On Hawaiian groups of some topological spaces

Abstract

The paper is devoted to study the structure of Hawaiian groups of some topological spaces. We present some behaviors of Hawaiian groups with respect to product spaces, weak join spaces, cone spaces, covering spaces and locally trivial bundles. In particular, we determine the structure of the n-dimensional Hawaiian group of the m-dimensional Hawaiian earring space, for all 1 ≤ m ≤ n.

On topological homotopy groups and relation to Hawaiian groups

“…If there exists a path γ from x 0 to x 1 , then γ # in Definition 3.5 induces an isomorphism from π n (X, x 0 ) onto π n (X, x 1 ). But there exist path connected spaces, namely HE n , n ≥ 2, such that L n (HE n , θ) ̸ ∼ = L n (HE n , a), where a ̸ = θ (see [3,Corollary 2.11]). By Theorem 4.7 and Corollary 3.9, γ # can analogously transfer L n (X, x 0 ) isomorphically onto L n (X, x 1 ), if γ and γ −1 are n-SLT paths.…”

“…It was proved that L n (X, x 0 ) = ϕ(H n (X, x 0 )), and hence it is a subgroup of ℵ 0 π n (X, x 0 ) (see [3,Theorem 2.7]). Therefore, one can consider the homomorphism ϕ as an epimorphism from…”

On a Van Kampen theorem for Hawaiian groups

“…Now by [9, Proposition 6.36], π n (X, x) ∼ = π n (X 1 ) ⊕ π n (X 2 ), and after a rearrangement, H n (X, x) ∼ = N π n (X 1 ) ⊕ N π n (X 2 ). We obtain the result, using [1,Theorem 2.5].…”

“…In addition, unlike homotopy groups, Hawaiian groups of pointed space (X, x 0 ) depend on the behaviour of X at x 0 , and then their structures depend on the choice of the base point. In this regard, there exist some examples of path connected spaces with non-isomorphic Hawaiian groups at different points, such as the n-dimensional Hawaiian earring, where n ≥ 2 (see [1,Corollary 2.11]).…”

“…{U l ; l ∈ N} be the local basis at θ defined in the proof of[1, Theorem 2.9], and also let {[f k ]} ∈ N π 1 (HE 1 , θ). Then {f k } converges uniformly to the constant 1-loop if and only if foreach l ∈ N, there exists K l ∈ N such that im(f k ) ⊆ U l whenever k ≥ K l .…”