Diagrammatically reducible complexes and Haken manifolds

Abstract

AbstractWe show that diagrammatically reducible two-complexes are characterized by the property: every finity subconmplex of the universal cover collapses to a one-complex. We use this to show that a compact orientable three-manifold with nonempty boundary is Haken if and only if it has a diagrammatically reducible spine. We also formulate an nanlogue of diagrammatic reducibility for higher dimensional complexes. Like Haken three-manifolds, we observe that ifn≥ 4 andMis compact connectedn-dimensional manifold with a traingulation, or a spine, satisfying this property, then the interior of the universal cover ofMis homeomorphic to Euclideann-space.