arXiv 0909.30862009-09math.GNmath.AT

Multiplication is discontinuous in the Hawaiian earring group (with the quotient topology)

Paul Fabel

备注：: 5 pages. A modified introduction and bibilography

英文摘要：

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 of whether the fundamental group (with the quotient topology) of a compact metric space is always a topological group with the standard operations.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A retraction theorem for topological fundamental groups with application to the Hawaiian earring

A characterization of regular topological fundamental groups yields a `no retraction theorem' for spaces constructed in similar fashion to the Hawaiian earring.

2005-02-10math.ATmath.GN

The Hawaiian earring group is topologically incomplete

The premier exhibition of the following phenomenon: The fundamental group of any Peano continuum constructed in similar fashion to the Hawaiian earring admits two natural distinct topological group structures. However despite being uncountable and regular, neither group is a Baire space and hence neither group admits a compatible complete metric.

2005-03-06math.GNmath.GT

The topological Hawaiian earring group does not embed in the inverse limit of free groups

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 and appears to contradict results of another author.

2005-11-30math.GNmath.GT

The fundamental group of the harmonic archipelago

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 description of the kernel, show that both pi1(HA) and the kernel are uncountable, and that pi1(HA) has the indiscrete topology.

2005-01-24math.ATmath.GN

On Hawaiian Groups of Some Topological Spaces

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 $1\leq m\leq n$.

2012-01-21math.ATmath.GN

On the discontinuity of the $π_1$-action

We show the classical $π_1$-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 $π_1(X)\timesπ_n(X)\toπ_n(X)$ fails to be continuous for a one-point union $X=A\vee \mathbb{H}_n$ where $A$ is an aspherical space such that $π_1(A)$ is a topological group and $\mathbb{H}_n$ is the $(n-1)$-connected, n-dimensional Hawaiian earring space $\mathbb{H}_n$ for which $π_n(\mathbb{H}_n)$ is a topological abelian group.

2018-07-20math.ATmath.GN

更多相关论文收起更多论文

On zero dimensional sequential spaces

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 open topology. In particular $G$ is $T_4$ and hence inclusion $G \rightarrow F_M (G)$ is a topological embedding into the free topological group $F_M (G)$ in the sense of Markov.

2020-07-04math.GN

Covering R-trees, R-free groups, and dendrites

We prove that every length space X is the orbit space (with the quotient metric) of an R-tree T via a free action of a locally free subgroup G(X) of isometries of X. The mapping f:T->X is a kind of generalized covering map called a URL-map and is universal among URL-maps onto X. T is the unique R-tree admitting a URL-map onto X. When X is a complete Riemannian manifold M of dimension n>1, the Menger sponge, the Sierpin'ski carpet or gasket, T is isometric to the so-called "universal" R-tree A_{c}, which has valency equal to the cardinality of the continuum at each point. In these cases, and wh…

2009-04-23math.GNmath.MG

Uncountable groups and the geometry of inverse limits of coverings

In this paper we develop a new approach to the study of uncountable fundamental groups by using Hurewicz fibrations with the unique path-lifting property (lifting spaces for short) as a replacement for covering spaces. In particular, we consider the inverse limit of a sequence of covering spaces of $X$. It is known that the path-connectivity of the inverse limit can be expressed by means of the derived inverse limit functor $\varprojlim^1$, which is, however, notoriously difficult to compute when the $π_1(X)$ is uncountable.To circumvent this difficulty, we express the set of path-components o…

2021-01-27math.GN