aa r X i v : . [ m a t h . L O ] N ov TIE-POINTS AND FIXED-POINTS IN N ∗ ALAN DOW AND SAHARON SHELAH
Abstract.
A point x is a (bow) tie-point of a space X if X \ { x } can be partitioned into (relatively) clopen sets each with x in itsclosure. Tie-points have appeared in the construction of non-trivialautohomeomorphisms of β N \ N (e.g. [10, 8]) and in the recent studyof (precisely) 2-to-1 maps on β N \ N . In these cases the tie-pointshave been the unique fixed point of an involution on β N \ N . Thispaper is motivated by the search for 2-to-1 maps and obtainingtie-points of strikingly differing characteristics. Introduction
A point x is a tie-point of a space X if there are closed sets A, B of X such that { x } = A ∩ B and x is an adherent point of each of A and B . We picture (and denote) this as X = A ⊲⊳ x B where A, B are the closed sets which have a unique common accumulation point x and say that x is a tie-point as witnessed by A, B . Let A ≡ x B meanthat there is a homeomorphism from A to B with x as a fixed point.If X = A ⊲⊳ x B and A ≡ x B , then there is an involution F of X (i.e. F = F ) such that { x } = fix( F ). In this case we will say that x is asymmetric tie-point of X .An autohomeomorphism F of β N \ N (or N ∗ ) is said to be trivial ifthere is a bijection f between cofinite subsets of N such that F = βf ↾ β N \ N . If F is a trivial autohomeomorphism, then fix( F ) is clopen;so of course β N \ N will have no symmetric tie-points in this case if allautohomeomorphisms are trivial.If A and B are arbitrary compact spaces, and if x ∈ A and y ∈ B are accumulation points, then let A ⊲⊳ x = y B denote the quotient space of Date : October 29, 2018.1991
Mathematics Subject Classification.
Key words and phrases. automorphism, Stone-Cech, fixed points.Research of the first author was supported by NSF grant No. NSF-. Theresearch of the second author was supported by The Israel Science Foundationfounded by the Israel Academy of Sciences and Humanities, and by NSF grant No.NSF- . This is paper number 916 in the second author’s personal listing. A ⊕ B obtained by identifying x and y and let xy denote the collapsedpoint. Clearly the point xy is a tie-point of this space.We came to the study of tie-points via the following observation. Proposition 1.1. If x, y are symmetric tie-points of β N \ N as wit-nessed by A, B and A ′ , B ′ respectively, then there is a 2-to-1 mappingfrom β N \ N onto the space A ⊲⊳ x = y B ′ . The proposition holds more generally if x and y are fixed points ofinvolutions F, F ′ respectively. That is, replace A by the quotient spaceof β N \ N obtained by collapsing all sets { z, F ( z ) } to single pointsand similary replace B ′ by the quotient space induced by F ′ . It is anopen problem to determine if 2-to-1 continuous images of β N \ N arehomeomorphic to β N \ N [5]. It is known to be true if CH [3] or PFA[2] holds.There are many interesting questions that arise naturally when con-sidering the concept of tie-points in β N \ N . Given a closed set A ⊂ β N \ N , let I A = { a ⊂ N : a ∗ ⊂ A } . Given an ideal I of subsets of N , let I ⊥ = { b ⊂ N : ( ∀ a ∈ I ) a ∩ b = ∗ ∅} and I + = { d ⊂ N : ( ∀ a ∈ I ) d \ a / ∈ I ⊥ } . If J ⊂ [ N ] ω , let J ↓ = S J ∈J P ( J ). Say that J ⊂ I isunbounded in I if for each a ∈ I , there is a b ∈ J such that b \ a isinfinite. Definition 1.1. If I is an ideal of subsets of N , set cf( I ) to be thecofinality of I ; b ( I ) is the minimum cardinality of an unbounded familyin I ; δ ( I ) is the minimum cardinality of a subset J of I such that J ↓ is dense in I .If β N \ N = A ⊲⊳ x B , then I B = I ⊥ A and x is the unique ultrafilteron N extending I + A ∩ I + B . The character of x in β N \ N is equal to themaximum of cf( I A ) and cf( I B ). Definition 1.2.
Say that a tie-point x has (i) b -type; (ii) δ -type; re-spectively (iii) b δ -type, ( κ, λ ) if β N \ N = A ⊲⊳ x B and ( κ, λ ) equals: (i)( b ( I A ) , b ( I B )) (ii) ( δ ( I A ) , δ ( I B )); and (iii) each of ( b ( I A ) , b ( I B )) and( δ ( I A ) , δ ( I B )). We will adopt the convention to put the smaller of thepair ( κ, λ ) in the first coordinate.Again, it is interesting to note that if x is a tie-point of b -type ( κ, λ ),then it is uniquely determined (in β N \ N ) by λ many subsets of N since x will be the unique point extending the family (( J A ) ↓ ) + ∩ (( J B ) ↓ ) + where J A and J B are unbounded subfamilies of I A and I B . Question 1.1.
Can there be a tie-point in β N \ N with δ -type ( κ, λ )with κ ≤ λ less than the character of the point? IE-POINTS AND FIXED-POINTS IN N ∗ Question 1.2.
