Homeomorphisms of Cech-Stone remainders: the zero-dimensional case - - PowerPoint PPT Presentation

Introduction A proof of a rigidity result Further work

Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional case

Paul McKenney Joint work with Ilijas Farah BLAST 2018

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Given a topological space X, let X ∗ = βX \ X denote its ˇ Cech-Stone remainder.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Given a topological space X, let X ∗ = βX \ X denote its ˇ Cech-Stone remainder. Question If X ∗ ≃ Y ∗, how similar do X and Y have to be?

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Given a topological space X, let X ∗ = βX \ X denote its ˇ Cech-Stone remainder. Question If X ∗ ≃ Y ∗, how similar do X and Y have to be? Theorem (Parovicenko) Assume the Continuum Hypothesis. Then for every zero-dimensional, locally compact, noncompact, Hausdorff space X, X ∗ ≃ ω∗.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Theorem (Parovicenko) Assume the Continuum Hypothesis. Then for every zero-dimensional, locally compact, noncompact, Hausdorff space X, X ∗ ≃ ω∗.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Theorem (Parovicenko) Assume the Continuum Hypothesis. Then for every zero-dimensional, locally compact, noncompact, Hausdorff space X, X ∗ ≃ ω∗. Sketch. Let C(X) denote the Boolean algebra of clopen subsets of X, and K(X) its ideal of compact-open sets.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Theorem (Parovicenko) Assume the Continuum Hypothesis. Then for every zero-dimensional, locally compact, noncompact, Hausdorff space X, X ∗ ≃ ω∗. Sketch. Let C(X) denote the Boolean algebra of clopen subsets of X, and K(X) its ideal of compact-open sets. Then by Stone duality, it’s enough to prove that C(X)/K(X) ≃ P(ω)/ fin.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Theorem (Parovicenko) Assume the Continuum Hypothesis. Then for every zero-dimensional, locally compact, noncompact, Hausdorff space X, X ∗ ≃ ω∗. Sketch. Let C(X) denote the Boolean algebra of clopen subsets of X, and K(X) its ideal of compact-open sets. Then by Stone duality, it’s enough to prove that C(X)/K(X) ≃ P(ω)/ fin. These Boolean algebras are both countably saturated and have size c = ℵ1.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Theorem (Parovicenko) Assume the Continuum Hypothesis. Then for every zero-dimensional, locally compact, noncompact, Hausdorff space X, X ∗ ≃ ω∗. Sketch. Let C(X) denote the Boolean algebra of clopen subsets of X, and K(X) its ideal of compact-open sets. Then by Stone duality, it’s enough to prove that C(X)/K(X) ≃ P(ω)/ fin. These Boolean algebras are both countably saturated and have size c = ℵ1. A back-and-forth argument finishes the proof.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

So under CH, ˇ Cech-Stone remainders are very malleable.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

So under CH, ˇ Cech-Stone remainders are very malleable. Theorem (Farah-McKenney) Assume OCA and MAℵ1. Let X and Y be zero-dimensional, locally compact Polish spaces, and suppose ϕ : X ∗ → Y ∗ is a

• homeomorphism. Then there are cocompact subsets of X and Y

which are homeomorphic, and moreover ϕ is induced by such a homeomorphism.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

So under CH, ˇ Cech-Stone remainders are very malleable. Theorem (Farah-McKenney) Assume OCA and MAℵ1. Let X and Y be zero-dimensional, locally compact Polish spaces, and suppose ϕ : X ∗ → Y ∗ is a

• homeomorphism. Then there are cocompact subsets of X and Y

which are homeomorphic, and moreover ϕ is induced by such a homeomorphism. This says that under OCA and MAℵ1, ˇ Cech-Stone remainders (in a certain class) are very rigid.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Notation: given a set V we write [V ]2 for the set of unordered pairs {v, w} (v = w) of elements of V .

