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Author(s)
Alexander V. Evako, Npk Novotek, Moscow, Russia
Abstract
In the present paper, we study basic properties of digital ndimensional manifolds and digital simply connected spaces. An important property of a digital nmanifold is that M is a digital nsphere if and only if for any point v of M, Mv is a digital ndisk. It is proved that a digital (n1)sphere S contained a digital nsphere M is a separating space of M. We show that a digital nmanifold can be converted to the compressed form by sequential contractions of simple pairs of adjacent points. We study structural features of digital simply connected spaces. In particular, we show that a digital (n1)sphere S in a digital simply connected nmanifold M is a separating space for M, and if a digital 3manifold M is locally simply connected, then M is a digital 3sphere.
Keywords
Digital Topology; Topological space; Separation; Digital simply connected space
Cite this paper
Alexander V. Evako,
Properties of Digital nDimensional Spheres and Manifolds. Separation of Digital Manifolds, SCIREA Journal of Mathematics. Vol.
3
, No.
1
,
2018
, pp.
29

56
.
References
[ 1 ]  L. Boxer, Continuous Maps on Digital Simple Closed Curves, Applied Mathematics. 1 (2010) 377386. 
[ 2 ]  HD. Cao, XP. Zhu, “A complete proof of the Poincaré and Geometrization conjectures – application of the Hamiltonperelman theory of the Ricci flow”, Asian J. Math., Vol. 10, No. 2, pp. 165–492, 2006. 
[ 3 ]  Li Chen,Yongwu Rong, Digital topological method for computing genus and the Betti numbers, Topology and its Applications. 157 (2010) 1931–1936 
[ 4 ]  X. Daragon, M. Couprie, G. Bertrand, Discrete surfaces and frontier orders, Journal of Mathematical Imaging and Vision 23 (3) (2005) 379399. 
[ 5 ]  X. Daragon, M. Couprie, G. Bertrand, Derived neighborhoods and frontier orders, Discrete Applied Mathematics. 147 (23) (2005) 227–243. 
[ 6 ]  U. Eckhardt, L.Latecki, Topologies for the digital spaces Z2 and Z3, Computer Vision and Image Understanding. 90 (2003) 295312. 
[ 7 ]  A.V. Evako, R. Kopperman, Y.V. Mukhin, Dimensional properties of graphs and digital spaces, Journal of Mathematical Imaging and Vision. 6 (1996) 109119. 
[ 8 ]  A.V. Evako, Topological properties of closed digital spaces, One method of constructing digital models of closed continuous surfaces by using covers. Computer Vision and Image Understanding. 102 (2006) 134144. 
[ 9 ]  A. V. Evako, Characterizations of simple points, simple edges and simple cliques of digital spaces. One method of topologypreserving transformations of digital spaces by deleting simple points and edges, Graphical Models. 73 (2011) 19. 
[ 10 ]  A. V. Evako, Classification of digital nmanifolds, Discrete Applied Mathematics, In press, DOI: 10.1016/j.dam.2014.08.023. 
[ 11 ]  A. V. Ivashchenko, Contractible transformations do not change the homology groups of graphs, Discrete Math. 126 (1994) 159170. 
[ 12 ]  B. Kleiner, J. Lott, “Notes on Perelman's papers”, http://www.arxiv.org/abs/math., DG/0605667, 25 May 2006. 
[ 13 ]  Annie Kurien K, M. S. Samuel, Homotopy properties of digital simple closed curves, International Journal of Mathematical Sciences and Applications. 1 (2) (2011) 447450. 
[ 14 ]  E.Melin, Locally finite spaces and the join operator, U.U.D.M. Report 2007:21ISSN 1101–3591. 
[ 15 ]  J. Milnor, “Toward the Poincaré conjecture and the classification of 3manifolds,” Notices of the AMS, Vol. 50, pp. 12261233, 2003. 
[ 16 ]  J. Morgan, G. Tian, "Ricci Flow and the Poincare Conjecture", http://arxiv.org/abs/math/0607607, 25 Jul 2006. 
[ 17 ]  G. Perelman, “The Entropy Formula for the Ricci Flow and its Geometric Applications”, http://arXiv.org/abs/math., DG/0211159, 11 Nov 2002. 
[ 18 ]  G. Perelman, “Ricci Flow with Surgery on ThreeManifolds”, http://arxiv.org/abs/math., DG/0303109, 10 Mar 2003. 
[ 19 ]  G. Perelman, “Finite Extinction Time for the Solutions to the Ricci Flow on Certain ThreeManifolds”, http://arxiv.org/abs/math., DG/0307245, 17 Jul 2003. 
[ 20 ]  M. B. Smyth, R.Tsaur, I. Stewarta, Topological graph dimension, Discrete Mathematics. 310 (2) ( 2010) 325–329. 