Can β N \ N have tie-points of δ -type ( ω , ω ) and( ω , ω )? Proposition 1.2. If β N \ N has symmetric tie-points of δ -type ( κ, κ ) and ( λ, λ ) , but no tie-points of δ -type ( κ, λ ) , then β N \ N has a 2-to-1image which is not homeomorphic to β N \ N . One could say that a tie-point x was radioactive in X (i.e. ▽ ⊲⊳ ) if X \ { x } can be similarly split into 3 (or more) relatively clopen setsaccumulating to x . This is equivalent to X = A ⊲⊳ x B such that x is atie-point in either A or B .Each point of character ω in β N \ N is a radioactive point (in partic-ular is a tie-point). P-points of character ω are symmetric tie-pointsof b δ -type ( ω , ω ), while points of character ω which are not P-pointswill have b -type ( ω, ω ) and δ -type ( ω , ω ). If there is a tie-point of b -type ( κ, λ ), then of course there are ( κ, λ )-gaps. If there is a tie-pointof δ -type ( κ, λ ), then p ≤ κ . Proposition 1.3. If β N \ N = A ⊲⊳ x B , then p ≤ δ ( I A ) .Proof. If J ⊂ I A has cardinality less than p , there is, by Solovay’sLemma (and Bell’s Theorem) an infinite set C ⊂ N such that C and N \ C each meet every infinite set of the form J \ ( S J ′ ) where { J }∪J ′ ∈ [ J ] <ω . We may assume that C / ∈ x , hence there are a ∈ I A and b ∈ I B such that C ⊂ a ∪ b . However no finite union from J covers a showingthat J ↓ can not be dense in I A . (cid:3) Although it does not seem to be completely trivial, it can be shownthat PFA implies there are no tie-points (the hardest case to eliminateis those of b -type ( ω , ω ))). Question 1.3.
Does p > ω imply there are no tie-points of b -type( ω , ω )?Analogous to tie-points, we also define a tie-set: say that K ⊂ β N \ N is a tie-set if β N \ N = A ⊲⊳ K B and K = A ∩ B , A = A \ K , and B = B \ K . Say that K is a symmetric tie-set if there is an involution F such that K = fix( F ) and F [ A ] = B . Question 1.4. If F is an involution on β N \ N such that K = fix( F )has empty interior, is K a (symmetric) tie-set? Question 1.5.
Is there some natural restriction on which compactspaces can (or can not) be homeomorphic to the fixed point set ofsome involution of β N \ N ?Again, we note a possible application to 2-to-1 maps. A. DOW AND S. SHELAH
Proposition 1.4.
Assume that F is an involution of β N \ N with K = fix( F ) = ∅ . Further assume that K has a symmetric tie-point x (i.e. K = A ⊲⊳ x B ), then β N \ N has a 2-to-1 continuous image whichhas a symmetric tie-point (and possibly β N \ N does not have such atie-point). Question 1.6. If F is an involution of N ∗ , is the quotient space N ∗ /F (in which each { x, F ( x ) } is collapsed to a single point) a homeomorphiccopy of β N \ N ? Proposition 1.5 (CH) . If F is an involution of β N \ N , then thequotient space N ∗ /F is homeomorphic to β N \ N .Proof. If fix( F ) is empty, then N ∗ /F is a 2-to-1 image of β N \ N , andso is a copy of β N \ N . If fix( F ) is not empty, then consider two copies,( N ∗ , F ) and ( N ∗ , F ), of ( N ∗ , F ). The quotient space of N ∗ /F ⊕ N ∗ /F obtained by identifying the two homeomorphic sets fix( F ) and fix( F )will be a 2-to-1-image of N ∗ , hence again a copy of N ∗ . Since N ∗ \ fix( F )and N ∗ \ fix( F ) are disjoint and homeomorphic, it follows easily thatfix( F ) must be a P-set in N ∗ . It is trivial to verify that a regular closedset of N ∗ with a P-set boundary will be (in a model of CH) a copy of N ∗ . Therefore the copy of N ∗ /F in this final quotient space is a copyof N ∗ . (cid:3) a spectrum of tie-sets We adapt a method from [1] to produce a model in which there aretie-sets of specified b δ -types. We further arrange that these tie-sets willthemselves have tie-points but unfortunately we are not able to makethe tie-sets symmetric. In the next section we make some progress ininvolving involutions. Theorem 2.1.
Assume GCH and that Λ is a set of regular uncountablecardinals such that for each λ ∈ Λ , T λ is a <λ -closed λ + -Souslin tree.There is a forcing extension in which there is a tie-set K (of b δ -type ( c , c ) ) and for each λ ∈ Λ , there is a tie-set K λ of b δ -type ( λ + , λ + ) suchthat K ∩ K λ is a single point which is a tie-point of K λ . Furthermore,for µ ≤ λ < c , if µ = λ or λ / ∈ Λ , then there is no tie-set of b δ -type ( µ, λ ) . We will assume that our Souslin trees are well-pruned and are ever ω -ary branching. That is, if T λ is a λ + -Souslin tree, we assume thatfor each t ∈ T , t has exactly ω immediate successors denoted { t ⌢ ℓ : ℓ ∈ ω } and that { s ∈ T λ : t < s } has cardinality λ + (and so hassuccessors on every level). A poset is <κ -closed if every directed subset IE-POINTS AND FIXED-POINTS IN N ∗ of cardinality less than κ has a lower bound. A poset is <κ -distributiveif the intersection of any family of fewer than κ dense open subsets isagain dense. For a cardinal µ , let µ − be the minimum cardinal suchthat ( µ − ) + ≥ µ (i.e. the predecessor if µ is a successor).The main idea of the construction is nicely illustrated by the follow-ing. Proposition 2.2.
