Dyadic Arc Space

A pair of integers (n,j) is dyadic unital if the dyadic rational \frac{2j-1}{2^n} lies in (0,1). For each dyadic unital pair (n,j) consider the upper semicircle:

\mathbb{D}(n,j)=\left\{(x,y)\in \mathbb{R}^2\mid\left(x-\frac{2j-1}{2^n}\right)^2+y^2=\left(\frac{1}{2^n}\right)^2\text{, }x\geq 0\right\}.

Let B=[0,1]\times \{0\} be the base arc. The dyadic arc-space is the union

\mathbb{D}=B\cup \bigcup_{(n,j)} \mathbb{D}(n,j)

over all dyadic unital pairs with the subspace topology inherited from \mathbb{R}^2 and with basepoint d_0=(0,0).

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

Utility: Used in [2] to characterize the existence of generalized universal coverings (as defined in [3] and extended in [1]).

Fundamental Group: \pi_1(\mathbb{D}) embeds as the subgroup of the natural inverse limit \varprojlim_{n}F_{2^{n}-1} of finitely generated free groups. This group may also be described in terms of a reduced path calculus.

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

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

Higher homotopy groups: \pi_n(\mathbb{D})=0 for n \geq 2, i.e. \mathbb{D} is aspherical.

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

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

Wild Set/Homotopy Type: The wild set is uncountable, namely, the arc \mathbf{w}(\mathbb{D})=B. Since \mathbb{D} is a one-dimensional Peano continuum any other 1-dim. Peano continuum with fundamental group isomorphic to \pi_1(\mathbb{D}) must be homotopy equivalent to \mathbb{D}.trianglecontinuum

The dyadic arc space is homotopy equivalent to the “triangle continuum” seen on the right, which appears in [3] and [4].

Other Properties:

  • Semi-locally simply connected: No, not at any point of B
  • 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

Other Note: If you collapse the base arc to a point, you get a space homeomorphic to the infinite earring.

References:

[1] J. Brazas, Generalized covering space theories, Theory and Appl. of Categories 30 (2015) 1132-1162

[2] Brazas, Fischer, Test map characterizations of local properties of fundamental groups. Journal of Topology and Analysis. 12 (2020) 37-85. arXiv.

[3] H. Fischer, A. Zastrow, Generalized universal covering spaces and the shape group, Fundamenta Mathematicae 197 (2007) 167–196.

[4] Z. Virk, A. Zastrow, A homotopically Hausdorff space which does not admit a generalized universal covering, Topology Appl. 160 (2013) 656–666.