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                 Working toward 0 = 1
     A non-trivial autohomeomorphism




ω ∗ , ω1 and non-trivial autohomeomorphisms
       ∗

      Quidquid latine dictum sit, altum videtur


                                K. P. Hart

                                Fakulta EMK
                                  TU Delft


           Praha, 8. srpen 2011: 15:40–16:10




                           K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                ∗
History
                          Working toward 0 = 1
              A non-trivial autohomeomorphism


A basic question


  All cardinals carry the discrete topology.
  Question (The Katowice Problem)
  Are there different infinite cardinals κ and λ such that κ∗ and λ∗
  are homeomorphic?
  Equivalently: are there different infinite cardinals κ and λ such
  that the Boolean algebras P(κ)/fin and P(λ)/fin are isomorphic?

  Gut reaction
  Of course not! That would be shocking.




                                    K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                         ∗
History
                         Working toward 0 = 1
             A non-trivial autohomeomorphism


A word of warning



  Remember: people thought that κ < λ would imply 2κ < 2λ .

  And had a hard time proving it.

  And then they learned that it was unprovable.

  Very unprovable: one can specify regular cardinals at will and
  create a model in which their 2-powers are equal.




                                   K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                        ∗
History
                          Working toward 0 = 1
              A non-trivial autohomeomorphism


A glimmer of hope




  If κ < λ then βκ and βλ are not homeomorphic
  (or P(κ) and P(λ) are not isomorphic)
  even if 2κ = 2λ .

  Their sets of isolated points (atoms) have different cardinalities.




                                    K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                         ∗
History
                         Working toward 0 = 1
             A non-trivial autohomeomorphism


More hope




  Theorem (Frankiewicz 1977)
  The minimum cardinal κ (if any) such that κ∗ is homeomorphic
  to λ∗ for some λ > κ must be ω.

  Theorem (Balcar and Frankiewicz 1978)
   ∗      ∗
  ω1 and ω2 are not homeomorphic.




                                   K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                        ∗
History
                         Working toward 0 = 1
             A non-trivial autohomeomorphism


Our gut was not completely wrong


  Corollary
  If ω1 κ < λ then κ∗ and λ∗ are not homeomorphic, and
  if ω2 λ then ω ∗ and λ∗ are not homeomorphic.


  So we are left with

  Question
  Are ω ∗ and ω1 ever homeomorphic?
               ∗




                                   K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                        ∗
History
                         Working toward 0 = 1
             A non-trivial autohomeomorphism


Consequences of ‘yes’



  Easiest consequence: 2ℵ0 = 2ℵ1 ;

  those are the respective weights of ω ∗ and ω1
                                               ∗


  (or cardinalities of P(ω)/fin and P(ω1 )/fin).

  So CH implies ‘no’.




                                   K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                        ∗
History
                         Working toward 0 = 1
             A non-trivial autohomeomorphism


Consequences of ‘yes’


  We consider the set ω × ω1 .

  We put
      Vn = {n} × ω1
      Hα = ω × {α}
      Bα = ω × α
      Eα = ω × [α, ω1 )

  Let γ : (ω × ω1 )∗ → (ω × ω)∗ be a homeomorphism.
                      ∗              ∗
  We can assume γ[Vn ] = {n} × ω for all n.



                                   K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                        ∗
History
                           Working toward 0 = 1
               A non-trivial autohomeomorphism


An ω1 -scale


                                       ∗      ∗
  Choose eα , for each α, such that γ[Eα ] = eα .
  Define fα : ω → ω by

                          fα (m) = min{n : m, n ∈ eα }

  Verify that fα ∗ fβ when α < β.
  If f : ω → ω then eα ∩ f is finite for many α and for those α we
  have f <∗ fα .
  So fα : α < ω1 is an ω1 -scale.

  And so MA + ¬CH implies ‘no’.



                                     K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                          ∗
History
                          Working toward 0 = 1
              A non-trivial autohomeomorphism


A strong Q-sequence

                                       ∗      ∗
  Choose hα , for each α, such that γ[Hα ] = hα .

  {hα : α < ω1 } is an almost disjoint family.
  And a very special one at that.

  Given xα ⊆ hα for each α there is x such that x ∩ hα =∗ xα for
  all α.

  Basically x ∗ = h[X ∗ ], where X is such that (X ∩ Hα )∗ = γ ← [xα ]
                                                                   ∗

  for all α.

  Such strong Q-sequences exist consistently (Stepr¯ns).
                                                   a



                                    K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                         ∗
History
                            Working toward 0 = 1
                A non-trivial autohomeomorphism


Even better (or worse?)



