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A chain complex and quadrilaterals for normal surfaces. (English) Zbl 1270.57061
Summary: We interpret a normal surface in a (singular) three-manifold in terms of the homology of a chain complex. This allows us to study the relation between normal surfaces and their quadrilateral coordinates. Specifically, we give a proof of an (unpublished) observation independently given by Casson and Rubinstein saying that quadrilaterals determine a normal surface up to vertex linking spheres. We also characterize the quadrilateral coordinates that correspond to a normal surface in a (possibly ideal) triangulation.
##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57Q15 Triangulating manifolds 52B70 Polyhedral manifolds
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