The higher topological complexity in digital images




topological complexity, digital topology, homotopy theory, digital topological complexity, image analysis


Y. Rudyak develops the concept of the topological complexity TC(X) defined by M. Farber. We study this notion in digital images by using the fundamental properties of the digital homotopy. These properties can also be useful for the future works in some applications of algebraic topology besides topological robotics. Moreover, we show that the cohomological lower bounds for the digital topological complexity TC(X,κ) do not hold.


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H. Arslan, I. Karaca and A. Oztel, Homology groups of $n-$ dimensional digital images, XXI Turkish National Mathematics Symposium (2008); B1-13.

A. Borat and T. Vergili, Digital lusternik-schnirelmann category, Turkish J. Math. 42, no.4 (2018), 1845-1852.

L. Boxer, Digitally continuous functions, Pattern Recognit. Lett. 15 (1994), 833-839.

L. Boxer, A classical construction for the digital fundamental group, J. Math. Im. Vis. 10 (1999), 51-62.

L. Boxer, Properties of digital homotopy, J. Math. Im. Vis. 22 (2005), 19-26.

L. Boxer, Homotopy properties of sphere-like digital images, J. Math. Im. Vis. 24 (2006), 167-175.

L. Boxer, Digital products, wedges, and covering spaces. J. Math. Im. Vis. 25 (2006), 169-171.

L. Boxer and I. Karaca, Fundamental groups for digital products, Adv. Appl. Math. Sci. 11, no. 4 (2012), 161-180.

L. Boxer and P. C. Staecker, Fundamental groups and Euler characteristics of sphere-like digital images, Appl. Gen. Topol. 17, no.2 (2016), 139-158.

L. Chen and J. Zhang, Digital manifolds: an intuitive definition and some properties, Proceedings of the Second ACM/SIGGRAPH Symposium on Solid Modeling and Applications (1993), 459-460.

L. Chen, Discrete surfaces and manifolds: a theory of digital-discrete geometry and topology, Rockville, MD, Scientific & Practical Computing, 2004.

L. Chen and Y. Rong, Digital topological method for computing genus and the betti numbers, Topol. Appl. 157, no. 12 (2010), 1931-1936.

A. Dranishnikov, Topological complexity of wedges and covering maps, Proc. Amer. Math. Soc. 142, no. 12 (2014), 4365-4376.

O. Ege and I. Karaca, Fundamental properties of simplicial homology groups for digital images, Am. J. Comp. Tech. Appl. 1 (2013), 25-43.

O. Ege and I. Karaca, Cohomology theory for digital images, Romanian J. Inf. Sci. Tech. 16, no.1 (2013), 10-28.

O. Ege and I. Karaca, Digital fibrations, Proc. Nat. Academy Sci. India Sec. A, 87 (2017), 109-114.

M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29, (2003), 211-221.

M. Farber, Invitation to Topological Robotics. Zur. Lect. Adv. Math., EMS, 2008.

M. Farber and M. Grant, Robot motion planning, weights of cohomology classes, and cohomology operations, Proc. Amer. Math. Soc. 136, no.9 (2008), 3339-3349.

M. Farber, S. Tabachnikov and S. Yuzvinsky, Topological robotics: motion planning in projective spaces, Int. Math. Res. Not. 34, (2003), 1850-1870.

S. E. Han, Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces, Inf. Sci. 177 (2007), 3314-3326.

G. T. Herman, Oriented surfaces in digital spaces, CVGIP: Graph. Models Im. Proc. 55 (1993), 381-396.

I. Karaca and M. Is, Digital topological complexity numbers, Turkish J. Math. 42, no. 6 (2018), 3173-3181.

I. Karaca and T. Vergili, Fiber bundles in digital images, Proceeding of 2nd International Symposium on Computing in Science and Engineering 700, no. 67 (2011), 1260-1265.

E. Khalimsky, Motion, deformation, and homotopy in finite spaces. Proceedings IEEE International Conference on Systems, Man, and Cybernetics (1987), 227-234.

T. Y. Kong, A digital fundamental group, Comp. Graph. 13 (1989), 159-166.

G. Lupton, J. Oprea and N. Scoville, Homotopy theory on digital topology, (2019), arXiv:1905.07783[math.AT].

Y. Rudyak, On higher analogs of topological complexity, Topol. Appl 157, no. 5 (2010), 916-920.

A. S. Schwarz, The genus of a fiber space, Amer. Math. Soc. Transl. 55, no. 2 (1966), 49-140.

E. Spanier, Algebraic Topology. New York, USA, McGraw-Hill, 1966.

T. tom Dieck, Algebraic Topology, Zurich, Switzerland: EMS Textbooks in Mathematics, EMS, 2008.




How to Cite

M. İs and I. Karaca, “The higher topological complexity in digital images”, Appl. Gen. Topol., vol. 21, no. 2, pp. 305–325, Oct. 2020.