The fundamental group of the harmonic archipelago

On the homology of the Harmonic Archipelago

We calculate the singular homology and Čech cohomology groups of the Harmonic archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda's proof that the first singular homology groups of these spaces are isomorphic.

Archipelago groups

The classical archipelago is a non-contractible subset of $\mathbb{R}^3$ which is homeomorphic to a disk except at one non-manifold point. Its fundamental group, $\mathcal{A}$, is the quotient of the topologist's product of $\mathbb Z$, the fundamental group of the shrinking wedge of countably many copies of the circle (the Hawaiian earring), modulo the corresponding free product. We show $\mathcal{A}$ is locally free, not indicable, and has the rationals both as a subgroup and a quotient group. Replacing $\mathbb Z$ with arbitrary groups yields the notion of archipelago groups. Surprisingly, …

双锥群同构于群岛群

The double cone group is isomorphic to the archipelago group

本文证明了Griffiths双锥空间的基群同构于和谐群岛的基群，指出该同构类包含多种描述形式的群。

Griffiths双锥群同构于三锥群

The Griffiths double cone group is isomorphic to the triple

该研究证明Griffiths双锥空间基本群与三锥及满足基数条件的多锥空间基本群同构，同构非构造且无法由连续映射实现，同时证实了关于其与调和群岛基本群同构的猜想。