Assume that β N \ N has no tie-sets of b δ -type ( κ , κ ) for some κ ≤ κ < c . Also assume that λ + < c is such that λ + is distinct from one of κ , κ and that T λ is a λ + -Souslin tree and { ( a t , x t , b t ) : t ∈ T λ } ⊂ ([ N ] ω ) satisfy that, for t < s ∈ T λ : (1) { a t , x t , b t } is a partition of N , (2) x t ⌢ j ∩ x t ⌢ ℓ = ∅ for j < ℓ , (3) x s ⊂ ∗ x t , a t ⊂ ∗ a s , and b t ⊂ ∗ b s , (4) for each ℓ ∈ ω , x t ⌢ ℓ +1 ⊂ ∗ a t ⌢ ℓ and x t ⌢ ℓ +2 ⊂ ∗ b t ⌢ ℓ ,then if ρ ∈ [ T λ ] λ + is a generic branch (i.e. ρ ( α ) is an element of the α -th level of T λ for each α ∈ λ + ), then K ρ = T α ∈ λ + x ∗ ρ ( α ) is a tie-set of β N \ N of b δ -type ( λ + , λ + ) , and there is no tie-set of b δ -type ( κ , κ ) . (5) Assume further that { ( a ξ , x ξ , b ξ ) : ξ ∈ c } is a family of partitionsof N such that { x ξ : ξ ∈ c } is a mod finite descending familyof subsets of N such that for each Y ⊂ N , there is a maximalantichain A Y ⊂ T λ and some ξ ∈ c such that for each t ∈ A Y , x t ∩ x ξ is a proper subset of either Y or N \ Y , then K = T ξ ∈ c x ∗ ξ meets K ρ in a single point z λ . (6) If we assume further that for each ξ < η < c , a ξ ⊂ ∗ a η and b ξ ⊂ ∗ b η , and for each t ∈ T λ , η may be chosen so that x t meetseach of ( a η \ a ξ ) and ( b η \ b ξ ) , then z λ is a tie-point of K ρ .Proof. To show that K ρ is a tie-set it is sufficient to show that K ρ ⊂ S α ∈ λ + a ∗ α ∩ S α ∈ λ + b ∗ α . Since T λ is a λ + -Souslin tree, no new subset of λ is added when forcing with T λ . Of course we use that ρ is T λ is generic,so assume that Y ⊂ N and that some t ∈ T λ forces that Y ∗ ∩ K ρ is notempty. We must show that there is some t < s such that s forces that a s ∩ Y and b s ∩ Y are both infinite. However, we know that x t ⌢ ℓ ∩ Y is infinite for each ℓ ∈ ω since t ⌢ ℓ (cid:13) T λ “ K ρ ⊂ x ∗ t ⌢ ℓ ”. Therefore, bycondition 4, for each ℓ ∈ ω , Y ∩ a t ⌢ ℓ and Y ∩ b t ⌢ ℓ are both infinite.Now let κ , κ be regular cardinals at least one of which is distinctfrom λ + . Recall that forcing with T λ preserves cardinals. Assume thatin V [ ρ ], K ⊂ N ∗ and N ∗ = C ⊲⊳ K D with b ( I C ) = δ ( I C ) = κ and b ( I D ) = δ ( I D ) = κ . In V , let { c γ : γ ∈ κ } be T λ -names for theincreasing cofinal sequence in I C and let { d ξ : ξ ∈ κ } be T λ -names forthe increasing cofinal sequence in I D . Again using the fact that T λ adds A. DOW AND S. SHELAH no new subsets of N and the fact that every dense open subset of T λ will contain an entire level of T λ , we may choose ordinals { α γ : γ ∈ κ } and { β ξ : ξ ∈ κ } such that each t ∈ T λ , if t is on level α γ it will force avalue on c γ and if t is on level β ξ it will force a value on d ξ . If κ < λ + ,then sup { α γ : γ ∈ κ } < λ + , hence there are t ∈ T λ which force avalue on each c γ . If λ + < κ , then there is some β < λ + , such that { ξ ∈ κ : β ξ ≤ β } has cardinality κ . Therefore there is some t ∈ T λ such that t forces a value on d ξ for a cofinal set of ξ ∈ κ . Of course, ifneither κ nor κ is equal to λ + , then we have a condition that decidedcofinal families of each of I C and I D . This implies that N ∗ already hastie-sets of b δ -type ( κ , κ ).If κ < κ = λ + , then fix t ∈ T λ deciding C = { c γ : γ ∈ κ } , and let D = { d ⊂ N : ( ∃ s > t ) s (cid:13) T λ “ d ∗ ⊂ D ” } . It follows easily that D = C ⊥ .But also, since forcing with T λ can not raise b ( D ) and can not lower δ ( D ), we again have that there are tie-sets of b δ -type in V .The case κ = λ + < κ is similar.Now assume we have the family { ( a ξ , x ξ , b ξ ) : ξ ∈ c } as in (5) and(6) and set K = T ξ x ∗ ξ , A = { K } ∪ S { a ∗ ξ : ξ ∈ c } , and B = { K } ∪ S { b ∗ ξ : ξ ∈ c } . It is routine to see that (5) ensures that the family { x ξ ∩ x ρ ( α ) : ξ ∈ c and α ∈ λ + } generates an ultrafilter when ρ meetseach maximal antichain A Y ( Y ⊂ N ). Condition (6) clearly ensuresthat A \ K and B \ K each meet ( x ξ ∩ x ρ ( α ) ) ∗ for each ξ ∈ c and α ∈ λ + . Thus A ∩ K ρ and B ∩ K ρ witness that z λ is a tie-point of K ρ . (cid:3) Let θ be a regular cardinal greater than λ + for all λ ∈ Λ. We willneed the following well-known Easton lemma (see [4, p234]).
Lemma 2.3.
Let µ be a regular cardinal and assume that P is aposet satisfying the µ -cc. Then any <µ -closed poset P remains <µ -distributive after forcing with P . Furthermore any <µ -distributiveposet remains <µ -distributive after forcing with a poset of cardinalityless than µ .Proof. Recall that a poset P is <µ -distributive if forcing with it doesnot add, for any γ < µ , any new γ -sequences of ordinals. Since P is <µ -closed, forcing with P does not add any new antichains to P .Therefore it follows that forcing with P preserves that P has the µ -ccand that for every γ < µ , each γ -sequence of ordinals in the forcingextension by P × P is really just a P -name. Since forcing with P × P is the same as P × P , this shows that in the extension by P , thereare no new P -names of γ -sequences of ordinals. IE-POINTS AND FIXED-POINTS IN N ∗ Now suppose that P is µ -distributive and that P has cardinalityless than µ . Let ˙ D be a P -name of a dense open subset of P . Foreach p ∈ P , let D p ⊂ P be the set of all q such that some extension of p forces that q ∈ ˙ D . Since p forces that ˙ D is dense and that ˙ D ⊂ D p ,it follows that D p is dense (and open). Since P is µ -distributive, T p ∈ P D p is dense and is clearly going to be a subset of ˙ D . Repeatingthis argument for at most µ many P -names of dense open subsets of P completes the proof. (cid:3) We recall the definition of Easton supported product of posets (see[4, p233]).
