Dihedral manifold approximate fibrations over the circle

TL;DR

This paper characterizes when certain high-dimensional manifolds admit free actions by the infinite dihedral group, linking it to equivariant fibrations over the circle and introducing a new equivariant sucking principle.

Contribution

It establishes a criterion for manifolds to admit dihedral group actions via equivariant fibrations, and develops an equivariant sucking principle for finite group actions.

Findings

Manifolds admit dihedral group actions iff they are cyclic covers of C_2-manifolds with equivariant fibrations.

Introduction of an equivariant sucking principle for finite group actions on Euclidean space.

Identification of invariant codimension-one submanifolds in the manifolds studied.

Abstract

Consider the cyclic group C_2 of order two acting by complex-conjugation on the unit circle S^1. The main result is that a finitely dominated manifold W of dimension > 4 admits a cocompact, free, discontinuous action by the infinite dihedral group D_\infty if and only if W is the infinite cyclic cover of a free C_2-manifold M such that M admits a C_2-equivariant manifold approximate fibration to S^1. The novelty in this setting is the existence of codimension-one, invariant submanifolds of M and W. Along the way, we develop an equivariant sucking principle for certain orthogonal actions of finite groups on Euclidean space.