  It is consistent to have
       d = ω1
       a strong Q-sequence
       2ℵ0 = 2ℵ1
  simultaneously (Chodounsky).

  (Actually second implies third.)




                                      K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                           ∗
History
                          Working toward 0 = 1
              A non-trivial autohomeomorphism

                         ∗
An autohomeomorphism of ω1



  Work with the set D = Z × ω1 — so now γ : D ∗ → ω ∗ .

  Define Σ : D → D by Σ(n, α) = n + 1, α .

  Then τ = γ ◦ Σ∗ ◦ γ −1 is an autohomeomorphism of ω ∗ .

  In fact, τ is non-trivial, i.e., there is no bijection σ : a → b between
  cofinite sets such that τ [x ∗ ] = σ[x ∩ a]∗ for all subsets x of ω




                                    K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                         ∗
History
                        Working toward 0 = 1
            A non-trivial autohomeomorphism


How does that work?




     {Hα : α < ω1 } is a maximal disjoint family of Σ∗ -invariant
        ∗

     clopen sets.
     Σ∗ [Vn ] = Vn+1 for all n
          ∗      ∗

     if Vn ⊆ C ∗ for all n then Eα ⊆ C for some α and hence
          ∗

     Hα ⊆ C ∗ for all but countably many α.
        ∗




                                  K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                       ∗
History
                          Working toward 0 = 1
              A non-trivial autohomeomorphism


How does that work?



  In ω we have sets hα , vn , bα and eα that mirror this:
         ∗
      {hα : α < ω1 } is a maximal disjoint family of τ -invariant
      clopen sets.
          ∗      ∗
      τ [vn ] = vn+1 for all n
      if vn ⊆ c ∗ for all n then eα ⊆ c ∗ for some α and hence
          ∗                       ∗

      hα ⊆ c ∗ for all but countably many α.
        ∗




                                    K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                         ∗
History
                          Working toward 0 = 1
              A non-trivial autohomeomorphism


How does that work?



  The assumption that τ = σ ∗ for some σ leads, via some
  bookkeeping, to a set c with the properties that
       vn ⊆∗ c for all n and
       hα ∗ c for uncountably many α (in fact all but countably
       many).
  which neatly contradicts what’s on the previous slide . . .




                                    K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                         ∗
History
                          Working toward 0 = 1
              A non-trivial autohomeomorphism


Some more details


  Assume we have a σ : a → b inducing the isomorphism (without
  loss of generality a = ω).
  Split ω into I and F — the unions of the Infinite and Finite orbits,
  respectively.
  An infinite orbit must meet an hα in an infinite set — and at most
  two of these.
  Why is ‘two’ even possible?
  If the orbit of n is two-sided infinite then both {σ k (n) : k                           0}∗
  and {σ k (n) : k 0}∗ are τ -invariant.




                                    K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                         ∗
History
                         Working toward 0 = 1
             A non-trivial autohomeomorphism


Some more details



  It follows that hα ⊆∗ F for all but countably many α and hence
  vn ∩ F is infinite for all n.

      each hα ∩ F is a union of finite orbits
      those finite orbits have arbitrarily large cardinality
      better still, the cardinalities converge to ω.
      Our set c is the union of I and half of each finite orbit.

  Certainly hα  c is infinite for our co-countably many α.




                                   K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                        ∗
History
                           Working toward 0 = 1
               A non-trivial autohomeomorphism


Finer detail




      Write each finite orbit as {σ k (n) : −l                     k       m}
      with n ∈ v0 and |m − l|                  1
      use   {σ k (n)   : −l/2         k       m/2} as a constituent of c.




                                     K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                          ∗
History
                        Working toward 0 = 1
            A non-trivial autohomeomorphism


Comment



 This tells us that a homeomorphism between ω ∗ and ω1 must be
                                                      ∗

 quite complicated.
 At least as complicated as a non-trivial autohomeomorphism.

 Veliˇkovi´’ analysis of autohomeomorphisms of ω ∗ shows that the
     c    c
 ones that are remotely describable (by Borel maps say) are trivial.
 Our Σ is very trivial, so it’s the putative homeomorphism γ that is
 badly describable.




                                  K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                       ∗
History
                          Working toward 0 = 1
              A non-trivial autohomeomorphism


Frustration


  This feels tantalisingly close to a proof that ω ∗ and ω1 are not
                                                          ∗

  homeomorphic, to me anyway, because.
      it seems that τ should be trivial on all (but countably many)
      hα ; reason: hα should be (the graph of) a function and so τ
      should be induced by the shift on hα
      an argument with complete accumulation points should then
      give enough triviality to make those shifts cohere
      which should then lead to a contradiction
  None of which I have been able to prove . . .



