Definable (co)homology, pro-torus rigidity, and (co)homological classification

Abstract

We show that the classical homology theory of Steenrod may be enriched with descriptive set-theoretic information. We prove that the resulting definable homology theory provides a strictly finer invariant than Steenrod homology for compact metrizable spaces up to homotopy. In particular, we show that pro-tori are completely classified up to homeomorphism by their definable homology. This is in contrast with the fact that, for example, there exist uncountably many pairwise non-homeomorphic solenoids with the same Steenrod homology groups. We similarly develop a definable cohomology theory which strengthens Čech cohomology and we show that it completely classifies complements of pro-tori up to homeomorphism. We also apply definable cohomology theory to the study of the space $\left[X,S^2\right]$ of homotopy classes of continuous functions from a solenoid complement $X$ to the $2$-sphere, which was initiated by Borsuk and Eilenberg in 1936. It was proved by Eilenberg and Steenrod in 1940 that the space $\left[X,S^2\right]$ is uncountable. We will strengthen this result, by showing that each orbit of the canonical action $\mathrm{Homeo}$ is countable, and hence that such an action has uncountably many orbits. This can be seen as a rigidity result, and will be deduced from a rigidity result for definable automorphisms of the Čech cohomology of $X$. We will also show that these results still hold if one replaces solenoids with pro-tori. We conclude by applying the machinery developed herein to bound the Borel complexity of several well-studied classification problems in mathematics, such as that of automorphisms of continuous-trace $C^{\ast }$-algebras up to unitary equivalence, or that of Hermitian line bundles, up to isomorphism, over a locally compact second countable space.

arXiv commentThese notes provide an early draft of what has now been corrected and expanded to the series of papers "The Definable Content of Homological Invariants I, II, and III" (see arXiv:2008.08782 and arXiv:2210.11098, for the first two papers in the series).The reader is strongly advised to refer to the series for a more detailed and rigorous treatment

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