HOMOTOPY AND THE FUNDAMENTAL GROUP OF A CIRCLE

• Main Author: Dihartomo Laweangi, Artmo
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/47741
• Institution: Institut Teknologi Bandung

Let ???? be a topological space, ????0????? and ????:[0,1] ?????. If ???? is continous in [0,1] with ????(0)= ????0=????(1) then ???? defines a loop with bases ????0. Let ???? and ???? two loops with bases , ????0. Let homotopy ????:[0,1]×[0,1]? ???? be a continous function such that, ????(????,0)=????(????), ????(????,1)=????(????) for all ???? and ????(0,????)= ????0=????(1,????) for all ????. Then ???? defines a homotopy. Further more ???? and ???? is said to be homotopic if there exist a homotopy between the two. Homotopiy defines an equivalence relation on the set of all loops with bases ????0. Let ????1(????,????0), be the set of equivalent class whit this homotopic relation. Let us define ?: ????1(????,????0)×????1(????,????0)?????1(????,????0) by [????]?[????]=[?], where ?:[0,1] ????? such that ?(????)=????(2????) for ?????[0,12] and ?(????)=????(2?????1)for ?????[12,1]. This operation introduces group structure on ????1(????,????0) then the group ????1(????,????0) is called fundamental group of ????. The goal of this study is to compute the fundamental group of a circle as an example. Furthermore, we would like to explore how can we use this concept to distinguish between two spaces whether they are homeomorphic or not