Digital Borsuk-Ulam theorem

Document Type : Research Paper

Authors

1 Department of Mathematics‎, ‎Pamukkale University‎, ‎P.O‎. ‎Box 20070‎, ‎Denizli‎, ‎Turkey.

2 Department of Mathematics‎, ‎Ege University‎, ‎P.O‎ .‎Box 35100‎, ‎Izmir‎, ‎Turkey.

Abstract

The aim of this paper is to compute a simplicial cohomology group of some specific digital images. Then we define ringand algebra structures of a digital cohomology with the cup product. Finally, we prove a special case of the Borsuk-Ulam theorem fordigital images.

Keywords

Main Subjects


‎J.W‎. ‎Alexander‎, ‎On the connectivity ring of an abstract space‎, ‎ Ann‎. ‎of Math‎. ‎(2)  37 (1936)‎, ‎no‎. ‎3‎, ‎698--708‎.
‎H‎. ‎Arslan‎, ‎I‎. ‎Karaca‎, ‎and A‎. ‎Oztel‎, ‎Homology groups of n-dimensional digital images XXI‎, ‎Proc‎. ‎Turkish National Mathematics Symposium‎, ‎B‎. ‎pp‎. ‎1-13‎, ‎2008‎.
‎G‎. ‎Bertrand‎, ‎Simple points‎, ‎topological numbers and geodesic neighborhoods in cubic grids‎, ‎ Pattern Recognit‎. ‎Lett.  15 (1994) 1003--1011‎.
‎G‎. ‎Bertrand and R‎. ‎Malgouyres‎, ‎Some topological properties of discrete surfaces‎, ‎ J‎. ‎Math‎. ‎Imaging Vision,  11 (1999) 207--221‎.
‎K‎. ‎Borsuk‎, ‎Drei sätze über die n-dimensionale euklidische sphäre‎, ‎ Fund‎. ‎Math.  20 (1933) 177--190‎.
‎L‎. ‎Boxer‎, ‎Digitally continuous functions‎, ‎ Pattern Recognit‎. ‎Lett.  15 (1994) 833--839‎.
‎L‎. ‎Boxer‎, ‎A classical construction for the digital fundamental group‎, ‎ J‎. ‎Math‎. ‎Imaging Vision  10 (1999)‎, ‎no‎. ‎1‎, ‎51--62‎.
‎L‎. ‎Boxer‎, ‎Homotopy properties of sphere-like digital images‎, ‎ J‎. ‎Math‎. ‎Imaging Vision  24 (2006)‎, ‎no‎. ‎2‎, ‎167--175‎.
‎L‎. ‎Boxer‎, ‎Digital products‎, ‎wedges‎, ‎and covering spaces‎, ‎ J‎. ‎Math‎. ‎Imaging Vision  25 (2006)‎, ‎no‎. ‎2‎, ‎159--171‎.
‎L‎. ‎Boxer‎, ‎Continuous maps on digital simple closed curves‎, ‎ Appl‎. ‎Math.  1 (2010) 377--386‎.
‎L‎. ‎Boxer‎, ‎I‎. ‎Karaca and A‎. ‎Oztel‎, ‎Topological invariants in digital images‎, ‎ J‎. ‎Math‎. ‎Sci‎. ‎Adv‎. ‎Appl.  11 (2011)‎, ‎no‎. ‎2‎, ‎109--140‎.
‎E‎. ‎Čech‎, ‎Multiplication on a complex‎, ‎ Ann‎. ‎of Math‎. ‎(2)  37 (1936)‎, ‎no‎. ‎3‎, ‎681--697‎.
‎L‎. ‎Chen‎, ‎Gradually varied surfaces and its optimal uniform approximation‎, ‎ Proc‎. ‎SPIE  2182 (1994) 300--307‎.
‎L.M‎Chen‎, ‎Digital and Discrete Geometry Theory and Algorithms‎, ‎Springer‎, ‎Berlin‎, ‎2014‎.
‎M.C‎. ‎Crabb and J‎. ‎Jaworowski‎, ‎Aspects of the Borsuk-Ulam theorem‎, ‎ J‎. ‎Fixed Point Theory Appl.  13 (2013) 459--488‎.
‎H‎. ‎Edelsbrunner and J.L‎. ‎Harer‎, ‎Computational Topology An Introduction‎, ‎American Mathematical Society‎, ‎Providence‎, ‎RI‎, ‎2010‎.
