The singular homology groups of compact CW-complexes are finitely generated, but the groups of compact metric spaces in general are very easy to generate infinitely and our understanding of these groups is far from complete even for the following compact subset of the plane, called the Hawaiian earring:

formula here

Griffiths [11] gave a presentation of the fundamental group of ℍ and the proof was completed by Morgan and Morrison [15]. The same group is presented as the free σ-product [smashp ]σℕℤ of integers ℤ in [4, Appendix]. Hence the first integral singular homology group H1(ℍ) is the abelianization of the group [smashp ]σℕℤ. These results have been generalized to non-metrizable counterparts ℍI of ℍ (see Section 3).

In Section 2 we prove that H1(X) is torsion-free and Hi(X) = 0 for each one-dimensional normal space X and for each i [ges ] 2. The result for i [ges ] 2 is a slight generalization of [2, Theorem 5]. In Section 3 we provide an explicit presentation of H1(ℍ) and also H1(ℍI) by using results of [4].

Throughout this paper, a continuum means a compact connected metric space and all maps are assumed to be continuous. All homology groups have the integers ℤ as the coefficients. The bouquet with n circles [xcup ]nj=1Cj is denoted by Bn. The base point (0, 0) of Bn is denoted by o for simplicity.