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Notation: given a set V we write [V ]2 for the set of unordered pairs {v, w} (v = w) of elements of V . The Open Coloring Axiom states:

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Notation: given a set V we write [V ]2 for the set of unordered pairs {v, w} (v = w) of elements of V . The Open Coloring Axiom states: For every separable metric space V and every open G ⊆ [V ]2, (where [V ]2 is identified with V × V minus the diagonal), either

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Notation: given a set V we write [V ]2 for the set of unordered pairs {v, w} (v = w) of elements of V . The Open Coloring Axiom states: For every separable metric space V and every open G ⊆ [V ]2, (where [V ]2 is identified with V × V minus the diagonal), either there is an uncountable A ⊆ V such that [A]2 ⊆ G (G has an uncountable complete subgraph), or

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Notation: given a set V we write [V ]2 for the set of unordered pairs {v, w} (v = w) of elements of V . The Open Coloring Axiom states: For every separable metric space V and every open G ⊆ [V ]2, (where [V ]2 is identified with V × V minus the diagonal), either there is an uncountable A ⊆ V such that [A]2 ⊆ G (G has an uncountable complete subgraph), or there is a partition V = ∞

n=1 Bn such that for all n,

[Bn]2 ∩ G = ∅. (G is countably chromatic.)

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Notation: given a set V we write [V ]2 for the set of unordered pairs {v, w} (v = w) of elements of V . The Open Coloring Axiom states: For every separable metric space V and every open G ⊆ [V ]2, (where [V ]2 is identified with V × V minus the diagonal), either there is an uncountable A ⊆ V such that [A]2 ⊆ G (G has an uncountable complete subgraph), or there is a partition V = ∞

n=1 Bn such that for all n,

[Bn]2 ∩ G = ∅. (G is countably chromatic.) Note: the set-theoretic strength of OCA is in the “for every separable metric V ” part. For instance, OCA is true in ZFC for analytic V ⊆ R (by a Cantor-Bendixon style argument).

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

OCA and MAℵ1 have been used to prove similar things before, most prominently

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

OCA and MAℵ1 have been used to prove similar things before, most prominently Theorem (Veliˇ ckovi´ c, 1993) Assume OCA and MAℵ1. Then every homeomorphism of ω∗ is induced by a bijection e : ω \ F1 → ω \ F2 where F1, F2 are finite.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

OCA and MAℵ1 have been used to prove similar things before, most prominently Theorem (Veliˇ ckovi´ c, 1993) Assume OCA and MAℵ1. Then every homeomorphism of ω∗ is induced by a bijection e : ω \ F1 → ω \ F2 where F1, F2 are finite. Theorem (Farah, 1996) (Same as our result but restricted to countable X and Y .)

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

OCA and MAℵ1 have been used to prove similar things before, most prominently Theorem (Veliˇ ckovi´ c, 1993) Assume OCA and MAℵ1. Then every homeomorphism of ω∗ is induced by a bijection e : ω \ F1 → ω \ F2 where F1, F2 are finite. Theorem (Farah, 1996) (Same as our result but restricted to countable X and Y .) Theorem (Farah, 2011) Assume OCA. Then every automorphism of B(ℓ2)/K(ℓ2) is induced by a linear isometry between closed subspaces of ℓ2 with finite codimension.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Theorem Assume OCA and MAℵ1. Let X and Y be zero-dimensional, locally compact, noncompact Polish spaces, and suppose ϕ : C(X)/K(X) → C(Y )/K(Y ) is an isomorphism. Then there are compact sets K ⊆ X and L ⊆ Y , and a homeomorphism e : Y \ L → X \ K, such that ϕ([A]) = [e−1(A)] for all A ∈ C(X).

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Theorem Assume OCA and MAℵ1. Let X and Y be zero-dimensional, locally compact, noncompact Polish spaces, and suppose ϕ : C(X)/K(X) → C(Y )/K(Y ) is an isomorphism. Then there are compact sets K ⊆ X and L ⊆ Y , and a homeomorphism e : Y \ L → X \ K, such that ϕ([A]) = [e−1(A)] for all A ∈ C(X). Proof.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Theorem Assume OCA and MAℵ1. Let X and Y be zero-dimensional, locally compact, noncompact Polish spaces, and suppose ϕ : C(X)/K(X) → C(Y )/K(Y ) is an isomorphism. Then there are compact sets K ⊆ X and L ⊆ Y , and a homeomorphism e : Y \ L → X \ K, such that ϕ([A]) = [e−1(A)] for all A ∈ C(X).

• Proof. (Sketch,

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Theorem Assume OCA and MAℵ1. Let X and Y be zero-dimensional, locally compact, noncompact Polish spaces, and suppose ϕ : C(X)/K(X) → C(Y )/K(Y ) is an isomorphism. Then there are compact sets K ⊆ X and L ⊆ Y , and a homeomorphism e : Y \ L → X \ K, such that ϕ([A]) = [e−1(A)] for all A ∈ C(X).

