for uncountable Paul Larson Miami University September 26, 2015 T - - PowerPoint PPT Presentation

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Automorphisms of P(λ)/Iκ for λ uncountable

Paul Larson Miami University September 26, 2015

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

joint with Paul McKenney

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Given an infinite cardinal κ, we let Iκ denote the ideal of sets of cardinality less than κ. Fin = Iℵ0; Ctble = Iℵ1 Given cardinals κ ≤ λ and A ⊆ λ, we let [A]λ,κ = {B ⊆ λ | |A △ B| < κ}

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

A function π: P(λ)/Iκ → P(χ)/Iρ is said to be trivial on A ⊆ λ if there exist B ∈ [λ]<κ and f : A \ B → χ such that π([C]λ,κ) = [f[C \ B]]χ,ρ for all C ⊆ A. The function π is trivial if it is trivial on λ.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Automorphisms of P(ω)/Fin Theorem.(W. Rudin, 1956) Assuming CH there exist 2ℵ1 many nontrivial automorphisms of P(ω)/Fin . Theorem.(Shelah, late 1970’s?) Consistently, all automorphisms

• f P(ω)/Fin are trivial.

Theorem.(Velickovic, 1993) Assuming PFA (MAℵ1 + OCA for λ ≤ ℵ1) all automorphisms of P(λ)/Fin are trivial, for all cardinals λ.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Question 1. Is every automorphism of P(λ)/Fin trivial on a cocountable set, for every uncountable cardinal λ? Question 1a. Is every automorphism of P(λ)/Fin trivial on an uncountable set, for every uncountable cardinal λ? Question 2. Is every automorphism of P(λ)/Iκ trivial, whenever κ ≤ λ are uncountable? Question 3. Must an automorphism of P(λ)/Fin be trivial if it is trivial on all countable sets (or all sets of cardinality ℵ1), for every uncountable cardinal λ?

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

A partial result on Questions 1 and 1a Theorem.(Shelah-Stepr¯ ans) If λ > 2ℵ0 is less than the first strongly inaccessible cardinal, then every automorphism of P(λ)/Fin is trivial on a subset of λ with complement of cardinality 2ℵ0. So : Question 1a has a positive answer for λ > 2ℵ0.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Question 4. (Turzanski/Katowice) Is is consistent that P(ω)/Fin and P(ω1)/Fin are isomorphic? Theorem.(Balcar-Frankiewicz) If λ and κ are distinct cardinals such that P(κ)/Fin ≃ P(λ)/Fin, then {κ, λ} = {ω, ω1}. One step of the proof shows that (**) if P(ω)/Fin and P(ω1)/Fin are isomorphic then d = ℵ1 (so MAℵ1 fails).

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

The same proof shows that if κ ≤ µ < λ are cardinals (with κ regular) such that P(µ)/Iκ ≃ P(λ)/Iκ then {µ, λ} = {κ, κ+}.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

(Folklore) Suppose that P(ω1)/Fin and P(ω)/Fin are isomorphic, and consider the automorphism (call it π) of P(ω1)/Fin conjugate to the shift on ω. It has no nontrivial fixed points, which shows that it is not cocountably trivial. Moreover, it has the property that for no (nontrivial) A ⊆∗ B is π([A]) = [B]. (It has no nontrivial expanding points.)

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Question 5. If π is an automorphism of P(ω1)/Fin, must there be an infinite, coinfinite A ⊂ ω1 such that π([A]Fin) = [A]Fin? That is, must π have a fixed point? Question 6. If π is an automorphism of P(ω1)/Fin, must there be infinite, coinfinite A ⊆ B ⊂ ω1 such that π([A]Fin) = [B]Fin? That is, must π have an expanding point?

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Theorem.(Hart) If P(ω)/Fin and P(ω1)/Fin are isomorphic then there is a nontrivial automorphism of P(ω)/Fin. Proof: Break ω1 into Z-chains and consider the automorphism

• f P(ω)/Fin induced by shifting the chains. It has uncountably

many minimal disjoint fixed points.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Question 7. Can there be an isomorphism from P(ω1)/Fin to P(ω)/Fin which is trivial on all countable sets?

