Related papers: The Hawaiian earring group is topologically incomp…
Endowed with quotient topology inherited from the space of based loops, the fundamental group of the Hawaiian earring fails to be metrizable. The fundamental group of any space which retracts to the Hawaiian earring is also nonmetrizable.
Endowed with natural topologies, the fundamental group of the Hawaiian earring continuously injects into the inverse limit of free groups. This note shows the injection fails to have a continuous inverse. Such a phenomenon was unexpected…
The harmonic archipelago HA is obtained by attaching a large pinched annulus to every pair of consecutive loops of the Hawaiian earring. We clarify the fundamental group pi1(HA) as a quotient of the Hawaiian earring group, provide a precise…
A characterization of regular topological fundamental groups yields a `no retraction theorem' for spaces constructed in similar fashion to the Hawaiian earring.
The natural quotient map q from the space of based loops in the Hawaiian earring onto the fundamental group provides a new example of a quotient map such that q x q fails to be a quotient map. This also settles in the negative the question…
We prove several new theorems regarding first homology. Some dichotomies for first homology of Peano continua are presented, as well as a notion of strong abelianization for arbitrary path connected metric spaces. We also show that the…
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…
We show a dichotomy for groups of cardinality less than continuum. The number of homomorphisms from the Hawaiian earring group to such a group $G$ is either the cardinality of $G$ in case $G$ is noncommutatively slender, or the number is…
The connected covering spaces of a connected and locally path-connected topological space $X$ can be classified by the conjugacy classes of those subgroups of $\pi_1(X,x)$ which contain an open normal subgroup of $\pi_1(X,x)$, when endowed…
The classical archipelago is a non-contractible subset of $\mathbb{R}^3$ which is homeomorphic to a disk except at one non-manifold point. Its fundamental group, $\mathcal{A}$, is the quotient of the topologist's product of $\mathbb Z$, the…
We will show that all inverse limits of finite rank free groups index by the natural numbers are isomorphic either to a finite rank free group or to a fixed universal group. In other words, any inverse system of finite rank free groups…
We show that certain algebraic structures lack freeness in the absence of the axiom of choice. These include some subgroups of the Baer-Specker group $\mathbb{Z}^{\omega}$ and the Hawaiian earring group. Applications to slenderness,…
The topological fundamental group $\pi_{1}^{top}$ is a topological invariant that assigns to each space a quasi-topological group and is discrete on spaces which are well behaved locally. For a totally path-disconnected, Hausdorff, unbased…
We show that the first homology group of a locally connected compact metric space is either uncountable or is finitely generated. This is related to Shelah's well-known result which shows that the fundamental group of such a space satisfies…
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…
We show the classical $\pi_1$-action on the $n$-th homotopy group can fail to be continuous for any $n$ when the homotopy groups are equipped with the natural quotient topology. In particular, we prove the action…
We develop tools to recognize sequential spaces with large inductive dimension zero. We show the Hawaiian earring group $G$ is 0 dimensional, when endowed with the quotient topology, inherited from the space of based loops with the compact…
If a Peano continuum $X$ is semilocally simply connected, then it has a finite polyhedral approximation whose fundamental group is isomorphic to that of $X$. In general, this fails to be true. It is known that the fundamental group of a…
Extending a theorem of Shelah we prove that fundamental groups of Peano continua (locally connected and connected metric compact spaces) are finitely presented if they are countable. The proof uses ideas from geometric group theory.
We describe homomorphisms $\varphi:H\rightarrow G$ for which the codomain is acylindrically hyperbolic and the domain is a topological group which is either completely metrizable or locally countably compact Hausdorff. It is shown that, in…