Borel equivalence relations and symmetric models Topologies for the - - PowerPoint PPT Presentation

Borel equivalence relations and symmetric models

Topologies for the Friedman-Stanley jumps Assaf Shani

UCLA

Logic colloquium, Udine, Italy July 2018

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Motivation

Given an equivalence relation E on a Polish space X, how does E behave generically?

Definition (Kanovei-Sabok-Zapletal 2013)

An analytic equivalence relation E is in the spectrum of the meager ideal if there is an equivalence relation F on a Polish space Y which is Borel bireducible with E, and furthermore for any non meager set C ⊆ Y , F ↾ C is Borel bireducible with E.

Example

The dichotomy theorems imply that E0, E1 and E ω

0 are in the

spectrum of the meager ideal, witnessed by the natural product topologies on their domains.

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Friedman-Stanley jumps

Definition

=+ on Rω is defined by the complete classification x0, x1, ... → {xi; i ∈ ω} .

Theorem (Kanovei-Sabok-Zapletal (2013))

Consider Rω with the natural product topology. If C is a comeager set, then =+↾ C is Borel bireducible to =+.

Definition

=++ on Rω2 is defined by the complete classification xi,j | i, j < ω → {{xi,j; j ∈ ω} ; i ∈ ω} .

Question (Zapletal)

Is =++ in the spectrum of the meager ideal?

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Results

=++ on Rω2 is defined by the complete classification xi,j | i, j < ω → {{xi,j; j ∈ ω} ; i ∈ ω} .

Question (Zapletal)

Is =++ in the spectrum of the meager ideal? Does =++≤=+ hold on comeager sets?

Proposition

=++ is Borel reducible to =+ on a comeager subset of Rω2.

Theorem (S.)

=++ is in the spectrum of the meager ideal.

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Borel equivalence relations and symmetric models

Theorem (S.)

Suppose E and F are Borel equivalence relations on X and Y respectively and x → Ax and y → By are classifications by countable structures of E and F respectively. Assume that f : X − → Y is a (partial) Borel reduction from E to

• F. Take x ∈ dom f in some generic extension and let A = Ax and

B = Bf (x). Then V (A) = V (B)

Example

Let A ⊆ R be a set of generic Cohen reals (a =+-invariant). The “basic Cohen model” V (A) is not of the form V (r) for any real r. It follows that =+ is not Borel reducible to =R (on any comeager set).

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A model of Monro (1973)

Let A1 be the Cohen set as above. Force over V (A1) to add a set A2 of infinitely many generic subsets of A1. Consider Monro’s model V (A1)(A2) = V (A2).

Proposition

V (A2) = V (B) for any set of reals B. Since A2 is an =++-invariant:

Corollary

=++ is not Borel reducible to =+.

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Topology for =++

Consider the equivalence relation F on Rω × (2ω)ω defined by the complete classification (x, z) → {{x(j); z(i)(j) = 1} ; i < ω} . . . . . . . . . . . . . x(3) 1 1 . . . x(2) 1 1 . . . x(1) 1 1 . . . x(0) 1 . . . → . . . . . . . . . x(3) − x(3) . . . x(2) x(2) − . . . − x(1) x(1) . . . − x(0) − . . . F is Borel bireducible with =++.

Corollary (of previous proof)

For any comeager set C ⊆ Rω × (2ω)ω, F ↾ C is not Borel reducible to =+.

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