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Preserving cardinalities A function π: P(λ)/Iκ → P(χ)/Iρ is said to be cardinality preserving if for each A ⊆ λ there exists a B ⊆ χ such that |A| = |B| and π([A]λ,κ) = [B]χ,ρ. For any pairs of cardinals κ < λ with κ regular, the existence of an isomorphism between P(κ+)/Iκ and P(κ)/Iκ is equivalent to the existence of an automorphism of P(λ)/Iκ which is not cardinality preserving.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Selectors A selector for a function π: P(λ)/Iκ → P(χ)/Iρ is a function ˆ π: P(λ) → P(χ) such that π([A]λ,κ) = [ˆ π(A)]χ,ρ for all A ⊆ λ.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Lemma 1. Let κ < µ ≤ λ be infinite cardinals, with κ regular, and let ˆ π be a selector for a cardinality preserving automorphism π of P(λ)/Iκ. Define πµ on P(λ)/Iµ by setting πµ([A]λ,µ) = [ˆ π(A)]λ,µ. Then πµ is an automorphism of P(λ)/Iµ. If π is not trivial on a set with compliment in Iµ, then πµ is nontrivial.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Theorem 1. Let κ ≤ µ be infinite cardinals, with κ is regular. Then every automorphism of P(2µ)/Iκ which is trivial on all sets of size µ+ is trivial. In the case κ = µ = ω: every automorphism of P(2ℵ0)/Fin which is trivial on all sets of cardinality at most ℵ1 is trivial.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

So MAℵ1 + OCA, which implies 2ℵ0 = ℵ2 and (by Velickovic) that all automorphisms of P(ω1)/Fin are trivial, implies (by Shelah-Stepr¯ ans) that all automorphisms of P(λ)/Fin are trivial, for all λ below the least strongly inaccessible cardinal.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Proof of Theorem 1.

Let κ ≤ µ be infinite cardinals, and suppose that π is an automorphism of P(2µ)/Iκ. Let ˆ π be a bijective selector for π, and let xβ : β < 2µ list P(µ). For each γ < µ, let Rγ = {β < 2µ | γ ∈ xβ}. For each α < 2µ, let yα = {γ < µ | α ∈ ˆ π−1(Rγ)}. Finally, for each α < 2µ, let h(α) = β if yα = xβ.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

For each a ∈ [2µ]≤µ+, let fa be a trivializing function on a. Then |(h ↾ a) △ fa| ≤ µ.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Assuming that π is not trivial, there exist pairwise disjoint aα (α < µ+) of cardinality at most κ+ such that, for each α, |(h ↾ aα) △ faα| ≥ κ. Let a =

α<µ+ aα. Then

|(h ↾ a) △ fa| ≤ µ and, for all α < µ+, |(fa ↾ aα) △ faα| < κ.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

In fact, using the fact that every automorphism of P(λ)/Iκ is determined by how it acts on sets of size κ, one can show Theorem 1’. For any infinite cardinal µ, every automorphism of P(2µ)/Iµ+ which is trivial on all sets of size µ+ is trivial.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

What about automorphisms of P(2ℵ0)/Fin which are trivial on countable sets?

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Given Γ ⊆ P(2ω), we let CSN(Γ) be the smallest cardinality of a family F ⊆ (2ω)ω × (2ω)ω such that

• 1. for every (f, g) ∈ F, {f(n) : n < ω} ∪ {g(n) : n < ω} is dense

in 2ω,

• 2. for all pairs (f, g), (f ′, g′) from F, if g = g′, then

{g(n) : n < ω} ∩ {g′(n) : n < ω} = ∅,

• 3. for every (f, g) ∈ F and n < ω, f(n) = g(n), and
• 4. for every set A ∈ Γ, the set

{(f, g) ∈ F : ∃∞n < ω |A ∩ {f(n), g(n)}| = 1} has cardinality smaller than that of F, if such a family F exists. If no such family exists, we set CSN(Γ) = (2ℵ0)+.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

CSN(open) ≥ cov(Meager) Theorem 2. If CSN(Borel) > ℵ1, then every cardinality preserving automorphism of P(2ℵ0)/Fin which is trivial on all countable sets is trivial. By Theorem 1, it suffices to show this for automorphisms of P(ω1)/Fin.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Suppose that π is a cardinality-preserving automorphism of P(ω1)/Fin which is trivial on countable sets. Let ˆ π be a bijective selector for π. Let xα (α < ω1) be distinct subsets of ω. For each n < ω, let Rn = {α < ω1 | n ∈ xα}. For each α < ω1, let yα = {n < ω | α ∈ ˆ π−1(Rn)}. Finally, for each α < ω1, let h(α) = β if yα = xβ (and 0 if there is no such β).

