Dihedral manifold approximate fibrations over the circle

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 _∞ 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 orthogonal actions of finite groups on Euclidean space.