Definition 2.1.
If Λ is a set of cardinals and { P λ : λ ∈ Λ } is a setof posets, then we will use Π λ ∈ Λ P λ to denote the collection of partialfunctions p such that(1) dom( p ) ⊂ Λ,(2) | dom( p ) ∩ µ | < µ for all regular cardinals µ ,(3) p ( λ ) ∈ P λ for all λ ∈ dom( p ).This collection is a poset when ordered by q < p if dom( q ) ⊃ dom( p )and q ( λ ) ≤ p ( λ ) for all λ ∈ dom( p ). Lemma 2.4.
For each cardinal µ , Π λ ∈ Λ \ µ + T λ is <µ + -closed and, if µ is regular, Π λ ∈ Λ ∩ µ T λ has cardinality at most <µ ≤ min(Λ \ µ ) . Lemma 2.5. If P is ccc and G ⊂ P × Π λ ∈ Λ T λ is generic, then in V [ G ] ,for any µ and any family A ⊂ [ N ] ω with |A| = µ : (1) if µ ≤ ω , then A is a member of V [ G ∩ P ] ; (2) if µ = λ + , λ ∈ Λ , then there is an A ′ ⊂ A of cardinality λ + such that A ′ is a member of V [ G ∩ ( P × T λ )] ; (3) if µ − / ∈ Λ , then there is an A ′ ⊂ A of cardinality µ which is amember of V [ G ∩ P ] . Corollary 2.6. If P is ccc and G ⊂ P × Π λ ∈ Λ T λ is generic, then forany κ ≤ µ < c such that either κ = µ or κ / ∈ { λ + : λ ∈ Λ } , if thereis a tie-set of b δ -type ( κ, µ ) in V [ G ] , then there is such a tie-set in V [ G ∩ P ] .Proof. Assume that β N \ N = A ⊲⊳ K B in V [ G ] with µ = b ( A ) and λ = b ( B ). Let J A ⊂ I A be an increasing mod finite chain, of order type µ , which is dense in I A . Similarly let J B ⊂ I B be such a chain of ordertype λ . By Lemma 2.5, J A and J B are subsets of [ N ] ω ∩ V [ G ∩ P ] = [ N ] ω .Choose, if possible µ ∈ Λ such that µ +1 = µ and λ ∈ Λ such that λ +1 = λ . Also by Lemma 2.5, we can, by passing to a subcollection,assume that J A ∈ V [ G ∩ ( P × T µ )] (if there is no µ , then let T µ denote A. DOW AND S. SHELAH the trivial order). Similarly, we may assume that J B ∈ V [ G ∩ ( P × T λ )].Fix a condition q ∈ G ⊂ ( P × Π λ ∈ Λ T λ ) which forces that ( J A ) ↓ is a ⊂ -dense subset of I A , that ( J B ) ↓ is a ⊂ -dense subset of I B , and that( I A ) ⊥ = I B .Working in the model V [ G ∩ P ] then, there is a family { ˙ a α : α ∈ µ } of T µ -names for the members of J A ; and a family { ˙ b β : β ∈ λ } of T λ -names for the members of J B . Of course if µ = λ and T µ is thetrivial order, then J A and J B are already in V [ G ∩ P ] and we have ourtie-set in V [ G ∩ P ].Otherwise, we assume that µ < λ . Set A to be the set of all a ⊂ N such that there is some q ( µ ) ≤ t ∈ T µ and α ∈ µ such that t (cid:13) T µ “ a = ˙ a α ”. Similarly let B be the set of all b ⊂ N such that thereis some q ( λ ) ≤ s ∈ T λ and β ∈ λ such that s (cid:13) T λ “ b = ˙ b β ”. It followsfrom the construction that, in V [ G ], for any ( a ′ , b ′ ) ∈ J A × J B , thereis an ( a, b ) ∈ A × B such that a ′ ⊂ ∗ a and b ′ ⊂ ∗ b . Therefore the idealgenerated by A ∪ B is certainly dense. It remains only to show that
B ⊂ ( A ) ⊥ . Consider any ( a, b ) ∈ A × B , and choose ( q ( µ ) , q ( λ )) ≤ ( t, s ) ∈ T µ × T λ such that t (cid:13) T µ “ a ∈ J A ” and s (cid:13) T λ “ b ∈ J B ”. Itfollows that for any condition ¯ q ≤ q with ¯ q ∈ ( P × Π λ ∈ Λ T λ ), ¯ q ( µ ) = t ,¯ q ( λ ) = s , we have that¯ q (cid:13) ( P × Π λ ∈ Λ T λ ) “ a ∈ J A and b ∈ J B ” . It is routine now to check that, in V [ G ∩ P ], A and B generate idealsthat witness that T { ( N \ ( a ∪ b )) ∗ : ( a, b ) ∈ A ×B} is a tie-set of b δ -type( µ, λ ). (cid:3) Let T be the rooted tree {∅} ∪ S λ ∈ Λ T λ and we will force an em-bedding of T into P ( N ) mod finite. In fact, we force a structure { ( a t , x t , b t ) : t ∈ T } satisfying the conditions (1)-(4) of Proposition2.2. Definition 2.2.