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Let fα (α < ω1) be functions witnessing that π is trivial on each α. Assuming that π is not trivial, we get an increasing sequence αξ : ξ < ω1 of elements of ω1 such that for each ξ there exists {βξ

i : i < ω} ⊆ [αξ, αξ+1)

such that the sequences h(βξ

i ) : i < ω and fαξ+1(βξ i ) : i < ω

satisfy conditions 1-3 in the defintion of CSN.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

For every Borel set B ⊆ P(ω), letting AB = {α | xα ∈ B}, h−1[AB] △ ˆ π−1(AB) is countable. Applying CSN(Borel) then gives a contradiction.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

A partial result on Question 2 A QB-set is a set A ⊂ P(ω) such that for all X ⊆ A there is a Borel B ⊆ P(ω) such that X = A ∩ B. The set A is a Q-set if the set B can be taken to be Fσ. Martin Axiom for partial orders of cardinality κ (MAκ) implies that every subset of P(ω) of cardinality κ is a Q-set (i.e., q0 > κ). We let z be the least cardinality of a non-QB-set.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Theorem 3. Given the existence of one QB-set (of cardinality λ), the existence of a nontrivial automorphism of P(λ)/Ctble is equivalent to the existence of two disjoint QB-sets of cardinality at most λ which intersect the same Borel sets uncountably. So, if z > λ, then every automorphism of P(λ)/Ctble is trivial.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Proof of Theorem 3. Let X = {xβ : β < λ} be a QB-set, and suppose that π is an automorphism of P(λ)/Ctble. Let ˆ π be a bijective selector for π. For each n ∈ ω, let Rn = {β < λ | n ∈ xβ}. For each α < λ, let yα = {n < ω | α ∈ ˆ π−1(Rn)}. Then Y = {yα : α < λ} is a QB-set, and for each Borel set B, π([{α < λ : yα ∈ Y ∩ B}]) = [{α < λ : xα ∈ X ∩ B}]. Letting Z = X ∩ Y, and defining h on Z by setting h(α) = β if yα = xβ, we have that h witnesses that π is trivial on Z, and that π is nontrivial on Y \ Z. Furthermore, Y \ Z and X \ Z intersect the same Borel sets uncountably.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Putting together Theorem 3 with Lemma 1, (**) and the fact that q0 ≤ d we get that if q0 > ℵ1, then every automorphism of P(R)/Fin is trivial on a cocountable set (again, this extends to all λ less than the least strongly inaccesible cardinal). We don’t know if we can replace q0 with z here (i.e., whether z > ℵ1 implies that P(ω1)/Fin and P(ω)/Fin are non-isomorphic). Velickovic has shown that MAℵ1 is consistent with the existence

• f nontrivial automorphisms of P(ω)/Fin.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Fixed points

• Lemma. Suppose that κ ≤ λ are infinite cardinals, and that π is

an automorphism of P(λ)/Iκ. Let π∗ be a selector for π. Let η be an infinite regular cardinal unequal to cfκ, and suppose that Aα : α < η is a sequence of subsets of λ such that |(Aα ∪ π∗(Aα) ∪ (π∗)−1(Aα)) \ Aβ| < κ for all α < β < η. Then {Aα : α < η} is a fixed point of π. It follows that π has nontrivial fixed points if either

◮ κ is regular and λ > κ+, or ◮ κ is uncountable and π is cardinality preserving.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

If κ is regular, π is a cardinality-preserving automorphism of P(κ+)/Iκ and S is the set of non-fixed points of π, then

◮ nonstationarily many members of S have cofinality less

than κ;

◮ there exist a club C ⊆ κ+ and a covering of C ∩ S by two

subsets, each of which carry a ladder system for which 2-uniformization holds.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Theorem.(Devlin-Shelah) If there is a partition of club subset of ω1 into two sets for which 2-uniformization holds, then 2ℵ0 = 2ℵ1. So : the existence of a cardinality-preserving automorphism of P(ω1)/Fin without nontrivial fixed points implies 2ℵ0 = 2ℵ1.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Uniformization Say that a collection of sets X is strongly uniformized if whenever yx ⊆ x for each x ∈ X, there is a set z such that z ∩ x =∗ yx for each x ∈ X. For a ladder system, being strongly uniformized is the same as 2-uniformization. A strong Q-set is a strongly uniformized MAD family in P(ω) (Steprans has shown that these are consistent with MA(σ-centered) but not MA(σ-linked)).

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

Given a set A, say that a collection of sets X = {xa : a ∈ A} is an A-strongly-uniformized if

◮ whenever a, b ∈ A are such that a ⊆ b and b \ a is infinite,

xa ⊆∗ xb and xb \ xa is infinite;

◮ whenever B ⊆ A is pairwise disjoint, {xa : a ∈ B} is

strongly uniformized.

TRIVIALITY TURZANKI’S QUESTION PARTIAL TRIVIALITY FIXED POINTS UNIFORMIZATION

If ˆ π is a selector for an isomorphism from P(ω1)/Fin to P(ω)/Fin, then {ˆ π(a) : a ⊆ ω1} is a P(ω1)-strongly uniformized set. If T is either of the two sets in the covering of C ∩ S as above, then there exists a P(T)-strongly uniformized set {xa : a ⊆ T} such that for each α ∈ T, x{α} is a cofinal subset of α. Question 8. Can either of these things exist?