Finite group actions on Kervaire manifolds
Abstract
The (4k+2)-dimensional Kervaire manifold is a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product of two (2k+1)-dimensional spheres. We show that a finite group of odd order acts freely on a Kervaire manifold if and only if it acts freely on the corresponding product of spheres. If the Kervaire manifold M is smoothable, then each smooth structure on M admits a free smooth involution. If k + 1 is not a 2-power, then the Kervaire manifold in dimension 4k+2 does not admit any free TOP involutions. Free "exotic" (PL) involutions are constructed on the Kervaire manifolds of dimensions 30, 62, and 126. Each smooth structure on the 30-dimensional Kervaire manifold admits a free Z/2 x Z/2 action.
Cite
@article{arxiv.1305.6546,
title = {Finite group actions on Kervaire manifolds},
author = {Diarmuid Crowley and Ian Hambleton},
journal= {arXiv preprint arXiv:1305.6546},
year = {2017}
}
Comments
36 pages. Final version: Advances in Mathematics (to appear)