                                    K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                         ∗
History
                        Working toward 0 = 1
            A non-trivial autohomeomorphism


More Info




  Website: http://fa.its.tudelft.nl/~hart




                                  K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                                       ∗
History
                Working toward 0 = 1
    A non-trivial autohomeomorphism

.




                          K. P. Hart    ω ∗ , ω1 and non-trivial autohomeomorphisms
                                               ∗

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omega*, omega1* and non-trivial autohomeomorphisms

  • 1. History Working toward 0 = 1 A non-trivial autohomeomorphism ω ∗ , ω1 and non-trivial autohomeomorphisms ∗ Quidquid latine dictum sit, altum videtur K. P. Hart Fakulta EMK TU Delft Praha, 8. srpen 2011: 15:40–16:10 K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 2. History Working toward 0 = 1 A non-trivial autohomeomorphism A basic question All cardinals carry the discrete topology. Question (The Katowice Problem) Are there different infinite cardinals κ and λ such that κ∗ and λ∗ are homeomorphic? Equivalently: are there different infinite cardinals κ and λ such that the Boolean algebras P(κ)/fin and P(λ)/fin are isomorphic? Gut reaction Of course not! That would be shocking. K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 3. History Working toward 0 = 1 A non-trivial autohomeomorphism A word of warning Remember: people thought that κ < λ would imply 2κ < 2λ . And had a hard time proving it. And then they learned that it was unprovable. Very unprovable: one can specify regular cardinals at will and create a model in which their 2-powers are equal. K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 4. History Working toward 0 = 1 A non-trivial autohomeomorphism A glimmer of hope If κ < λ then βκ and βλ are not homeomorphic (or P(κ) and P(λ) are not isomorphic) even if 2κ = 2λ . Their sets of isolated points (atoms) have different cardinalities. K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 5. History Working toward 0 = 1 A non-trivial autohomeomorphism More hope Theorem (Frankiewicz 1977) The minimum cardinal κ (if any) such that κ∗ is homeomorphic to λ∗ for some λ > κ must be ω. Theorem (Balcar and Frankiewicz 1978) ∗ ∗ ω1 and ω2 are not homeomorphic. K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 6. History Working toward 0 = 1 A non-trivial autohomeomorphism Our gut was not completely wrong Corollary If ω1 κ < λ then κ∗ and λ∗ are not homeomorphic, and if ω2 λ then ω ∗ and λ∗ are not homeomorphic. So we are left with Question Are ω ∗ and ω1 ever homeomorphic? ∗ K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 7. History Working toward 0 = 1 A non-trivial autohomeomorphism Consequences of ‘yes’ Easiest consequence: 2ℵ0 = 2ℵ1 ; those are the respective weights of ω ∗ and ω1 ∗ (or cardinalities of P(ω)/fin and P(ω1 )/fin). So CH implies ‘no’. K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 8. History Working toward 0 = 1 A non-trivial autohomeomorphism Consequences of ‘yes’ We consider the set ω × ω1 . We put Vn = {n} × ω1 Hα = ω × {α} Bα = ω × α Eα = ω × [α, ω1 ) Let γ : (ω × ω1 )∗ → (ω × ω)∗ be a homeomorphism. ∗ ∗ We can assume γ[Vn ] = {n} × ω for all n. K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 9. History Working toward 0 = 1 A non-trivial autohomeomorphism An ω1 -scale ∗ ∗ Choose eα , for each α, such that γ[Eα ] = eα . Define fα : ω → ω by fα (m) = min{n : m, n ∈ eα } Verify that fα ∗ fβ when α < β. If f : ω → ω then eα ∩ f is finite for many α and for those α we have f <∗ fα . So fα : α < ω1 is an ω1 -scale. And so MA + ¬CH implies ‘no’. K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 10. History Working toward 0 = 1 A non-trivial autohomeomorphism A strong Q-sequence ∗ ∗ Choose hα , for each α, such that γ[Hα ] = hα . {hα : α < ω1 } is an almost disjoint family. And a very special one at that. Given xα ⊆ hα for each α there is x such that x ∩ hα =∗ xα for all α. Basically x ∗ = h[X ∗ ], where X is such that (X ∩ Hα )∗ = γ ← [xα ] ∗ for all α. Such strong Q-sequences exist consistently (Stepr¯ns). a K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 11. History Working toward 0 = 1 A non-trivial autohomeomorphism Even better (or