Let be the disk of radius centered at with basepoint . Let so that is the usual infinite earring. Let be a closed disk contained in the open set .

Embed in the xy-plane of . If is the center of be the cone in over with vertex . The (metric) Harmonic Archipelago is

More simply, view as a subspace of and push up a hill of height between each pair of consecutive circles and .

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

**Relative CW construction:** Let be a homeomorphism that generates . For each , attach a 2-cell to by the attaching loop .

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

**Fundamental Group:** Using the relative CW construction, it is clear that is isomorphic to the quotient where is the normal closure of in . One of the consequences of this is that in .

Alternatively, let be the map, which is determined by , i.e. shifting the -th circle to the -st circle and the identity elsewhere. Then there is a canonical isomorphim to the direct limit

.

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

**Higher homotopy groups:** for , i.e. is aspherical.

**Homology groups: **

**Cech homotopy groups:** for all .

**Cech homology groups:** for all .

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

**Cech cohomology groups:**

**Wild Set/Homotopy Type:** The wild set 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**–:**No**-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[1] W. Bogley, A. Sieradski, *Universal path spaces*, Unpublished manuscript.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.