• Proof. (Sketch, of a special case,

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Theorem Assume OCA and MAℵ1. Let X and Y be zero-dimensional, locally compact, noncompact Polish spaces, and suppose ϕ : C(X)/K(X) → C(Y )/K(Y ) is an isomorphism. Then there are compact sets K ⊆ X and L ⊆ Y , and a homeomorphism e : Y \ L → X \ K, such that ϕ([A]) = [e−1(A)] for all A ∈ C(X).

• Proof. (Sketch, of a special case, using a big black box.)

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Theorem Assume OCA and MAℵ1. Let X and Y be zero-dimensional, locally compact, noncompact Polish spaces, and suppose ϕ : C(X)/K(X) → C(Y )/K(Y ) is an isomorphism. Then there are compact sets K ⊆ X and L ⊆ Y , and a homeomorphism e : Y \ L → X \ K, such that ϕ([A]) = [e−1(A)] for all A ∈ C(X).

• Proof. (Sketch, of a special case, using a big black box.)

Let X = ˙ ∞

n=1[Tn] where each Tn is a finitely-branching tree.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Theorem Assume OCA and MAℵ1. Let X and Y be zero-dimensional, locally compact, noncompact Polish spaces, and suppose ϕ : C(X)/K(X) → C(Y )/K(Y ) is an isomorphism. Then there are compact sets K ⊆ X and L ⊆ Y , and a homeomorphism e : Y \ L → X \ K, such that ϕ([A]) = [e−1(A)] for all A ∈ C(X).

• Proof. (Sketch, of a special case, using a big black box.)

Let X = ˙ ∞

n=1[Tn] where each Tn is a finitely-branching tree.

Given s ∈ Tn we write [s] for the set of x ∈ [Tn] extending s.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Theorem Assume OCA and MAℵ1. Let X and Y be zero-dimensional, locally compact, noncompact Polish spaces, and suppose ϕ : C(X)/K(X) → C(Y )/K(Y ) is an isomorphism. Then there are compact sets K ⊆ X and L ⊆ Y , and a homeomorphism e : Y \ L → X \ K, such that ϕ([A]) = [e−1(A)] for all A ∈ C(X).

• Proof. (Sketch, of a special case, using a big black box.)

Let X = ˙ ∞

n=1[Tn] where each Tn is a finitely-branching tree.

Given s ∈ Tn we write [s] for the set of x ∈ [Tn] extending s. Let P be the set of all partitions Q of X into sets of the form [s].

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Given Q ∈ P there is a natural embedding σQ : P(Q)/ fin ֒ → C(X)/K(X)

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Given Q ∈ P there is a natural embedding σQ : P(Q)/ fin ֒ → C(X)/K(X) P is ordered by eventual refinement: Q1 ≺∗ Q2 ⇐ ⇒ ∀∞U ∈ Q1 ∃V ∈ Q2 V ⊇ U

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Given Q ∈ P there is a natural embedding σQ : P(Q)/ fin ֒ → C(X)/K(X) P is ordered by eventual refinement: Q1 ≺∗ Q2 ⇐ ⇒ ∀∞U ∈ Q1 ∃V ∈ Q2 V ⊇ U If Q1 ≺∗ Q2 then there is a an embedding τQ2Q1 : P(Q2)/ fin → P(Q1)/ fin such that σQ2 = σQ1 ◦ τQ2Q1.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Given Q ∈ P there is a natural embedding σQ : P(Q)/ fin ֒ → C(X)/K(X) P is ordered by eventual refinement: Q1 ≺∗ Q2 ⇐ ⇒ ∀∞U ∈ Q1 ∃V ∈ Q2 V ⊇ U If Q1 ≺∗ Q2 then there is a an embedding τQ2Q1 : P(Q2)/ fin → P(Q1)/ fin such that σQ2 = σQ1 ◦ τQ2Q1. Lemma C(X)/K(X) is the direct limit of the algebras P(Q)/ fin along the connecting maps τQ2Q1.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

For each Q ∈ P, define ϕQ = ϕ ◦ σQ. ϕQ : P(Q)/ fin ֒ → C(Y )/K(Y )

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

For each Q ∈ P, define ϕQ = ϕ ◦ σQ. ϕQ : P(Q)/ fin ֒ → C(Y )/K(Y ) Theorem (Veliˇ ckovi´ c, essentially) Assume OCA and MAℵ1. Then every embedding ψ : P(ω)/ fin ֒ → C(Y )/K(Y ) is of the form ψ([A]) = [e−1(A)] for some continuous function e : Y → ω.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

