Abstract
In the present paper, we study basic properties of digital n-dimensional manifolds and digital simply connected spaces. An important property of a digital n-manifold is that M is a digital n-sphere if and only if for any point v of M, M-v is a digital n-disk. It is proved that a digital (n-1)-sphere S contained a digital n-sphere M is a separating space of M. We show that a digital n-manifold 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 (n-1)-sphere S in a digital simply connected n-manifold M is a separating space for M, and if a digital 3-manifold M is locally simply connected, then M is a digital 3-sphere.

Keywords
Digital Topology; Topological space; Separation; Digital simply connected space

Cite this paper
Alexander V. Evako, Properties of Digital n-Dimensional Spheres and Manifolds. Separation of Digital Manifolds , SCIREA Journal of Mathematics. Volume 3, Issue 1, February 2018 | PP. 29-56.

References

[ 1 ]	L. Boxer, Continuous Maps on Digital Simple Closed Curves, Applied Mathematics. 1 (2010) 377-386.
[ 2 ]	H-D. Cao, X-P. Zhu, “A complete proof of the Poincaré and Geometrization conjectures – application of the Hamilton-perelman 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) 379-399.
[ 5 ]	X. Daragon, M. Couprie, G. Bertrand, Derived neighborhoods and frontier orders, Discrete Applied Mathematics. 147 (2-3) (2005) 227–243.
[ 6 ]	U. Eckhardt, L.Latecki, Topologies for the digital spaces Z2 and Z3, Computer Vision and Image Understanding. 90 (2003) 295-312.
[ 7 ]	A.V. Evako, R. Kopperman, Y.V. Mukhin, Dimensional properties of graphs and digital spaces, Journal of Mathematical Imaging and Vision. 6 (1996) 109-119.
[ 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) 134-144.
[ 9 ]	A. V. Evako, Characterizations of simple points, simple edges and simple cliques of digital spaces. One method of topology-preserving transformations of digital spaces by deleting simple points and edges, Graphical Models. 73 (2011) 1-9.
[ 10 ]	A. V. Evako, Classification of digital n-manifolds, 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) 159-170.
[ 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) 447-450.
[ 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 3-manifolds,” Notices of the AMS, Vol. 50, pp. 1226-1233, 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 Three-Manifolds”, 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 Three-Manifolds”, 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.