The poset Q is defined as the set of elements q =( n q , T q , f q ) where n q ∈ N , T q ∈ [ T ] <ω , and f q : n q × T q → { , , } .The idea is that x t will be S q ∈ G { j ∈ n q : f q ( j, t ) = 0 } , a t will be S q ∈ G { j ∈ n q : f q ( j, t ) = 1 } and b t = N \ ( a t ∪ x t ). We set q < p if n q ≥ n p , T q ⊃ T p , f q ⊃ f p and for t, s ∈ T p and i ∈ [ n p , n q )(1) if t < s and f q ( i, t ) ∈ { , } , then f q ( i, s ) = f q ( i, t );(2) if t < s and f q ( i, s ) = 0, then f q ( i, t ) = 0;(3) if t ⊥ s , then f q ( i, t ) + f q ( i, s ) > j ∈ { , } and { t ⌢ ℓ, t ⌢ ( ℓ + j ) } ⊂ T p and f q ( i, t ⌢ ( ℓ + j )) = 0,then f q ( i, t ⌢ ℓ ) = j .The next lemma is very routine but we record it for reference. IE-POINTS AND FIXED-POINTS IN N ∗ Lemma 2.7.
The poset Q is ccc and if G ⊂ Q is generic, the family X T = { ( a t , x t , b t ) : t ∈ T } satisfies the conditions of Proposition 2.2. We will need some other combinatorial properties of the family X T . Definition 2.3.
For any ˜ T ∈ [ T ] <ω , we define the following ( Q -names).(1) for i ∈ N , [ i ] ˜ T = { j ∈ N : ( ∀ t ∈ ˜ T ) i ∈ x t iff j ∈ x t } ,(2) the collection fin( ˜ T ) is the set of [ i ] ˜ T which are finite.We abuse notation and let fin( ˜ T ) ⊂ n abbreviate fin( ˜ T ) ⊂ P ( n ). Lemma 2.8.
For each q ∈ Q and each ˜ T ⊂ T q , fin( ˜ T ) ⊂ n q and for i ≥ n q , [ i ] ˜ T is infinite. Definition 2.4.
A sequence S W = { ( a ξ , x ξ , b ξ ) : ξ ∈ W } is a tower of T -splitters if for ξ < η ∈ W and t ∈ T :(1) { a ξ , x ξ , b ξ } is a partition of N ,(2) a ξ ⊂ ∗ a η , b ξ ⊂ ∗ b η ,(3) x t ∩ x ξ is infinite. Definition 2.5. If S W is a tower of T -splitters and Y is a subset N ,then the poset Q ( S W , Y ) is defined as follows. Let E Y be the (possiblyempty) set of minimal elements of T such that there is some finite H ⊂ W such that x t ∩ Y ∩ T ξ ∈ H x ξ is finite. Let D Y = E ⊥ Y = { t ∈ T : ( ∀ s ∈ E Y ) t ⊥ s } . A condition q ∈ Q ( S W , Y ) is a tuple ( n q , a q , x q , b q , T q , H q )where(1) n q ∈ N and { a q , x q , b q } is a partition of n q ,(2) T q ∈ [ T ] <ω and H q ∈ [ W ] <ω ,(3) ( a ξ \ a η ), ( b ξ \ b η ), and ( x η \ x ξ ) are all contained in n q for ξ < η ∈ H q .We define q < p to mean n p ≤ n q , T p ⊂ T q , H p ⊂ H q , and(4) for t ∈ T p ∩ D Y , x t ∩ ( x q \ x p ) ⊂ Y ,(5) x q \ x p ⊂ T ξ ∈ H p x ξ ,(6) a q \ a p is disjoint from b max( H p ) ,(7) b q \ b p is disjoint from a max( H p ) . Lemma 2.9. If W ⊂ γ , S W is a tower of T -splitters, and if G is Q ( S W , Y ) -generic, then S W ∪{ ( a γ , x γ , b γ ) } is also a tower of T -splitterswhere a γ = S { a q : q ∈ G } , x γ = S { x q : q ∈ G } , and b γ = S { b q : q ∈ G } . In addition, for each t ∈ D Y , x t ∩ x ξ ⊂ ∗ Y (and x t ∩ x ξ ⊂ ∗ N \ Y for t ∈ E Y ). Lemma 2.10. If W does not have cofinality ω , then Q ( S W , Y ) is σ -centered. As usual with ( ω , ω )-gaps, Q ( S W , Y ) may not (in general) be cccif W has a cofinal ω sequence.Let 0 / ∈ C ⊂ θ be cofinal and assume that if C ∩ γ is cofinal in γ andcf( γ ) = ω , then γ ∈ C . Definition 2.6.
Fix any well-ordering ≺ of H ( θ ). We define a fi-nite support iteration sequence { P γ , ˙ Q γ : γ ∈ θ } ⊂ H ( θ ). We abusenotation and use Q rather than ˙ Q from definition 2.2. If γ / ∈ C ,then let ˙ Q γ be the ≺ -least among the list of P γ -names of ccc posets in H ( θ ) \ { ˙ Q ξ : ξ ∈ γ } . If γ ∈ C , then let ˙ Y γ be the ≺ -least P γ -name ofa subset N which is in H ( θ ) \ { ˙ Y ξ : ξ ∈ C ∩ γ } . Set ˙ Q γ to be the P γ name of Q ( S C ∩ γ , ˙ Y γ ) adding the partition { ˙ a γ , ˙ x γ , ˙ b γ } and, where S C ∩ γ is the P γ -name of the T -splitting tower { ( a ξ , x ξ , b ξ ) : ξ ∈ C ∩ γ } .We view the members of P θ as functions p with finite domain (orsupport) denoted dom( p ).The main difficulty to the proof of Theorem 2.1 is to prove that theiteration P θ is ccc. Of course, since it is a finite support iteration, thiscan be proven by induction at successor ordinals. Lemma 2.11.
