A general Universal Coefficient Theorem, and applications

In this paper we isolate a general Universal Coefficient Theorem in the context of abelian categories. We then apply it in the left heart of the quasi-abelian category of abelian Polish groups to obtain short exact sequences relating Steenrod homology and \v{C}ech cohomology, regarded as functors to such a category. These are then used to (1) intrinsically characterize the phantom subgroups of \v{C}ech cohomology via a recursive purely algebraic formula, which can be see as a higher order generalization of the Milnor exact sequence; (2) describe phantom subgroups of \v{C}ech cohomology in terms of higher order phantom maps, in the context of the homotopical description of \v{C}ech cohomology; (3) classify up to homotopy (phantom) maps on higher order versions of solenoid complements, and measure from the viewpoint of Borel complexity theory the complexity of such a classification problem.