Polishable Borel equivalence relations S lawomir Solecki Cornell - - PowerPoint PPT Presentation

Polishable Borel equivalence relations

S lawomir Solecki

Cornell University Research supported by NSF grant DMS–1700426

June 2018

Outline

Outline of Topics

1

Introduction

2

Polishable equivalence relations

3

Canonical approximations

4

Polishable equivalence relations continued

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Introduction

Introduction

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Introduction

Scope: E a Borel equivalence relation on a Polish space E induced by a continuous action of a Borel group Aim: introduce a notion of Polishable equivalence relations Requirements: (1) orbit equivalence relations of continuous Polish group actions ⊆ Polishable equivalence relations ⊆ idealistic equivalence relations; (2) approximability by transfinite sequences of “simple” Polishable equivalence relations.

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Introduction

Recall the notion of uniformity V on a set X: V is a closed upwards family of symmetric sets, whose intersection is the diagonal, and such that for each V ∈ V there exists W ∈ V with W ◦ W ⊆ V . V induces a topology t(V) whose neighborhood basis (not necessarily open) at x ∈ X is {Vx : V ∈ V}. Weil: t(V) is metrizable if and only if V has a countable basis.

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Introduction

The following basic framework will show up on several occasions. X a set, τ a topology on X, V a uniformity on X, and Γ a group of transformations of X. We say that τ, V, and Γ are compatible if — τ is Polish; — t(V) is completely metrizable; — Γ is countable; — τ ⊆ t(V); — functions in Γ are τ-homeomorphisms; — functions in Γ are V-uniform.

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Introduction

Let a topology τ and a unformity V, both on X, be compatible. We say that V has a Borel basis (with respect to τ) if it has a basis consisting

• f subsets of X × X that are Borel with respect to τ × τ.

Lemma If V has a Borel basis and τ is Polish, then V has a Borel basis consisting

• f sets that are open with respect to t(V) × t(V).

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Introduction

Let a uniformity V and a group of transformations Γ, both on X, be compatible. For V ∈ V and γ ∈ Γ, let γ ≤ V iff γx ∈ Vx for each x ∈ X. Γ is dense in V if, for each V ∈ V, there exists W ∈ V such that, for each x ∈ X, {γx : γ ∈ Γ, γ ≤ V } is t(V)-dense in Wx.

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Introduction

Examples

• 1. Let (X, d) be a metric space. The uniformity induced by d is the

upward closure of the family of all sets of the form {(x, y) ∈ X × X : d(x, y) < r} for r ∈ R, r > 0.

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Introduction

• 2. Let a be a continuous action of a Polish group G on a Polish space

(X, τ). The action a induces the uniformity Va = the upward closure of { ˆ V : 1 ∈ V = V −1 open in G}, where ˆ V = {(x, y) ∈ X × X : ∃g ∈ V gx = y}. If Γa is the group of transformations of X induced by a countable dense subgroup of G, then τ, Va and Γa are compatible. Γa is dense in Va. If Ea is the orbit equivalence relation, then Ea ∈ Va.

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Polishable equivalence relations

Polishable equivalence relations

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Polishable equivalence relations

E an equivalence relation on a Polish space (X, τ) E is Polishable if there is a uniformity V on X and a group Γ of transformations of X such that — τ, V and Γ are compatible; — V has a Borel basis with respect to τ; and, for each x ∈ X, — [x]E is Gδ with respect to t(V); — Γx is a t(V)-dense subset of [x]E. So t(V) is a Polish topology when restricted to each E-class; the assignment of the Polish topologies to E-classes (the witnessing of completeness and separability) is global/uniform.

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Polishable equivalence relations

Kechris–Louveau: E an equivalence relation on a Polish space X. E is idealistic if there is an assignment C → I(C) that with each equivalence class C of E associates a σ-ideal I(C) of subsets of C such that C ∈ I(C) and, for each Borel set A ⊆ X × X, the set {x ∈ X : Ax ∩ [x]E ∈ I([x]E)} is Borel.

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Polishable equivalence relations

Theorem (i) If E be a Polishable Borel equivalence relation, then E is idealistic. (ii) Let a be a continuous action of a Polish group on a Polish space. If the orbit equivalence relation Ea is Borel, then it is Polishable.

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Polishable equivalence relations

For (i): if a uniformity V witnesses that E is Polishable, then the assignment C → {M ⊆ C : M meager in C with respect to t(V)}, where C is an E-equivalence class, witnesses that E is idealistic. For (ii): The uniformity Va and the group Γa witness Polishability of Ea. Proof uses work of Becker–Kechris.

