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 $1\leq m\leq n$.
In this paper, we introduce a kind of homology which we call Hawaiian homology to study and classify pointed topological spaces. The Hawaiian homology group has advantages of Hawaiian groups. Moreover, the first Hawaiian homology group is isomorphic to the abelianization of the first Hawaiian group for path-connected and locally path-connected topological spaces. Since Hawaiian homology has concrete elements and abelian structure, its calculations are more routine. Thus we use Hawaiian homology groups to compare Hawaiian groups, and then we obtain some information about Hawaiian groups of some…
By generalizing the whisker topology on the $n$th homotopy group of pointed space $(X, x_0)$, denoted by $π_n^{wh}(X, x_0)$, we show that $π_n^{wh}(X, x_0)$ is a topological group if $n \ge 2$. Also, we present some necessary and sufficient conditions for $π_n^{wh}(X,x_0)$ to be discrete, Hausdorff and indiscrete. Then we prove that $L_n(X,x_0)$ the natural epimorphic image of the Hawaiian group $\mathcal{H}_n(X, x_0)$ is equal to the set of all classes of convergent sequences to the identity in $π_n^{wh}(X, x_0)$. As a consequence, we show that $L_n(X, x_0) \cong L_n(Y, y_0)$ if $π_n^{wh}(X, …
We construct a space $\mathbb{P}$ for which the canonical homomorphism $π_1(\mathbb{P},p) \rightarrow \checkπ_1(\mathbb{P},p)$ from the fundamental group to the first Čech homotopy group is not injective, although it has all of the following properties: (1) $\mathbb{P}\setminus\{p\}$ is a 2-manifold with connected non-compact boundary; (2) $\mathbb{P}$ is connected and locally path connected; (3) $\mathbb{P}$ is strongly homotopically Hausdorff; (4) $\mathbb{P}$ is homotopically path Hausdorff; (5) $\mathbb{P}$ is 1-UV$_0$; (6) $\mathbb{P}$ admits a simply connected generalized covering space …
A space $X$ is "sequentially $n$-connected" at $x\in X$ if for every $0\leq k\leq n$ and sequence of maps $f_1,f_2,f_3,\dots:S^k\to X$ that converges toward a point $x\in X$, the maps $f_m$ contract by a sequence of null-homotopies that converge toward $x$. We use this property, in conjunction with the Whitney Covering Lemma, as a foundation for developing new methods for characterizing higher homotopy groups of finite dimensional Peano continua. Among many new computations, a culminating result of this paper is: if $Y$ is a space obtained by attaching an infinite shrinking sequence $A_1,A_2,A…
D. K. Biss（Topology and its Applications 124（2002）355-371）介绍了拓扑基本群并给出了这个概念的一些有趣的基本性质。本文将上述概念推广到同伦群，并试图证明拓扑同伦群的一些类似的基本性质。我们还进一步研究了拓扑同伦群的拓扑结构，得到了拓扑结构离散的充要条件。此外，我们还证明了研究拓扑同伦群可能比拓扑基本群更有用。
We obtain all coset $n$-valued topological groups on $S^3$ and $\mathbb{R}P^3$, arising from compact Lie groups Sp(1) and SO(3) and there finite groups of automorphisms.
We show that the parametrised topological complexity of Cohen, Farber and Weinberger gives an invariant of group epimorphisms. We extend various bounds for the topological complexity of groups to obtain bounds for the parametrised topological complexity of epimorphisms. Several applications are given, including an alternative computation of the parametrised topological complexity of the planar Fadell--Neuwirth fibrations which avoids calculations involving cup products. We also prove a homotopy invariance result for parametrised topological complexity of fibrations over different bases.
Jeonbuk National University（全北国立大学）；Fairfield University（费尔菲尔德大学）
In this article, we develop the basic theory of digital topological groups. The basic definitions directly lead to two separate categories, based on the details of the continuity required of the group multiplication. We define $\NP_1$- and $\NP_2$-digital topological groups, and investigate their properties and algebraic structure. The $\NP_2$ category is very restrictive, and we give a complete classification of $\NP_2$-digital topological groups. We also give many examples of $\NP_1$-digital topological groups. We define digital topological group homomorphisms, and describe the digital count…
