# The Double Earring

The double earring consists of two copies of the  usual infinite earring $\mathbb{E}$ adjoined by an arc. Specifically, one many construct it as:

$X=\left(\bigcup_{n\in\mathbb{N}}C_n\right)\cup \left([-1,0]\times \{0\}\right)\cup \left(\bigcup_{n\in\mathbb{N}}D_n\right)$

where $C_n$ and $D_n$ are the circle of radius $1/n$ centered at $(1/n,0)$ and $(-1-1/n,0)$ respectively.

The double earring

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

Fundamental Group: Using the van Kampen Theorem, one can show that $\pi_1(X)$ is isomorphic to the free product of the earring group $\pi_1(\mathbb{E})$ with itself, i.e. $\pi_1(\mathbb{E})\ast \pi_1(\mathbb{E})$. It is an interesting and non-obvious fact that this fundamental group is not isomorphic to $\pi_1(\mathbb{E})$ itself.

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

Higher homotopy groups: $\pi_n(X)=0$ for $n \geq 2$, i.e. this space is aspherical.

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

$H_1(X)$ is isomorphic (but not naturally) to $H_1(\mathbb{E})$.

Cech Homotopy groups: $\check{\pi}_n(X)=\begin{cases} \varprojlim_{n}F_{2n}, & n=1 \\ 0, & n \neq 1 \end{cases}$

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

We have a summand of $\mathbb{Z}^{\mathbb{N}}$ in $\check{H}_1(X)$ for both copies of $\mathbb{E}$ in $X$. However, $\mathbb{Z}^{\mathbb{N}}\cong \mathbb{Z}^{\mathbb{N}}\oplus \mathbb{Z}^{\mathbb{N}}$.

Wild Set/Homotopy Type: The 1-wild set is the two-point discrete set $\mathbf{w}(X)={(0,0),(-1,0)}$. Since the homotopy type of the 1-wild set is a homotopy invariant, one can see from this that $X$ is not homotopy equivalent to $\mathbb{E}$. In fact, any map $f:X\to \mathbb{E}$ which induces an injection on $\pi_1$ must map both wild points of $X$ to the wild-point of $\mathbb{E}$ and then $f$ cannot induce a surjection on $\pi_1$ using standard alternating infinite product arguments.

Other Properties:

• Semi-locally simply connected: No, not at $(0,0)$ nor $(-1,0)$.
• 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:

[1] J.W. Cannon, G.R. Conner, On the fundamental groups of one-dimensional spaces, Topology Appl. 153 (2006) 2648–2672.