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:
Griffiths  gave a presentation of the fundamental group of ℍ and the proof
was completed by Morgan and Morrison . 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 .
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.