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Diagrammatically reducible complexes and Haken manifolds

Part of: Topological manifolds Special aspects of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  09 April 2009

J. M Corson  and
B. Trace
B. Trace
Affiliation:
Department of Mathematics University of AlabamaBox 870350 Tuscaloosa, AL 35487-0350USA e-mail: jcorson@mathdept.as.ua.edu e-mail: btrace@mathdept.as.ua.edu
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Abstract

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We 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 if n ≥ 4 and M is compact connected n-dimensional manifold with a traingulation, or a spine, satisfying this property, then the interior of the universal cover of M is homeomorphic to Euclidean n-space.

Keywords

Diagrammatically reducibleManifoldHaken manifoldSpinecollapsecovering spacetwo-complex

MSC classification

Secondary: 57M20: Two-dimensional complexes 57N10: Topology of general $3$-manifolds 20F06: Cancellation theory; application of van Kampen diagrams
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 1 , August 2000 , pp. 116 - 126
Copyright
Copyright © Australian Mathematical Society 2000

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