‎O‎. ‎Ege and I‎. ‎Karaca‎, ‎Fundamental properties of digital simplicial homology groups‎, ‎ Amer‎. ‎J‎. ‎Comput‎. ‎Technol‎. ‎Appl.  1 (2013)‎, ‎no‎. ‎2‎, ‎25--42‎.
‎O‎. ‎Ege and I‎. ‎Karaca‎, ‎Cohomology theory for digital images‎, ‎ Rom‎. ‎J‎. ‎Inform‎. ‎Sci‎. ‎Technol.  16 (2013)‎, ‎no‎. ‎1‎, ‎10--28‎.
‎S.E‎. ‎Han‎, ‎Non-product property of the digital fundamental group‎, ‎ Inform‎. ‎Sci.  171 (2005)‎, ‎no‎. ‎1-3‎, ‎73--91‎.
‎S.E‎. ‎Han‎, ‎Connected sum of digital closed surfaces‎, ‎ Inform‎. ‎Sci.  176 (2006)‎, ‎no‎. ‎3‎, ‎332--348‎.
‎S.E‎. ‎Han‎, ‎Minimal simple closed 18-surfaces and a topological preservation of 3D surfaces‎, ‎ Inform‎. ‎Sci.  176 (2006)‎, ‎no‎. ‎2‎, ‎120--134‎.
‎S.E‎. ‎Han‎, ‎Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces‎, ‎ Inform‎. ‎Sci.  177 (2007)‎, ‎no‎. ‎16‎, ‎3314--3326‎.
‎A‎. ‎Hatcher‎, ‎Algebraic Topology‎, ‎Cambridge Univ‎. ‎Press‎, ‎2002‎.
‎G.T‎. ‎Herman‎, ‎Oriented surfaces in digital spaces‎, ‎ CVGIP‎: ‎Graphical Models and Image Processing  55 (1993) 381--396‎.
‎I‎. ‎Karaca and G‎. ‎Burak‎, ‎Simplicial relative cohomology rings of digital images‎, ‎ Appl‎. ‎Math‎. ‎Inf‎. ‎Sci. 8 (2014)‎, ‎no‎. ‎5‎, ‎2375--2387‎.
‎A‎. ‎Kolmogoroff‎, ‎Homologiering des komplexes und des lokal bikompakten raumes‎, ‎ Matem‎. ‎Sb.  1 (1936) 701--705‎.
‎T.Y‎. ‎Kong‎, ‎A digital fundamental group‎, ‎ Computers $&$ Graphics  13 (1989)‎, ‎no‎. ‎2‎, ‎159--166‎.
‎R‎. ‎Kopperman‎, ‎R‎. ‎Meyer and R.G‎. ‎Wilson‎, ‎A Jordan surface theorem for three-dimensional digital spaces‎, ‎ Discrete Comput‎. ‎Geom.  6 (1991)‎, ‎no‎. ‎2‎, ‎155--161‎.
‎R‎. ‎Malgouyres and G‎. ‎Bertrand‎, ‎A new local property of strong $n$-surfaces‎, ‎ Pattern Recognit‎. ‎Lett.  20 (1999) 417--428‎.
‎J.R‎. ‎Munkres‎, ‎Elements of Algebraic Topology‎, ‎Addison-Wesley‎, ‎1984‎.
‎V.V‎. ‎Prasolov‎, ‎Elements of Homology Theory‎, ‎American Mathematical Society‎, ‎Providence‎, ‎RI‎, ‎2007‎.
‎A‎. ‎Rosenfeld‎, ‎Digital topology‎, ‎ Amer‎. ‎Math‎. ‎Monthly  86 (1979)‎, ‎no‎. ‎8‎, ‎621--630‎.
‎A‎. ‎Rosenfeld‎, ‎Continuous functions on digital images‎, ‎ Pattern Recognit‎. ‎Lett.  4 (1986) 177--184‎.
‎J.J‎. ‎Rotman‎, ‎An Introduction to Algebraic Topology‎, ‎Springer-Verlag‎, ‎New York‎, ‎1998‎.
‎S‎. ‎Roy and W‎. ‎Steiger‎, ‎Some combinatorial and algorithmic applications of the Borsuk-Ulam Theorem‎, ‎ Graphs Combin.  23 (2007) 331--341‎.
‎E‎. ‎Spanier‎, ‎Algebraic Topology‎, ‎McGraw-Hill‎, ‎New York‎, ‎1966‎.
‎E‎. ‎Ünver Demir and I‎. ‎Karaca‎, ‎Simplicial homology groups of certain digital surfaces‎, ‎Hacet‎. ‎J‎. ‎Math‎. ‎Stat. 44 (2015)‎, ‎no‎. ‎5‎, ‎1011--1022‎.
‎H‎. ‎Whitney‎, ‎On products in a complex‎, ‎ Ann‎. ‎of Math‎. ‎(2)  39 (1938)‎, ‎no‎. ‎2‎, ‎397--432‎.