The k-fundamental group of a closed k-surface

摘要

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 SCkin1,li be a simple closed k(i)-curve with l(i) elements in Z(ni), i epsilon {1, 21}. Then, the Cartesian product SCn(k1)(1,l1) x SCk2n2,l2 C Z(1)(2)(n)(+n) is not always a closed k-surface with some k-adjacency of Z(n1+n2). Thus, we provide a condition for SCk1n1,l1 x SCk2n2,l2 to be a (simple) closed k-surface with some k-adjacency depending on the ki-adjacency, i C f 1, 21. Besides, even if SCk x SC,,"" is not a closed k-surface, we show that the k-fundamental group of SCk1n1,l1 x SCn(k2)(2,l2) can be calculated by both a k-homotopic thinning and a strong k-deformation retract. (C) 2007 Elsevier Inc. All rights reserved.