A general Universal Coefficient Theorem, and applications

2020/09/22 by Luigi Caputi, Martino Lupini, Lupini, Martino · 1 citation
Mathematics · Computer Science · #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2009.10805

Abstract

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 Čech cohomology, regarded as functors to such a category. These are then used to (1) intrinsically characterize the phantom subgroups of Čech 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 Čech cohomology in terms of higher order phantom maps, in the context of the homotopical description of Čech 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.

Cited by

• Banach and Fréchet groups, and the derivation and automorphism lengths of C*-algebras
• Definable K-homology of separable C*-algebras

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