Extensions of generalized homology theories (Q1103914)

Let E denote a CW-spectrum and \(E_*(-)\) the corresponding homology theory on the category of based finite CW-complexes. The following extension problems are discussed: Can \(E_*(-)\) be extended over the category of based compact metric spaces or over the strong shape category? Under what circumstances is such an extension uniquely determined? In particular homology theories satisfying the strong wedge axiom allow interesting answers. The results provide a new proof of the main theorem of \textit{D. S. Kahn}, \textit{J. Kaminker} and \textit{C. Schochet} [Mich. Math. J. 24, 203-224 (1977; Zbl 0384.55001)] on the Brown-Douglas- Fillmore homology \(\epsilon_*\).