The Singular Homology of the Hawaiian Earring

The singular homology groups of compact CW-complexes are finitelygenerated, but the groups of compact metric spaces in generalare very easy to generate infinitely and our understanding ofthese groups is far from complete even for the following compactsubset of the plane, called the Hawaiian earring: Griffiths 11] gave a presentation of the fundamental groupof H and the proof was completed by Morgan and Morrison 15].The same group is presented as the free -product of integers Z in 4, Appendix]. Hence the firstintegral singular homology group H₁(H) is the abelianizationof the group . These results have been generalized to non-metrizable counterparts H_I of H(see Section 3). In Section 2 we prove that H₁(X) is torsion-free and H_i(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, Theorem5]. In Section 3 we provide an explicit presentation of H₁(H)and also H₁(H_I) by using results of 4]. Throughout this paper, a continuum means a compact connectedmetric space and all maps are assumed to be continuous. Allhomology groups have the integers Z as the coefficients. Thebouquet with n circles is denoted by B_n. The base point (0, 0) of B_n is denoted by o forsimplicity.