The k-fundamental group of a closed k-surface

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 SCkini,li be a simple closed ki-curve with li elements in ZniZni, i∈{1,2}i∈{1,2}. Then, the Cartesian product SCk1n1,l1×SCk2n2,l2⊂Zn1+n2 is not always a closed k-surface with some k  -adjacency of Zn1+n2Zn1+n2. Thus, we provide a condition for SCk1n1,l1×SCk2n2,l2 to be a (simple) closed k-surface with some k-adjacency depending on the ki-adjacency, i∈{1,2}i∈{1,2}. Besides, even if SCk1n1,l1×SCk2n2,l2 is not a closed k-surface, we show that the k  -fundamental group of SCk1n1,l1×SCk2n2,l2 can be calculated by both a k-homotopic thinning and a strong k-deformation retract.