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Canonical approximations

Canonical approximations

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Canonical approximations

(X, τ) a topological space An operation on subsets of X × X For A ⊆ X × X symmetric, let Aτ = {(x, y) ∈ X × X : y ∈ Ax and x ∈ Ay}. Aτ is symmetric. An operation on families of symmetric subsets of X × X For an upward closed family U of symmetric subsets of X × X, let Uτ = the upward closure of {Aτ : A ∈ U}.

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Canonical approximations

τ a topology on X, V a uniformity on X; assume they are compatible. Let α ≤ ω1. (Vξ)0<ξ<α, where each Vξ is a uniformity on X with t(Vξ) completely metrizable, is called a canonical approximation of V if — V1 = Vτ and Vξ = (V)t(Vξ−1), for all successor ξ < α; — Vλ =

ξ<λ Vξ, for all limit λ < α.

Notation: τ0 = τ, τξ = t(Vξ) for 0 < ξ < α.

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Canonical approximations

Main issues: — termination at V of canonical approximations to V; — uniqueness of canonical approximations to V (trivial); — existence of canonical approximations to V.

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Canonical approximations

Termination Theorem Let τ, V be compatible. Let (Vξ)0<ξ<β be a canonical approximation of V, with β ≤ ω1, and let α < β. If V has an open basis consisting of sets in

ξ<α Π0 1+ξ with respect to τ,

then V = Vα.

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Canonical approximations

Lemma (Main Lemma) Let τα, α ≤ ω1, be completely metrizable topologies on X such that — τα ⊆ τβ for α < β ≤ ω1 and — for each α < ω1, if F is τξ-closed for some ξ < α, then intτα(F) = intτω1(F). If τω1 has an open basis consisting of sets in

ξ<α Π0 1+ξ with respect to

τ0, then τω1 = τα.

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Canonical approximations

Existence Theorem Assume that τ, V, and Γ are compatible, and Γ is dense in V. There exists a canonical approximation (Vξ)0<ξ<ω1 of V. Moreover, for each 0 < ξ < ω1, — Vξ has a basis consisting of Π0

1+ξ·2 sets with respect to τ × τ;

— τ, Vξ, and Γ are compatible and Γ is dense in Vξ.

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Canonical approximations

Lemma Assume that τ, V, Γ are compatible and Γ is dense in V. Then Vτ is a uniformity.

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Canonical approximations

Summary Corollary Assume: τ, V, and Γ compatible, V has a Borel basis with respect to τ, and Γ is dense in V. Then: for some α < ω1, there exists a canonical approximation (Vξ)0<ξ≤α

• f V with Vα = V.

Moreover: Vξ has a basis consisting of Π0

1+ξ·2 sets with respect to τ × τ,

for each 0 < ξ ≤ α.

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Polishable equivalence relations cont’d

Polishable equivalence relations continued

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Polishable equivalence relations cont’d

τ, V, and Γ, all on X, compatible Assume Γ is dense in V, and consider the canonical approximation (Vξ)0<ξ<ω1 to V. E an equivalence relation on X Define an ω1 sequence of equivalence relations Eξ = E τξ. Then E0 ⊇ E1 ⊇ · · · ⊇ Eξ ⊇ · · · ⊇ E.

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Polishable equivalence relations cont’d

Theorem Adopt notation and assumptions as above. Assume additionally that E is Polishable as witnessed by V and Γ, and E ∈ V. If E is Borel, then Eβ = E, for some 0 < β < ω1. Moreover, for each ξ < ω1, — Eξ is Polishable as witnessed by Vξ and Γ; — Eξ is Gδ with respect to τξ × τξ; — Eξ is Π0

1+ξ·2 with respect to τ × τ.

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Polishable equivalence relations cont’d

Corollary Let a be a continuous action of a Polish group on a Polish space. Let (Vξ)0<ξ<ω1 be the canonical approximation to Va. If Ea is Borel, then Eβ = Ea, for some 0 < β < ω1. Moreover, for each ξ < ω1, — Eξ is Polishable as witnessed by Vξ and Γa; — Eξ is Gδ with respect to τξ × τξ; — Eξ is Π0

1+ξ·2 with respect to τ × τ;

— Vξ has a basis consisting of sets that are Π0

1+ξ·2 with respect to τ × τ;

— Vβ = Va.

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