# The Earring Space $\mathbb{E}=\bigcup_{n\in\mathbb{N}}\left\{(x,y)\in\mathbb{R}^2\mid (x-1/n)^2+y^2=1/n^2\right\}$

Other Names: infinite earring, 1-dimensional earring, shrinking wedge of circles, shrinking bouquet of circles, clamshell. This space has often been referred to as the “Hawaiian earring” in the literature; however, there is substantial movement by experts to stop using this term.

Topological Properties:  Planar, 1-dimensional, path-connected, locally path-connected, compact metric space.

Other constructions:

• Reduced suspension of $0$-dimensional earring, i.e. the convergent sequence space $\{0,\dots,1/3,1/2,1\}$
• infinite wedge $\bigvee_{\mathbb{N}}S^1$ viewed as subspace of infinite dimensional torus $\prod_{\mathbb{N}}S^1$ with the product topology.
• one-point compactification of a countable disjoint union of open arcs.

Universal Property: given any sequence $\{\alpha_n\}_{n\in\mathbb{N}}\to c_{x_0}$ of loops based at $x_0$ converging (in the compact-open topology) to the constant loop at $x_0$, there exists a unique map $f:\mathbb{E}\to X$ such that $f\circ\ell_n=\alpha_n$ where $\ell_n:S^1\to\mathbb{E}$ is the canonical counterclockwise loop traversing the n-th circle. This universal property is a special case of the loop space-suspension adjuction.

Fundamental Group: $\pi_1(\mathbb{E})$ embeds as the subgroup of the natural inverse limit $\varprojlim_{n}F_n$ of finitely generated free groups consisting of locally eventually constant elements. This group is $\pi_1(\mathbb{E})$ is sometimes called the free- $\sigma$-product and denoted $\#_{\mathbb{N}}\mathbb{Z}$. See this post and others for more details.

Fundamental Group Properties: Uncountable, Residually free, torsion free, locally free, locally finite.

Higher homotopy groups: $\pi_n(\mathbb{E})=0$ for $n\geq 2$, i.e. $\mathbb{E}$ is aspherical.

Homology groups: $\widetilde{H}_n(\mathbb{E})=\begin{cases} \prod_{\mathbb{N}}\mathbb{Z}\oplus\prod_{\mathbb{N}}\mathbb{Z}/\bigoplus_{\mathbb{N}}\mathbb{Z}, & n=1 \\ 0, & n \neq 1 \end{cases}$

Cech homotopy groups: $\check{\pi}_n(\mathbb{E})=\begin{cases} \varprojlim_{n}F_{n}, & n=1 \\ 0, & n\neq 1 \end{cases}$

Cech homology groups: $\check{H}_n(\mathbb{E})=\begin{cases} \mathbb{Z}, & n=0 \\ \prod_{\mathbb{N}}\mathbb{Z}, & n=1 \\ 0, & n \geq 2 \end{cases}$

Cech cohomology groups: $\check{H}_n(\mathbb{E})=\begin{cases} \mathbb{Z}, & n=0 \\ \bigoplus_{\mathbb{N}}\mathbb{Z}, & n=1 \\ 0, & n \geq 2 \end{cases}$

Wild Set/Homotopy Type: The wild set is the single-point set $\mathbf{w}(\mathbb{E})=\{(0,0)\}$. The infinite earring represents the unique homotopy type of 1-dimensional Peano continua with a single wild point.

Other Properties:

• Semi-locally simply connected: No, not at $(0,0)$.
• Traditional Universal Covering Space: No
• Generalized Universal Covering Space: Yes
• Homotopically Hausdorff: Yes
• Strongly (freely) homotopically Hausdorff: Yes
• Homotopically Path-Hausdorff: Yes
• $1$ $UV_0$: Yes
• $\pi_1$-shape injective: Yes

References:

There are a couple hundred papers involving the 1-dimensional earring. Here are a couple key ones:

 J.W. Cannon, G.R. Conner, The combinatorial structure of the Hawaiian earring group, Topol. Appl. 106 (2000) 225–271.

 K. Eda, Free σ-products and noncommutatively slender groups, J. Algebra 148 (1992) 243–263.

 K. Eda, K. Kawamura, The singular homology of the Hawaiian Earring, J. Lond. Math. Soc. (2) 62 (2000) 305–310.

 J.W. Morgan, I. Morrison, A van Kampen theorem for weak joins, Proc. Lond. Math. Soc. (3) 53 (1986) 562–576.