Borel equivalence relations and cardinal algebras

Macdonald, Henry L.; Kechris, Alexander S.

doi:10.4064/fm242-4-2016

Cited by 12 publications

(12 citation statements)

References 15 publications

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“…Consider the Borel function g 1 : B 1 → A 1 corresponding to g, and the induced group isomorphism g 2 : B 2 → A 2 . Then we have that, in particular, g 1 is a Borel reduction from the B 0 -coset equivalence relation [36,Lemma 3.8] (which applies to such equivalence relations as remarked in [36,Lemma 3.8] after Definition 3.5), there exists a Borel reduction…”

Section: Definable Homological Algebramentioning

confidence: 91%

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

Bergfalk,

Lupini,

Panagiotopoulos

2019

Preprint

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We show that the classical homology theory of Steenrod may be enriched with descriptive settheoretic 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 nonhomeomorphic solenoids with the same Steenrod homology groups.We develop an analogous theory of definable cohomology for locally compact second countable spaces, one which may be regarded as refining Čech cohomology theory. We prove that definable cohomology is a strictly finer invariant than Čech cohomology for locally compact second countable spaces. In particular, we show that there exists an uncountable family of solenoid complements (complements of solenoids in the 3-sphere) that have the same Čech cohomology groups, and that are completely classified up to homotopy equivalence and homeomorphism by their definable cohomology.We also apply definable cohomology theory to the study of the space X, S 2 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 X, S 2 is uncountable. We will strengthen this result, by showing that each orbit of the canonical action Homeo (X) X, S 2 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 wellstudied classification problems in mathematics, such as that of automorphisms of continuous-trace C * -algebras up to unitary equivalence, or that of Hermitian line bundles, up to isomorphism, over a locally compact second countable space.

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Section: Definable Homological Algebramentioning

confidence: 91%

“…As application of Lemma 2.9 we have that every Polish exact sequence 0 → A → B → C → 0 is Borel homotopy equivalent to 0 → 0 → C → C → 0. The next lemma, which is a consequence of [36,Lemma 3.8], can be viewed as a generalization of this fact. Lemma 2.12.…”

Section: Definable Homological Algebramentioning

confidence: 92%

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

Bergfalk,

Lupini,

Panagiotopoulos

2019

Preprint

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“…We present here the notion of definable set and definable group as in [Lup20b,Lup20a]. We begin with recalling the notion of idealistic equivalence relation from [Kec94]; see also [Gao09,Definition 5.4.9] and [KM16]. We will consider as in [Lup20b] a slightly more generous notion.…”

Section: Groups With a Polish Cover And Their Solecki Subgroupsmentioning

confidence: 99%

The classification problem for extensions of torsion abelian groups

Lupini¹

2021

Preprint

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“…We also consider the notion of Polish category, which is a category whose hom-sets are Polish spaces and composition of morphisms is a continuous function, and establish some of its basic properties. Furthermore, we recall the notion of idealistic equivalence relation on a standard Borel space and some of its fundamental properties as established in [KM16,MR12]. We then define precisely the notion of (semi)definable set and (semi)definable group.…”

Section: Polish Spaces and Definable Groupsmentioning

confidence: 99%

Definable $\mathrm{K}$-homology of separable C*-algebras

Lupini¹

2020

Preprint

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In this paper we show that the K-homology groups of a separable C*-algebra can be enriched with additional descriptive set-theoretic information, and regarded as definable groups. Using a definable version of the Universal Coefficient Theorem, we prove that the corresponding definable K-homology is a finer invariant than the purely algebraic one, even when restricted to the class of UHF C*-algebras, or to the class of unital commutative C*-algebras whose spectrum is a 1-dimensional connected subspace of R 3 .

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Borel equivalence relations and cardinal algebras

Cited by 12 publications

References 15 publications

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

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

The classification problem for extensions of torsion abelian groups

Definable $\mathrm{K}$-homology of separable C*-algebras

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