Sciences and Engineering Research Council of Canada
Funding Acknowledgements
Research partially supported by NSERO Discovery Grant A4000. The authors would like to thank the Max Planck Institut fur Mathematik and the Hausdorff Research Institute for Mathematics in Bonn for hospitality and support.