On Special Subgroups of Fundamental Group

Suppose a is alpha nonzero cardinal number, I is an ideal on arc connected topological space X, and beta(alpha)(I)(X) is the subgroup of pi(1)(X) (the first fundamental group of X) generated by homotopy classes of alpha(I)loops. The main aim of this text is to study beta(alpha)(I)(X)s and compare them. Most interest is in alpha is an element of{omega, c} and I is an element of {P-fin(X), {empty set}}, where P-fin(X) denotes the collection of all finite subsets of X. We denote beta(alpha)({empty set})(X) with beta(alpha)(X). We prove the following statements: for arc connected topological spaces X and Y if beta(alpha)(X) is isomorphic to beta(alpha)(Y) for all infinite cardinal number alpha, then pi(1)(X) is isomorphic to pi(1)(Y); there are arc connected topological spaces X and Y such that pi(1)(X) is isomorphic to pi(1)(Y) but beta(omega)(X) is not isomorphic to beta(omega)(Y); for arc connected topological space X we have beta(omega)(X) subset of beta(c)(X) subset of pi(1)(X); for Hawaiian earring X, the sets beta(omega)(X), beta(c)(X), and pi(1)(X) are pairwise distinct. So beta(alpha)(X)s and beta(alpha)(I)(X)s will help us to classify the class of all arc connected topological spaces with isomorphic fundamental groups.