For each Q ∈ P, define ϕQ = ϕ ◦ σQ. ϕQ : P(Q)/ fin ֒ → C(Y )/K(Y ) Theorem (Veliˇ ckovi´ c, essentially) Assume OCA and MAℵ1. Then every embedding ψ : P(ω)/ fin ֒ → C(Y )/K(Y ) is of the form ψ([A]) = [e−1(A)] for some continuous function e : Y → ω. So we get continuous functions eQ : Y → Q inducing the embeddings ϕQ.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

The fact that the eQ’s induce the same isomorphism on different subalgebras implies that they are coherent in the following way.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

The fact that the eQ’s induce the same isomorphism on different subalgebras implies that they are coherent in the following way. Given Q1, Q2 define Q1 ∨ Q2 to be the finest partition which is coarser than both Q1 and Q2.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

The fact that the eQ’s induce the same isomorphism on different subalgebras implies that they are coherent in the following way. Given Q1, Q2 define Q1 ∨ Q2 to be the finest partition which is coarser than both Q1 and Q2. Define sQ1,Q1∨Q2 : Q1 → Q1 ∨ Q2 by defining sQ1,Q1∨Q2(U) to be the unique element V of Q1 ∨ Q2 such that V ⊇ U.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

The fact that the eQ’s induce the same isomorphism on different subalgebras implies that they are coherent in the following way. Given Q1, Q2 define Q1 ∨ Q2 to be the finest partition which is coarser than both Q1 and Q2. Define sQ1,Q1∨Q2 : Q1 → Q1 ∨ Q2 by defining sQ1,Q1∨Q2(U) to be the unique element V of Q1 ∨ Q2 such that V ⊇ U. Then for any Q1, Q2 ∈ P, the set ∆(eQ1, eQ2) = {y ∈ Y | sQ1,Q1∨Q2(eQ1(y)) = sQ2,Q1∨Q2(eQ2(y))} is compact.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

The fact that the eQ’s induce the same isomorphism on different subalgebras implies that they are coherent in the following way. Given Q1, Q2 define Q1 ∨ Q2 to be the finest partition which is coarser than both Q1 and Q2. Define sQ1,Q1∨Q2 : Q1 → Q1 ∨ Q2 by defining sQ1,Q1∨Q2(U) to be the unique element V of Q1 ∨ Q2 such that V ⊇ U. Then for any Q1, Q2 ∈ P, the set ∆(eQ1, eQ2) = {y ∈ Y | sQ1,Q1∨Q2(eQ1(y)) = sQ2,Q1∨Q2(eQ2(y))} is compact. (We will say that eQ1 and eQ2 cohere exactly if ∆(eQ1, eQ2) = ∅.)

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Define G ⊆ [P]2 to be the set of {Q1, Q2} such that eQ1 and eQ2 do not cohere exactly.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Define G ⊆ [P]2 to be the set of {Q1, Q2} such that eQ1 and eQ2 do not cohere exactly. Then G is open when P is given a certain separable metric topology.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Define G ⊆ [P]2 to be the set of {Q1, Q2} such that eQ1 and eQ2 do not cohere exactly. Then G is open when P is given a certain separable metric topology. Recall: OCA says that either

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Define G ⊆ [P]2 to be the set of {Q1, Q2} such that eQ1 and eQ2 do not cohere exactly. Then G is open when P is given a certain separable metric topology. Recall: OCA says that either there is an uncountable A ⊆ P such that [A]2 ⊆ G, or

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Define G ⊆ [P]2 to be the set of {Q1, Q2} such that eQ1 and eQ2 do not cohere exactly. Then G is open when P is given a certain separable metric topology. Recall: OCA says that either there is an uncountable A ⊆ P such that [A]2 ⊆ G, or there is a partition P =

n Bn such that for each n,

[Bn]2 ∩ G = ∅.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Lemma Assume MAℵ1. Then there is no uncountable A ⊆ P such that [A]2 ⊆ G.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Lemma Assume MAℵ1. Then there is no uncountable A ⊆ P such that [A]2 ⊆ G. Sketch. WLOG |A| = ℵ1. MAℵ1 implies that there is some Q ∈ P which is ≺∗ every Q′ ∈ A. Then using the coherence of eQ with all of the eQ′’s, along with a pigeonhole argument, we can find Q1, Q2 ∈ A such that eQ1 and eQ2 cohere exactly.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Lemma Assume MAℵ1. Then there is no uncountable A ⊆ P such that [A]2 ⊆ G. Sketch. WLOG |A| = ℵ1. MAℵ1 implies that there is some Q ∈ P which is ≺∗ every Q′ ∈ A. Then using the coherence of eQ with all of the eQ′’s, along with a pigeonhole argument, we can find Q1, Q2 ∈ A such that eQ1 and eQ2 cohere exactly. Lemma If P =

n Bn then one of the Bn’s is cofinal in (P, ≻∗).

