The k-fundamental group of a closed k-surface

  • Authors:

  • Sang-Eon Han

  • Affiliations:

  • Department of Computer and Applied Mathematics, Honam University, Gwangju 506-714, Republic of Korea

  • Venue:
  • Information Sciences: an International Journal
  • Year:
  • 2007

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Abstract

In this paper, we deal with the problem of computing the digital fundamental group of a closed k-surface by using various properties of both a (simple) closed k-surface and a digital covering map. To be specific, let SC"k"""i^n^"^i^,^l^"^i be a simple closed k"i-curve with l"i elements in Z^n^"^i, i@?{1,2}. Then, the Cartesian product SC"k"""1^n^"^1^,^l^"^1xSC"k"""2^n^"^2^,^l^"^2@?Z^n^"^1^+^n^"^2 is not always a closed k-surface with some k-adjacency of Z^n^"^1^+^n^"^2. Thus, we provide a condition for SC"k"""1^n^"^1^,^l^"^1xSC"k"""2^n^"^2^,^l^"^2 to be a (simple) closed k-surface with some k-adjacency depending on the k"i-adjacency, i@?{1,2}. Besides, even if SC"k"""1^n^"^1^,^l^"^1xSC"k"""2^n^"^2^,^l^"^2 is not a closed k-surface, we show that the k-fundamental group of SC"k"""1^n^"^1^,^l^"^1xSC"k"""2^n^"^2^,^l^"^2 can be calculated by both a k-homotopic thinning and a strong k-deformation retract.