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
  • 1UV_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.