# 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:

[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.