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

Lemma Assume MAℵ1. Then there is no uncountable A ⊆ P such that [A]2 ⊆ G. Sketch. WLOG |A| = ℵ1. MAℵ1 implies that there is some Q ∈ P which is ≺∗ every Q′ ∈ A. Then using the coherence of eQ with all of the eQ′’s, along with a pigeonhole argument, we can find Q1, Q2 ∈ A such that eQ1 and eQ2 cohere exactly. Lemma If P =

n Bn then one of the Bn’s is cofinal in (P, ≻∗).

Sketch. (P, ≻∗) is countably-directed.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

By OCA, then, there is a set B ⊆ P which is cofinal with respect to ≻∗ and for which the functions {eQ | Q ∈ B} cohere exactly.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

By OCA, then, there is a set B ⊆ P which is cofinal with respect to ≻∗ and for which the functions {eQ | Q ∈ B} cohere exactly. Define a function e by e(y) ∈

• Q∈B

eQ(y)

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

By OCA, then, there is a set B ⊆ P which is cofinal with respect to ≻∗ and for which the functions {eQ | Q ∈ B} cohere exactly. Define a function e by e(y) ∈

• Q∈B

eQ(y) Lemma e is a homeomorphism from a cocompact subset of Y to a cocompact subset of X, and for every A ∈ C(X), ϕ([A]) = [e−1(A)]

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work Setup Coherence OCA and MA

By OCA, then, there is a set B ⊆ P which is cofinal with respect to ≻∗ and for which the functions {eQ | Q ∈ B} cohere exactly. Define a function e by e(y) ∈

• Q∈B

eQ(y) Lemma e is a homeomorphism from a cocompact subset of Y to a cocompact subset of X, and for every A ∈ C(X), ϕ([A]) = [e−1(A)]

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Theorem (Gelfand Duality) Compact Hausdorff spaces are dual to unital, commutative C*-algebras, via X → C(X, C)

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Theorem (Gelfand Duality) Compact Hausdorff spaces are dual to unital, commutative C*-algebras, via X → C(X, C) A C*-algebra is a complex Banach space A with a product · and an involution ∗, satisfying some axioms relating those operations to each other.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Theorem (Gelfand Duality) Compact Hausdorff spaces are dual to unital, commutative C*-algebras, via X → C(X, C) A C*-algebra is a complex Banach space A with a product · and an involution ∗, satisfying some axioms relating those operations to each other. Theorem (Farah) Assume OCA. Then every automorphism of B(ℓ2)/K(ℓ2) is induced by a linear isometry between closed subspaces of ℓ2 with finite codimension.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Theorem (Gelfand Duality) Compact Hausdorff spaces are dual to unital, commutative C*-algebras, via X → C(X, C) A C*-algebra is a complex Banach space A with a product · and an involution ∗, satisfying some axioms relating those operations to each other. Theorem (Farah) Assume OCA. Then every automorphism of B(ℓ2)/K(ℓ2) is induced by a linear isometry between closed subspaces of ℓ2 with finite codimension. B(ℓ2)/K(ℓ2) and C(X ∗) are both C*-algebras of a special kind called corona algebras.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Theorem (M.-Vignati) Assume OCA∞ and MAℵ1. Then every isomorphism between two corona algebras (with certain approximation properties) is definable.

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Theorem (M.-Vignati) Assume OCA∞ and MAℵ1. Then every isomorphism between two corona algebras (with certain approximation properties) is definable. Corollary Assume OCA∞ and MAℵ1. Let X = ˙ Kn and Y = ˙ Ln where Kn and Ln are compact Hausdorff spaces. Then every homeomorphism X ∗ ≃ Y ∗ is induced by a sequence of homeomorphisms Kn ≃ Ln (after possibly permuting the indices.)

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional

Introduction A proof of a rigidity result Further work

Thank you!

Paul McKenney Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional