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
  • 1UV_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.