A canonical splitting of the first homology group of Peano continua

Abstract

The first singular homology of a Peano continuum $X$ with torsion-free first Cech homology, ${\check{H}}_1(x)$, splits as $H_1(X)={\check{H}}_1(X)\oplus K$ where $K$ is the homology shape kernel of $X$. Consequently if a Peano continuum $X$ is a subspace of ${\mathbb{R}}^3$, then $H_1(X)={\mathbb{Z}}^{\lambda }\oplus K$ where $K$ is the homology shape kernel of $X$ and $\lambda$ is a countable cardinal. In the process we construct cotorsion quotients of subgroups of the first homology which correspond to path-connected fibrations of $X$.