Digital Hurewicz theorem and digital homology theory

In this paper, we develop homology groups for digital images based on cubical singular homology theory for topological spaces. Using this homology, we obtain two main results that make this homology different from already existing homologies of digital images. We prove digital analog of Hurewicz theorem for digital cubical singular homology. We also show that the homology functors developed in this paper satisfy properties that resemble the Eilenberg-Steenrod axioms of homology theory, in particular, the homotopy and the excision axioms. We finally define axioms of digital homology theory.

___

  • [1] Arslan H, Karaca I, Oztel A. Homology groups of n-dimensional digital images XXI. In: Turkish National Mathematics Symposium; 2008. pp. B1-13
  • [2] Boxer L. Digitally continuous functions. Pattern Recognition Letters 1994; 15: 833-839. doi: 10.1016/0167- 8655(94)90012-4
  • [3] Boxer L. A classical construction for the digital fundamental group. Journal of Mathematical Imaging and Vision 1999; 10:51-62.
  • [4] Boxer L. Homotopy properties of sphere-like digital images. Journal Mathematical Imaging and Vision 2006; 24:167- 175. doi: 10.1007/s10851-005-3619-x
  • [5] Boxer L. Continuous maps on digital simple closed curves. Applied Mathematics 2010; 1: 377-386. doi: 10.4236/am.2010.15050
  • [6] Boxer L, Karaca I, Oztel A. Topological invariants in digital images. Journal of Mathematical Sciences: Advances and Applications 2011; 11: 109-140.
  • [7] Ege O, Karaca I. Fundamental properties of digital simplicial homology groups. American Journal of Computer Technology and Application 2013; 1: 25-42.
  • [8] Ege O, Karaca I. Some Properties of Digital H spaces. Turkish Journal of Electrical Engineering & Computer Sciences 2016; 24: 1930-1941.
  • [9] Ege O, Karaca I. Digital Fibrations. In: Proceedings Of The National Academy Of Sciences India Section A-Physical Sciences 2017; vol.87, pp.109-114.
  • [10] Escribano C, Giraldo A, Sastre MA. Digitally continuous multivalued functions, morphological operations and thinning algorithms. Journal of Mathematical Imaging and Vision 2012; 42:76–91. doi: 10.1007/s10851-011-0277-z
  • [11] Karaca I, Ege O. Cubical homology in digital images. International Journal of Information and Computer Science 2012; 1: 178-187.
  • [12] Kong TY. A digital fundamental group. Computers & Graphics 1989; 13(2): 159-166. doi: 10.1016/0097- 8493(89)90058-7
  • [13] Kong TY, Rosenfeld A (eds.). Topological algorithms for digital image processing. Elsevier, Amsterdam, 1996. doi: 10.1016/s0923-0459(96)x8001-7
  • [14] Kim IS, Han SE. Digital covering theory and its applications. Honam Mathematical Journal 2008; 30(4): 589-602. doi: 10.5831/hmj.2008.30.4.589
  • [15] Lee DW. Digital singular homology groups of digital images. Far East Journal of Mathematical Sciences 2014; 88: 39-63.
  • [16] Massey WS. A Basic Course in Algebraic Topology. Springer Verlag, 1991. doi: 10.1007/978-1-4939-9063-4
  • [17] Munkres JR. Elements of Algebraic Topology. Addison-Wesley, 1984. doi: 10.1201/9780429493911
  • [18] Rosenfeld A. Digital topology. The American Mathematical Monthly 1979; 621-630. doi: 10.2307/2321290
  • [19] Rosenfeld, A. ‘Continuous’ functions on digital pictures. Pattern Recognition Letters 1986; 4: 177-184. doi: 10.1016/0167-8655(86)90017-6
  • [20] Rotman JJ. An introduction to algebraic topology. Springer Science and Business Media, 1988. doi: 10.1007/978- 1-4612-4576-6
  • [21] Slapal J. A digital analogue of the Jordan curve theorem. Discrete Applied Mathematics 2004; 139(1-3): 231-251. doi: 10.1016/j.dam.2002.11.003