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 $\mathbb{HA}$

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:

 W. Bogley, A. Sieradski, Universal path spaces, Unpublished manuscript.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

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

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 Unpublished manuscript.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

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