Connection between continuous and digital n-manifolds and the Poincare conjecture

2006/08/24 by Alexander Evako, Evako, Alexander
Computer Science · Mathematics · #Algebraic Topology (math.AT) #Computer Vision and Pattern Recognition (cs.CV) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #cs.CV #cs.DM #math.AT

paper · pdf · doi:10.48550/arxiv.cs/0608093

39 pages, 26 figures

arxiv created 2006/08/24 · arxiv updated 2009/12/01

Abstract

We introduce LCL covers of closed n-dimensional manifolds by n-dimensional disks and study their properties. We show that any LCL cover of an n-dimensional sphere can be converted to the minimal LCL cover, which consists of 2n+2 disks. We prove that an LCL collection of n-disks is a cover of a continuous n-sphere if and only if the intersection graph of this collection is a digital n-sphere. Using a link between LCL covers of closed continuous n-manifolds and digital n-manifolds, we find conditions where a continuous closed three-dimensional manifold is the three-dimensional sphere. We discuss a connection between the classification problems for closed continuous three-dimensional manifolds and digital three-manifolds.

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