Fundamental groups

This research work the fundamental groups which take an important part of the algebraictopology.The fundamental group of an arcwise-connected set X is the group formed by the sets ofequivalence classes of the set of all loops, under the equivalence relation of homotopy.A path that begins and ends at p is called a loop at p. Given points x 0 and x1 of the space X, apath in X from x 0 to x1 is a continuos map f: [0,1] → X , such that f(0) = x 0 and f(1) = x1 .The identity element of this group is the set of all paths homotopic to the degenerate pathconsisting of the point . The fundamental groups of homeomorphic spaces are isomorphic. Infact, the fundamental group only depends on the homotopy type of X.The group product a∗b, of loop a and loop b is given by the path of a followed by the path of b.The identity element is represented by the constant path, and the inverse of a is given bytraversing a in the opposite direction. The fundamental group is independent of the choiceThe fundamental group whose every loop is homotopic to constant loop called trivialfundamental group. A space with a trivial fundamental group is called simply connected.The properties of the fundamental groups can be proved by those explanations and thoseproperties are used in the solution of the homomorphism problems.Keywords: Arcwise connected, homotopy, loop, equivalence relation, path,homotopic.JURY:1. Prof. Dr. Semin Akdoğan Date: 07.07.20062. Doç. Dr. Ayşe Kara (Supervisor) Page: 553. Doç. Dr. Ömer Gökvi