The higher topological complexity in digital images. (English) Zbl 1453.55001
In the framework of topological robotics, higher topological complexities have been introduced by Y. B. Rudyak [Topology Appl. 157, No. 5, 916–920 (2010; Zbl 1187.55001)] which generalize Farber’s topological complexity. In [Turk. J. Math. 42, No. 6, 3173–3181 (2018; Zbl 1438.68268)], the present authors introduced an analog of topological complexity in the framework of digital topology as a digital homotopy invariant of digital images. In the present article, the same authors define analogs of higher topological complexities in digital topology.
In addition, the authors provide a simple example showing that unlike in the topological setting, the zero-divisor cup length in digital cohomology provides no lower bound for digital topological complexity.
In Sections 2 and 3, the authors recall basic notions of digital topology and digital homotopy theory, respectively, in a detailed and mostly self-contained way. Section 4 contains the definition of digital higher topological complexities and the computation of some simple examples. The aforementioned cohomology example is given in Section 5, which is independent from the rest of the article. A concluding Section 6 summarizes the results.
In addition, the authors provide a simple example showing that unlike in the topological setting, the zero-divisor cup length in digital cohomology provides no lower bound for digital topological complexity.
In Sections 2 and 3, the authors recall basic notions of digital topology and digital homotopy theory, respectively, in a detailed and mostly self-contained way. Section 4 contains the definition of digital higher topological complexities and the computation of some simple examples. The aforementioned cohomology example is given in Section 5, which is independent from the rest of the article. A concluding Section 6 summarizes the results.
Reviewer: Stephan Mescher (Lepizig)
MSC:
| 55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
| 68U10 | Computing methodologies for image processing |
| 68T40 | Artificial intelligence for robotics |
| 62H35 | Image analysis in multivariate analysis |
Keywords:
topological complexity; digital topology; homotopy theory; digital topological complexity; image analysisReferences:
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