For each γ ∈ C such that C ∩ γ has cofinality ω , P γ +1 is ccc.Proof. We proceed by induction. For each α , define p ∈ P ∗ α if p ∈ P α and there is an n ∈ N such that(1) for each β ∈ dom( p ) ∩ C , with H β = dom( p ) ∩ C ∩ β , thereare subsets a β , x β , b β of n and T β ∈ [ T ] <ω such that p ↾ β (cid:13) P β “ p ( β ) = ( n, a β , x β , b β , T β , H β )”Assume that P ∗ β is dense in P β and let p ∈ P β +1 . To show that P ∗ β +1 isdense in P β +1 we must find some p ∗ ≤ p in P ∗ β +1 . If β / ∈ C and p ∗ ∈ P ∗ β is below p ↾ β , then p ∗ ∪ { ( β, p ( β ) } is the desired element of P ∗ β +1 . Nowassume that β ∈ C and assume that p ↾ β ∈ P ∗ β and that p ↾ β forcesthat p ( β ) is the tuple ( n , a, x, b, ˜ T , ˜ H ). By an easy density argument,we may assume that ˜ H ⊂ dom( p ). Let n ∗ be the integer witnessingthat p ↾ β ∈ P ∗ β . Let ζ be the maximum element of dom( p ) ∩ C ∩ β and let p ↾ ζ (cid:13) P ζ “ p ( ζ ) = ( n ∗ , a ζ , x ζ , b ζ , T ζ , H ζ )” as per the definitionof P ∗ ζ +1 . Notice that since ˜ H ⊂ H ζ we have that p ↾ β (cid:13) P β “( n ∗ , a ∗ , x, b ∗ , T ζ ∪ ˜ T , H ζ ∪ { ζ } ) ≤ p ( β )”where a ∗ = a ∪ ([ n , n ∗ ) \ b ζ ) and b ∗ = b ∪ ([ n , n ∗ ) ∩ b ζ ). Defining p ∗ ∈ P β +1 by p ∗ ↾ β = p ↾ β and p ∗ ( β ) = ( n ∗ , a ∗ , x, b ∗ , T ζ ∪ ˜ T , H ζ ∪ { ζ } ) IE-POINTS AND FIXED-POINTS IN N ∗ completes the proof that P ∗ β +1 is dense in P β +1 , and by induction, thatthis holds for β = γ .Now assume that { p α : α ∈ ω } ⊂ P ∗ γ +1 . By passing to a subcollec-tion, we may assume that(1) the collection { T p α ( γ ) : α ∈ ω } forms a ∆-system with root T ∗ ;(2) the collection { dom( p α ) : α ∈ ω } also forms a ∆-system withroot R ;(3) there is a tuple ( n ∗ , a ∗ , x ∗ , b ∗ ) so that for all α ∈ ω , a p α ( γ ) = a ∗ , x p α ( γ ) = x ∗ , and b p α ( γ ) = b ∗ .Since C ∩ γ has a cofinal sequence of order type ω , there is a δ ∈ γ such that R ⊂ δ and, we may assume, (dom( p α ) \ δ ) ⊂ min(dom( p β ) \ δ )for α < β < ω . Since P δ is ccc, there is a pair α < β < ω such that p α ↾ δ is compatible with p β ↾ δ . Define q ∈ P γ +1 by(1) q ↾ δ is any element of P δ which is below each of p α ↾ δ and p β ↾ δ ,(2) if δ ≤ ξ ∈ γ ∩ dom( p α ), then q ( ξ ) = p α ( ξ ),(3) if δ ≤ ξ ∈ dom( p β ) \ C , then q ( ξ ) = p β ( ξ ),(4) if δ ≤ ξ ∈ dom( p β ) ∩ C , then q ( ξ ) = ( n ∗ , a p β ( ξ ) , x p β ( ξ ) , b p β ( ξ ) , T p β ( ξ ) , H p β ( ξ ) ∪ H p α ( γ ) ) . The main non-trivial fact about q is that it is in P γ +1 which dependson the fact that, by induction on η ∈ C ∩ γ , q ↾ η forces that( a η \ a ξ ) ∪ ( b η \ b ξ ) ∪ ( x ξ \ x η ) ⊂ n ∗ for ξ ∈ C ∩ η. It now follows trivially that q is below each of p α and p β . (cid:3) Proof of Theorem 2.1.
This completes the construction of the ccc poset P ( P θ as above). Let G ⊂ ( P × Π λ ∈ Λ T λ ) be generic. It follows that V [ G ∩ P ] is a model of Martin’s Axiom and c = θ . Furthermore byapplying Lemma 2.4 with µ = ω and Lemma 2.3, we have that P =Π λ ∈ Λ T λ is ω -distributive in the model V [ G ∩ P ]. Therefore all subsetsof N in the model V [ G ] are also in the model V [ G ∩ P ].Fix any λ ∈ Λ and let ρ λ denote the generic branch in T λ given by G .Let G λ denote the generic filter on P × Π { T µ : λ = µ ∈ Λ } and workin the model V [ G λ ]. It follows easily by Lemma 2.4 and Lemma 2.3,that T λ is a λ + -Souslin tree in this model. Therefore by Proposition2.2, K λ = T α<λ + x ∗ ρ λ ( α ) is a tie-set of b δ -type ( λ + , λ + ) in V [ G ]. By thedefinition of the iteration in P , it follows that condition (4) of Lemma2.2 is also satisfied, hence the tie-set K = T ξ ∈ C x ∗ ξ meets K λ in a singlepoint z λ . A simple genericity argument confirms that conditions (5)and (6) of Proposition 2.2 also holds, hence z λ is a tie-point of K λ . It follows from Corollary 2.6 that there are no unwanted tie-sets in β N \ N in V [ G ], at least if there are none in V [ G ∩ P ]. Since p = c in V [ G ∩ P ], it follows from Proposition 1.3 that indeed there are no suchtie-sets in V [ G ∩ P ]. (cid:3) Unfortunately the next result shows that the construction does notprovide us with our desired variety of tie-points (even with variationsin the definition of the iteration). We do not know if b δ -type can beimproved to δ -type (or simply exclude tie-points altogether). Proposition 2.12.
