THE SINGULAR HOMOLOGY OF THE HAWAIIAN EARRING

Abstract

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 ]^σ_ℕℤ of integers ℤ in [4, Appendix]. Hence the first integral singular homology group H₁(ℍ) is the abelianization of the group [smashp ]^σ_ℕℤ. 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 [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(ℍ_I) 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 ]ⁿ_j=1C_j is denoted by B_n. The base point (0, 0) of B_n is denoted by o for simplicity.