worse?) It is consistent to have d = ω1 a strong Q-sequence 2ℵ0 = 2ℵ1 simultaneously (Chodounsky). (Actually second implies third.) K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 12. History Working toward 0 = 1 A non-trivial autohomeomorphism ∗ An autohomeomorphism of ω1 Work with the set D = Z × ω1 — so now γ : D ∗ → ω ∗ . Define Σ : D → D by Σ(n, α) = n + 1, α . Then τ = γ ◦ Σ∗ ◦ γ −1 is an autohomeomorphism of ω ∗ . In fact, τ is non-trivial, i.e., there is no bijection σ : a → b between cofinite sets such that τ [x ∗ ] = σ[x ∩ a]∗ for all subsets x of ω K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 13. History Working toward 0 = 1 A non-trivial autohomeomorphism How does that work? {Hα : α < ω1 } is a maximal disjoint family of Σ∗ -invariant ∗ clopen sets. Σ∗ [Vn ] = Vn+1 for all n ∗ ∗ if Vn ⊆ C ∗ for all n then Eα ⊆ C for some α and hence ∗ Hα ⊆ C ∗ for all but countably many α. ∗ K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 14. History Working toward 0 = 1 A non-trivial autohomeomorphism How does that work? In ω we have sets hα , vn , bα and eα that mirror this: ∗ {hα : α < ω1 } is a maximal disjoint family of τ -invariant clopen sets. ∗ ∗ τ [vn ] = vn+1 for all n if vn ⊆ c ∗ for all n then eα ⊆ c ∗ for some α and hence ∗ ∗ hα ⊆ c ∗ for all but countably many α. ∗ K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 15. History Working toward 0 = 1 A non-trivial autohomeomorphism How does that work? The assumption that τ = σ ∗ for some σ leads, via some bookkeeping, to a set c with the properties that vn ⊆∗ c for all n and hα ∗ c for uncountably many α (in fact all but countably many). which neatly contradicts what’s on the previous slide . . . K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 16. History Working toward 0 = 1 A non-trivial autohomeomorphism Some more details Assume we have a σ : a → b inducing the isomorphism (without loss of generality a = ω). Split ω into I and F — the unions of the Infinite and Finite orbits, respectively. An infinite orbit must meet an hα in an infinite set — and at most two of these. Why is ‘two’ even possible? If the orbit of n is two-sided infinite then both {σ k (n) : k 0}∗ and {σ k (n) : k 0}∗ are τ -invariant. K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 17. History Working toward 0 = 1 A non-trivial autohomeomorphism Some more details It follows that hα ⊆∗ F for all but countably many α and hence vn ∩ F is infinite for all n. each hα ∩ F is a union of finite orbits those finite orbits have arbitrarily large cardinality better still, the cardinalities converge to ω. Our set c is the union of I and half of each finite orbit. Certainly hα c is infinite for our co-countably many α. K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 18. History Working toward 0 = 1 A non-trivial autohomeomorphism Finer detail Write each finite orbit as {σ k (n) : −l k m} with n ∈ v0 and |m − l| 1 use {σ k (n) : −l/2 k m/2} as a constituent of c. K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 19. History Working toward 0 = 1 A non-trivial autohomeomorphism Comment This tells us that a homeomorphism between ω ∗ and ω1 must be ∗ quite complicated. At least as complicated as a non-trivial autohomeomorphism. Veliˇkovi´’ analysis of autohomeomorphisms of ω ∗ shows that the c c ones that are remotely describable (by Borel maps say) are trivial. Our Σ is very trivial, so it’s the putative homeomorphism γ that is badly describable. K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 20. History Working toward 0 = 1 A non-trivial autohomeomorphism Frustration This feels tantalisingly close to a proof that ω ∗ and ω1 are not ∗ homeomorphic, to me anyway, because. it seems that τ should be trivial on all (but countably many) hα ; reason: hα should be (the graph of) a function and so τ should be induced by the shift on hα an argument with complete accumulation points should then give enough triviality to make those shifts cohere which should then lead to a contradiction None of which I have been able to prove . . . K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 21. History Working toward 0 = 1 A non-trivial autohomeomorphism More Info Website: http://fa.its.tudelft.nl/~hart K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗
  • 22. History Working toward 0 = 1 A non-trivial autohomeomorphism . K. P. Hart ω ∗ , ω1 and non-trivial autohomeomorphisms ∗