Wild Circle

The wild circle, denoted here simply as X, consists of the unit circle S^1 with a shrinking sequence of copies of the infinite earring \mathbb{E} attached along a countable dense subset of S^1. Specifically, one many construct it as an inverse limit of finite graphs.

The wild circle

The particular relevance of this space is that it is the simplest (and in a sense universal) space with a non-simply connected 1-wild set.

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

Fundamental Group: The fundamental group of the wild circle is best understood as a subgroup of an inverse limit \varprojlim_{n}F_n of free groups. Since this space is 1-dimensional for many purposes it is best to represent homotopy classes by their reduced representatives, which are unique up to reparameterization. According to Eda’s homotopy classification theorem it is not isomorphic to the fundamental group of any one-dimensional Peano continuum with a simply connected 1-wild set.

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_{n}, & 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}

Wild Set/Homotopy Type: The 1-wild set is the unit circle \mathbf{w}(X)= S^1, which is not simply connected. 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}, the double earring, or dyadic arc space. In fact, if Y is any metric space with \mathbf{w}(Y) not simply connected, then there is a map f:X\to Y witnessing this phenomenon.

Other Properties:

  • Semi-locally simply connected: No, not at the points of S^1.
  • 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.

[2] K. Eda, Homotopy types of one-dimensional Peano continua, Fund. Math. 209 (2010) 27-42.