Harmonic Archipelago

Let D_n\subseteq\mathbb{R}^2 be the disk of radius 1/n centered at (1/n,0) with basepoint b_0=(0,0). Let C_n=\partial D_n so that \mathbb{E}=\bigcup_{n\in\mathbb{N}}C_{n} is the usual infinite earring. Let E_n be a closed disk contained in the open set D_n\backslash D_{n+1}.

Embed D in the xy-plane of \mathbb{R}^3. If (a_n,b_n) is the center of K_n be the cone in \mathbb{R}^3 over \partial E_n with vertex (a_,b_n,1). The (metric) Harmonic Archipelago is

\mathbb{HA}=\left(D\backslash \bigcup_{n\in\mathbb{N}}int(E_n)\right)\cup\bigcup_{n\in\mathbb{N}}K_n

More simply, view \mathbb{E} as a subspace of D and push up a hill of height 1 between each pair of consecutive circles C_n and C_{n+1}.

The harmonic archipelago

The harmonic archipelago \mathbb{HA}

Topological Properties of the construction in 3-space:  2-dimensional, path-connected, locally path-connected, metric space. Not compact.

Relative CW construction: Let \ell_{n}:S^1\to C_n be a homeomorphism that generates \pi_1(C_n,b_0). For each n\in\mathbb{N}, attach a 2-cell e_{n}^{2} to \mathbb{E} by the attaching loop \ell_{n}\cdot\ell_{n+1}^{-}.

Topological Properties of the relative CW construction:  2-dimensional, path-connected, locally path-connected, paracompact Hausdorff space. Not compact. Not first countable at b_0. Homotopy theory is easier to do in this construction since every compact set may intersect at most finitely many of the attached 2-cells e_{n}^{2}. The metric and relative CW constructions are homotopy equivalent.

Fundamental Group: Using the relative CW construction, it is clear that \pi_1(\mathbb{HA},b_0) is isomorphic to the quotient \pi_1(\mathbb{E},b_0)/N where N is the normal closure of \{[\ell_{n}][\ell_{n+1}]^{-1}\mid n\in\mathbb{N}\} in \pi_1(\mathbb{E},b_0). One of the consequences of this is that 1\neq [\ell_1]=[\ell_2]=[\ell_3]=\cdots in \pi_1(\mathbb{HA},b_0).

Alternatively, let f_n:\mathbb{E}\to \mathbb{E} be the map, which is determined by f_{n}\circ \ell_{m}=\begin{cases} \ell_m , & \text{if }m\neq n\\ \ell_{n+1} & \text{if }m= n \end{cases}, i.e. shifting the n-th circle to the (n+1)-st circle and the identity elsewhere. Then there is a canonical isomorphim to the direct limit

\pi_1(\mathbb{HA},b_0)\cong \varinjlim_{n}(\pi_1(\mathbb{E},b_0),(f_n)_{\#}).

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

Higher homotopy groups: \pi_n(\mathbb{HA},b_0)=0 for n \geq 2, i.e. \mathbb{HA} is aspherical.

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

Cech homotopy groups: \check{\pi}_n(\mathbb{HA})=0 for all n\geq 1.

Cech homology groups: \check{H}_n(\mathbb{HA})=0 for all n\geq 1.

The Cech homotopy and homology groups are trivial even though nerves of carefully chosen open covers of \mathbb{HA} are each homotopy equivalent to an infinite wedge of 2-spheres.

Cech cohomology groups: \check{H}^n(\mathbb{HA})= \begin{cases} \prod_{\mathbb{N}}\mathbb{Z}/\bigoplus_{\mathbb{N}}\mathbb{Z}, & n=2\\ 0, & n \neq 2 \end{cases}

Wild Set/Homotopy Type: The wild set \mathbf{w}(\mathbb{HA})={b_0} is a single point.

Other Properties:

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

References:

[1] W. Bogley, A. Sieradski, Universal path spaces, Unpublished manuscript.
http://people.oregonstate.edu/~bogleyw/research/ups.pdf

[2] G. Conner, W. Hojka, M. Meilstrup, Archipelago Groups. Proceedings of the American Mathematical Society. 143 no. 11 (2015) 4973–4988.

[3] P. Fabel, The fundamental group of the harmonic archipelago. Unpublished manuscript. Arxiv.

[4] U.H. Karimov, D. Repovš, On the homology of the harmonic archipelago, Cent. Eur. J. Math. 10 (2012) 863–872.