On the homology of the Harmonic Archipelago

TL;DR

This paper computes the singular homology and Čech cohomology of the Harmonic Archipelago, revealing it is not homotopy equivalent to the Griffiths space, despite their first homology groups being isomorphic.

Contribution

It provides the first detailed homology and cohomology calculations for the Harmonic Archipelago, distinguishing it from the Griffiths space.

Findings

Harmonic Archipelago's homology groups are explicitly calculated.

It is shown that the Harmonic Archipelago is not homotopy equivalent to the Griffiths space.

First homology groups of these spaces are isomorphic, but higher invariants differ.

Abstract

We calculate the singular homology and \v{C}ech 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.