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

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In the following text, for finite discrete X with at least two elements, nonempty countable Γ, and φ:Γ→Γ we prove the generalized shift dynamical system (XΓ,σφ) is densely chaotic if and only if φ:Γ→Γ does not have any (quasi--)periodic point. Hence the class of all densely chaotic generalized shifts on XΓ is intermediate between the class of all Devaney chaotic generalized shifts on XΓ and the class of all Li--Yorke chaotic generalized shifts on XΓ. In addition, these inclusions are proper for infinite countable Γ. Moreover we prove (XΓ,σφ) is Li--Yorke sensitive (resp. sensitive, strongly sensitive, asymptotic sensitive, syndetically sensitive, cofinitely sensitive, multi--sensitive, ergodically sensitive, spatiotemporally chaotic, Li--Yorke chaotic) if and only if φ:Γ→Γ has at least one non--quasi--periodic point.