Finite group actions on Kervaire manifolds

TL;DR

This paper investigates finite group actions on Kervaire manifolds, establishing conditions for free actions, existence of free involutions, and constructing exotic involutions in specific dimensions.

Contribution

It characterizes when finite groups act freely on Kervaire manifolds and constructs explicit free involutions, including exotic examples, in certain dimensions.

Findings

Finite groups of odd order act freely iff they do so on the product of spheres.

Smoothable Kervaire manifolds admit free smooth involutions in all smooth structures.

Certain dimensions lack free TOP involutions if not a 2-power.

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.