The definable content of homological invariants II: \v{C}ech cohomology and homotopy classification

TL;DR

This paper introduces a definable version of cech cohomology that refines classical invariants, providing complete homotopy classification for certain spaces and analyzing the complexity of homotopy maps using descriptive set theory.

Contribution

It develops a new definable cohomology functor factoring through groups with Polish covers, enhancing classical invariants and enabling detailed classification of homotopy types.

Findings

Definable cohomology functors are complete invariants for certain homotopy types.

Classical cohomology can be constant on uncountably many non-homotopy equivalent spaces.

The classification problem for maps from solenoid complements to spheres is hyperfinite but not smooth.

Abstract

This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the \v{C}ech cohomology functors $\check{\mathrm{H}}^n$ on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor $\check{\mathrm{H}}^n_{\mathrm{def}}$ taking values in the category $\mathsf{GPC}$ of groups with a Polish cover (a category first introduced in this work's predecessor), followed by (ii) a forgetful functor from $\mathsf{GPC}$ to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of $d$-spheres or $d$-tori for any $d\geq 1$, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors $\check{\mathrm{H}}^n_{\mathrm{def}}$ to show that a seminal problem in the development of algebraic topology, namely Borsuk and Eilenberg's 1936 problem of classifying, up to homotopy, the maps from a solenoid complement $S^3\backslash\Sigma$ to the $2$-sphere, is essentially hyperfinite but not smooth. In the course of this work, we record Borel definable versions of a number of classical results bearing on both the combinatorial and homotopical formulations of \v{C}ech cohomology; in aggregate, this work may be regarded as laying foundations for the descriptive set theoretic study of the homotopy relation on the space of maps from a locally compact Polish space to a polyhedron, a relation which embodies a substantial variety of classification problems arising throughout mathematics.