Singular Homology of One-Dimensional Spaces

Abstract

If X is a one-dimensional separable metric space, then wk(X) = 0 for all > 1 (see [2]). Hence, such a space X is a K(r, 1), and w1(X) determines the singular homology of X. The principal result of this paper is that the integral singular homology groups Hk(X, Z) vanish for all k>1. In Section 2 we show that the k homology group of a group w is a direct limit of the kth homology groups of the finitely generated subgroups of r. We also show that if w is an inverse limit of free groups of finite rank, then w is locally free; i.e., its finitely generated subgroups are free. These two results are combined to show that if wr is such an inverse limit, then Hk(r, Z) = 0 for all > 1. Sections 3 and 4 are devoted to showing that, if X is the Menger uni