The Singular Homology of the Hawaiian Earring
Katsuya Eda, Kazuhiro Kawamura · Journal of the London Mathematical Society · 2000
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: H = ∪ m = 1 ∞ C m , where C m = { ( x , y ) ∈ R 2 | ( x - 1 m ) 2 + y 2 = 1 m 2 } 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 ⨳ σ N Z of integers Z in [4, Appendix]. Hence the first integral singular homology group H1(H) is the abelianization of the group ⨳ σ N Z . 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 ⩾ 2. The result for i ⩾ 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 ∪ j = 1 n C j is denoted by Bn. The base point (0, 0) of Bn is denoted by o for simplicity.