# Earring Mapping Torus

Let $\mathbb{E}=\bigcup_{n\in\mathbb{N}}C_n$ where $C_n$ is the circle of radius $1/n$ with basepoint $b_0=(0,0)$. Let $\sigma:\mathbb{E}\to\mathbb{E}$ be the shift map, which takes $C_n$ homeomorphically onto $C_{n+1}$. The earring mapping torus is the mapping torus $T_f$ of the map $f$. In particular,

$T_f=\mathbb{E}\times [0,1]/\sim$ where $(x,0)\sim (f(x),1)$.

Topological Properties:  2-dimensional, path-connected, locally path-connected, compact metric space. Embeds in $\mathbb{R}^3$.

Fundamental Group: Identify $\mathbb{E}$ with the image of $\mathbb{E}\times \{0\}$ in $T_f$. Let $\ell_n$ be the loop in $\mathbb{E}$ traversing the n-th circle. If $\alpha(t)=(b_0,t)$ is the loop going once around the circle formed by the basepoint in the mapping torus, then we have the relation $[\ell_n]=[\alpha][\ell_{n+1}][\alpha^{-}]$ in $\pi_1(T_f,b_0)$. Using the van Kampen theorem and the fact that neighborhoods of $\mathbb{E}$ deformation retract onto $\mathbb{E}$, one can show that $\pi_1(T_f,b_0)\cong \left(\pi_1(\mathbb{E},b_0)\ast\langle [\alpha]\rangle\right)/N$ where $N$ is the normal subgroup generated by $\{[\alpha]^{-1}[\beta][\alpha]=\sigma_{\#}([\beta])\mid , [\beta]\in \pi_1(\mathbb{E}),\,n\in\mathbb{N}\}$.

The inner black loop represents the image of $\alpha$.

Fundamental Group Properties: Uncountable, torsion free, locally free. Not residually free.

Higher homotopy groups: $\pi_n(T_f,b_0)=0$ for $n \geq 2$, i.e. $T_f$ is aspherical.

Homology groups: One non-intuitive thing about the singular homology of this space is that $H_1$ is infinite cyclic, but $H_2$ is highly non-trivial. $\widetilde{H}_n(T_f)=\begin{cases} 0 , & n\neq 1,2 \\ \mathbb{Z}, & n = 1 \\ \{a\in H_1(\mathbb{E})\mid \sigma_{\ast}(a)=a\}, & n=2 \end{cases}$

Cech Homotopy groups: $\check{\pi}_n(T_f)=\begin{cases} \mathbb{Z}, & n=1 \\ 0, & n \neq 1 \end{cases}$. Moreover, $T_f$ is shape equivalent to $S^1$.

Cech Homology groups: $\check{H}_n(T_f)=\begin{cases} \mathbb{Z}, & n=1 \\ 0, & n \neq 1 \end{cases}$

Wild Set/Homotopy Type: The wild set $\mathbf{w}(T_f)=Im(\alpha)$ homeomorphic to $S^1$.

Other Properties:

• Semi-locally simply connected: No.
• Traditional Universal Covering Space: No
• Generalized Universal Covering Space: Yes
• Homotopically Hausdorff: Yes
• Strongly (freely) Homotopically Hausdorff: No
• Homotopically Path-Hausdorff: Yes
• $1$$UV_0$: Yes
• $\pi_1$-shape injective: No

Other notes: If you collapse the circle $Im(\alpha)$, then you get an infinitely shrinking spiral, which is contractible.