REMARKS ON SIMPLY k-CONNECTIVITY AND k-DEFORMATION RETRACT IN DIGITAL TOPOLOGY

Abstract

Abstract. To study a deformation of a digital space from the view-point of digital homotopy theory, we have often used the notions ofa weak k-deformation retract [20] and a strong k-deformation re-tract [10, 12, 13]. Thus the papers [10, 12, 13, 16] firstly developedthe notion of a strong k-deformation retract which can play an im-portant role in studying a homotopic thinning of a digital space.Besides, the paper [3] deals with a k-deformation retract and itshomotopic property related to a digital fundamental group. Thus,as a survey article, comparing among a k-deformation retract in [3],a strong k-deformation retract in [10, 12, 13], a weak deformationk-retract in [20] and a digital k-homotopy equivalence [5, 24], we ob-serve some relationships among them from the viewpoint of digitalhomotopy theory. Furthermore, the present paper deals with someparts of the preprint [10] which were not published in a journal (seeProposition 3.1). Finally, the present paper corrects Boxer’s paper[3] as follows: even though the paper [3] referred to the notion of adigital homotopy equivalence (or a same k-homotopy type) whichis a special kind of a k-deformation retract, we need to point outthat the notion was already developed in [5] instead of [3] and fur-ther corrects the proof of Theorem 4.5 of Boxer’s paper [3] (see theproof of Theorem 4.1 in the present paper). While the paper [4]refers some properties of a deck transformation group (or an auto-morphism group) of digital covering space without any citation, thestudy was early done by Han in his paper (see the paper [14]).Received May 15, 2014. Accepted June 10, 2014.2010 Mathematics Subject Classification. 55Q70, 52Cxx, 55P15, 68R10, 68U05.Key words and phrases. simply k-connected, digital isomorphism, strong k-deformation, weak k-deformation retract, k-homotopy equivalence (same k-homotopytype), digital topology.