In the model constructed in Theorem 2.1, there areno tie-points with b δ -type ( κ , κ ) for any κ ≤ κ < c ,Proof. Assume that β N \ N = A ⊲⊳ x B and that δ ( I A ) = κ and δ ( I B ) = κ . It follows from Corollary 2.6 that we can assume that κ = κ = λ + for some λ ∈ Λ. Also, following the proof of Corollary 2.6, there are P × T λ -names J A = { ˜ a α : α ∈ λ + } and P × T λ + -names J B = { ˜ b β : β ∈ λ + } such that the valuation of these names by G result in increasing(mod finite) chains in I A and I B respectively whose downward closuresare dense. Passing to V [ G ∩ P ], since T λ has the θ -cc, there is a Booleansubalgebra B ∈ [ P ( N )] <θ such that each ˜ a α and ˜ b β is a name of amember of B . Furthermore, there is an infinite C ⊂ N such that C / ∈ x and each of b ∩ C and b \ C are infinite for all b ∈ B . Since C / ∈ x , thereis a Y ⊂ N (in V [ G ]) such that C ∩ Y ∈ I A and C \ Y ∈ I B . Nowchoose t ∈ T λ which forces this about C and Y . Back in V [ G ∩ P ], set A = { b ∈ B : ( ∃ t ≤ t ) t (cid:13) T λ “ b ∈ J A ∪ J B ” } . Since V [ G ∩ P ] satisfies p = θ and A ↓ is forced by t to be dense in[ N ] ω , there must be a finite subset A ′ of A which covers C . It alsofollows easily then that there must be some a, b ∈ A ′ and t , t eachbelow t such that t (cid:13) T λ + “ a ∈ J A ”, t (cid:13) T λ + “ b ∈ J B ”, and a ∩ b isinfinite. The final contradiction is that we will now have that t failsto force that C ∩ a ⊂ ∗ Y and C ∩ b ⊂ ∗ ( N \ Y ). (cid:3) T -involutions In this section we strengthen the result in Theorem 2.1 by makingeach K ∩ K λ a symmetric tie-point in K λ (at the expense of weakeningMartin’s Axiom in V [ G ∩ P ]). This is progress in producing involutionswith some control over the fixed point set but we are still not able tomake K the fixed point set of an involution. A poset is said to be σ -linked if there is a countable collection of linked (elements are pairwisecompatible) which union to the poset. The statement MA( σ − linked) IE-POINTS AND FIXED-POINTS IN N ∗ is, of course, the assertion that Martin’s Axiom holds when restrictedto σ -linked posets.Our approach is to replace T -splitting towers by the following notion.If f is a (partial) involution on N , let min( f ) = { n ∈ N : n < f ( n ) } and max( f ) = { n ∈ N : f ( n ) < n } (hence dom( f ) is partitioned intomin( f ) ∪ fix( f ) ∪ max( f )). Definition 3.1.
A sequence T = { ( A ξ , f ξ ) : ξ ∈ W } is a tower of T -involutions if W is a set of ordinals and for ξ < ν ∈ W and t ∈ T (1) A ν ⊂ ∗ A ξ ;(2) f ξ = f ξ and f ξ ↾ ( N \ fix( f ξ )) ⊂ ∗ f η ;(3) f ξ [ x t ] = ∗ x t and fix( f ξ ) ∩ x t is infinite;(4) f ξ ([ n, m )) = [ n, m ) for n < m both in A ξ .Say that T , a tower of T -involutions, is full if K = K T = T { fix( f ξ ) ∗ : ξ ∈ W } is a tie-set with β N \ N = A ⊲⊳ K B where A = K ∪ S { min( f ξ ) ∗ : ξ ∈ W } and B = K ∪ S { max( f ξ ) ∗ : ξ ∈ W } .If T is a tower of T -involutions, then there is a natural involution F T on S ξ ∈ W ( N \ fix( f ξ )) ∗ , but this F T need not extend to an involution onthe closure of the union - even if the tower is full.In this section we prove the following theorem. Theorem 3.1.
Assume GCH and that Λ is a set of regular uncountablecardinals such that for each λ ∈ Λ , T λ is a <λ -closed λ + -Souslin tree.Let T denote the tree sum of { T λ : λ ∈ Λ } . There is forcing extension inwhich there is T , a full tower of T -involutions, such that the associatedtie-set K has b δ -type ( c , c ) and such that for each λ ∈ Λ , there is atie-set K λ of b δ -type ( λ + , λ + ) such that F T does induce an involutionon K λ with a singleton fixed point set { z λ } = K ∩ K λ . Furthermore,for µ ≤ λ < c , if µ = λ or λ / ∈ Λ , then there is no tie-set of b δ -type ( µ, λ ) . Question 3.1.
