Fundamental groups and applications

The objective of this thesis is to disclose the fundemental groups which has an important role in algebraic topology by means of definitions, theories and applications. f and g being two continuous descriptions, there is a continuous F homotopy description between these two descriptions in order for these two descriptions to be homotopic. p-based loop is the route which considers p point as the start and end point. The path, on the other hand, is the description which directs the 0 point of the closed range [0,1] to x_0 point, and 1 point to x_1 point for x_0,x_1∈X. This description is called as the route from x_0 to x_1. On X space, the x_0-based loops form a group under * operation, and this group is called the x_0-based fundamental group of X and indicated with π_1 (X,x_0). The fundamental groups of homeomorphic spaces are isomorphic. The fundamental groups are connected to the homotopy type of X. The multiplication of the groups of f loop and g loop f*g , is indicated in the form of consecutive f and g routes. Unit element is the fixed route; opposite f is the route opposite to this route. One of the fundamental groups of the circle is Z . The fundamental group of punctured R^2 is Z '. The fundamental group, each loop of which is homotopic to the fixed loop is called trivial group. The space which has a trivial fundamental group is called space with simple connection. Based on these definitions, the characteristics of the fundamental group can be proved and these proofs are used in the solution of many homeomorphism problems. Key words: Homotopy, loop, route, fundamental group, covering space, homotopic descriptions, space with simple connection