Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces

Fixed Point Theorems for Digital Images

“…Thus the recent works [19,20,21] studied Brouwer, Nielsen and Banach fixed point theorems from the viewpoint of digital topology which corrects the papers [7,8,9] written by Ege et al. Furthermore, the recent works [18,19,20,21] precede many fixed point properties related to The rest of the paper is organized as follows: Section 2 provides basic notions from digital topology. Section 3 investigates some properties of digital k-contractibility inherited from a digital k-homotopy and studies digital homotopy axiom associated with the digital homology in [1,4,8].…”

“…If x 0 = x and x l = y, then the length of the simple k-path, denoted by l k (x, y), is the number l. A simple closed k-curve with l elements in Z n , denoted by SC n,l k [24,11] (see Figure 1 [24]. For a digital image (X, k), as a generalization of N * k (p) [24] the digital kneighborhood of x 0 ∈ X with radius 1 is defined in X to be the following subset of X [11,14] N…”

“…3) In Section 4, in relation to the study of Lefschetz fixed point theorem, we use the notions of both a digital simplicial complex derived from a digital image (X, k) [12,14,1,8] and a digital homology group H k q (X) introduced in [1] (for more details, see [1,7,8]). In terms of this approach, the following q-th digital simplicial group is established in [1,8].…”

“…Since an nD digital image (X, k) is viewed as a set X ⊂ Z n with one of the k-adjacency relations of (2.2) (or a digital k-graph in [11,14]), in relation to the classification of nD digital images, we use the term a (k 0 , k 1 )-isomorphism as in [12] (see also [22]) rather than a (k 0 , k 1 )-homeomorphism as in [6]. Definition 1.…”

“…Owing to this finding, we correct some errors in [7,8,9] and improves the paper (O. Ege, I. Karaca in The Bulletin of the Belgian Mathematical SocietySimon Stevin, Applications of the Lefschetz Number to Digital Images, Vol.21(5) (2014), 823-839). Besides, we need to point out that Ege et al [8] used some of figures such as M SC 4 [11], M SS 6 [13] and M SS 18 [13,14] and a digital wedge [11] without citation. Section 5 concludes the paper with some remarks.…”