Manifolds that fail to be co-dimension 2 fibrators necessarily cover themselves

Abstract

AbstractLetNbe a closeds-Hopfiann-manifold with residually finite, torsion free π1(N) and finiteH1(N). Suppose that either πk(N) is finitely generated for allk≥ 2, or πk(N) ≅ 0 for 1 <k<n– 1, orn≤ 4. We show that ifNfails to be a co-dimension 2 fibrator, thenNcyclically covers itself, up to homotopy type.