Can the tower T in Theorem 3.1 be constructed so that F T extends to an involution of β N \ N with fix( F ) = K T ?We introduce T -tower extending forcing. Definition 3.2. If T = { ( A ξ , f ξ ) : ξ ∈ W } is a tower of T -involutionsand Y is a subset of N , we define the poset Q = Q ( T , Y ) as follows.Let E Y be the (possibly empty) set of minimal elements of T such thatthere is some finite H ⊂ W such that x t ∩ Y ∩ T ξ ∈ H fix( f ξ ) is finite.Let D Y = E ⊥ Y = { t ∈ T : ( ∀ s ∈ E Y ) t ⊥ s } . A tuple q ∈ Q if q = ( a q , f q , T q , H q ) where: (1) H q ∈ [ W ] <ω , T q ∈ [ T ] <ω , and n q = max( a q ) ∈ A α q where α q = max( H q ),(2) f q is an involution on n q ,(3) ( A α q \ n q ) ⊂ A ξ for each ξ ∈ H q ,(4) fin( T q ) ⊂ n q ,(5) f ξ ↾ ( N \ (fix( f ξ ) ∪ n q )) ⊂ f α q for ξ ∈ H q ,(6) f α q [ x t \ n q ] = x t \ n q for t ∈ T q ,We define p < q if n p ≤ n q , and for t ∈ T p and i ∈ [ n p , n q ):(7) a p = a q ∩ n p , T p ⊂ T q , and H p ⊂ H q ,(8) a q \ a p ⊂ A α p ,(9) f α p ( i ) = i implies f q ( i ) = f α p ( i ),(10) f q ([ n, m )) = [ n, m ) for n < m both in a q \ a p ,(11) f q ( x t ∩ [ n p , n q )) = x t ∩ [ n p , n q ),(12) if t ∈ D p and i ∈ x t ∩ fix( f q ), then i ∈ Y It should be clear that the involution f introduced by Q ( T , Y ) sat-isfies that for each t ∈ D Y , fix( f ) ∩ x t ⊂ ∗ Y , and, with the help ofthe following density argument, that T ∪ { ( γ, A, f ) } is again a tower of T -involutions where A is the infinite set introduced by the first coor-dinates of the conditions in the generic filter. Lemma 3.2. If W ⊂ γ , Y ⊂ N , and T = { ( A ξ , f ξ ) : ξ ∈ W } is a towerof T -involutions and p ∈ Q ( T , Y ) , then for any ˜ T ∈ [ T ] <ω , ζ ∈ W , andany m ∈ N , there is a q < p such that n q ≥ m , ζ ∈ H q , T q ⊃ ˜ T , and fix( f q ) ∩ ( x t \ n p ) is not empty for each t ∈ T p .Proof. Let β denote the maximum α p and ζ and let η denote the min-imum. Choose any n q ∈ A α q \ m large enough so that(1) f α p [ x t \ n q ] = x t \ n q for t ∈ ˜ T ,(2) f η ↾ ( N \ ( n q ∪ fix( f η ))) ⊂ f β ,(3) A β \ A η is contained in n q ,(4) n q ∩ [ i ] T p ∩ fix( f α p ) is non-empty for each i ∈ N such that [ i ] T p is in the finite set { [ i ] T p : i ∈ N } \ fin( T p ),(5) if i ∈ x t ∩ n q \ n p for some t ∈ D Y ∩ T p , then Y meets [ i ] T p ∩ n q \ n p in at least two points.Naturally we also set H q = H p ∪ { ζ } and T q = T p ∪ ˜ T . The choiceof n q is large enough to satisfy (3), (4), (5) and (6) of Definition 3.2.We will set a q = a p ∪ { n q } ensuring (1) of Definition 3.2. Thereforefor any f q ⊃ f p which is an involution on n q , we will have that q =( a q , f q , T q , H q ) is in the poset. We have to choose f q more carefully toensure that q ≤ p . Let S = [ n p , n q ) ∩ fix( f α p ), and S ′ = [ n p , n q ) \ S . Wechoose ¯ f an involution on S and set f q = f p ∪ ( f α p ↾ S ′ ) ∪ ¯ f . We leave IE-POINTS AND FIXED-POINTS IN N ∗ it to the reader to check that it suffices to ensure that ¯ f sends [ i ] T p ∩ S to itself for each t ∈ T p and that fix( ¯ f ) ∩ x t ⊂ Y for each t ∈ T p ∩ D Y .Since the members of { [ i ] T p ∩ S : i ∈ N } are pairwise disjoint we candefine ¯ f on each separately.For each [ i ] T p ∩ S which has even cardinality, choose two points y i , z i from it so that if there is a p ∈ D Y ∩ T p such that [ i ] T p ⊂ x t , then { y i , z i } ⊂ Y . Let ¯ f be any involution on [ i ] T p ∩ S so that y i , z i are theonly fixed points. If [ i ] T p ∩ S has odd cardinality then choose a point y i from it so that if [ i ] T p is contained in x t for some t ∈ D y ∩ T p , then y i ∈ Y ∩ [ i ] T p ∩ S . Set ¯ f ( y i ) = y i and choose ¯ f to be any fixed-pointfree involution on [ i ] T p ∩ S \ { y i } . (cid:3) Let P θ now be the finite support iteration defined as in Definition 2.6except for two important changes. For γ ∈ C , we replace T -splittingtowers by the obvious inductive definition of towers of T -involutionswhen we replace the posets ˙ Q ( S C ∩ γ , ˙ Y γ ) by ˙ Q ( T C ∩ γ , ˙ Y γ ). For γ / ∈ C werequire that (cid:13) P γ “ ˙ Q γ is σ -linked.”Special (parity) properties of the family { x t : t ∈ T } are needed toensure that (cid:13) P γ “ ˙ Q ( S C ∩ γ , ˙ Y γ ) is ccc ” even for cases when cf( γ ) is not ω .The proof of Theorem 3.1 is virtually the same as the proof of The-orem 2.1 (so we skip) once we have established that the iteration isccc. Lemma 3.3.
For each γ ∈ C , P γ +1 is ccc.Proof. We again define P ∗ α to be those p ∈ P α for which there is an n ∈ N such that for each β ∈ dom( p ) ∩ C , there are n ∈ a β ⊂ n +1, f β ∈ n n , T β ∈ [ T ] <ω , and H β = dom( p ) ∩ C ∩ β such that p ↾ β (cid:13) P β “ p ( β ) = ( a β , f β , T β , H β )”. However, in this proof we must also makesome special assumptions in coordinates other than those in C . Foreach ξ ∈ γ \ C , we fix a collection { ˙ Q ( ξ, n ) : n ∈ ω } of P ξ -names sothat 1 (cid:13) P ξ “ ˙ Q ξ = [ n ˙ Q ( ξ, n ) and ( ∀ n ) ˙ Q ( ξ, n ) is linked.”The final restriction on p ∈ P ∗ α is that for each ξ ∈ α \ C , there is a k ξ ∈ ω such that p ↾ ξ (cid:13) P ξ “ p ( ξ ) ∈ ˙ Q ( ξ, k ξ )”.Just as in Lemma 2.11, Lemma 3.2 can be used to show by inductionthat P ∗ α is a dense subset of P α . This time though, we also demand thatdom( f p (0) ) = n × T p (0) is such that T β ⊂ T p (0) for all β ∈ dom( p ) ∩ C and some extra argument is needed because of needing to decide valuesin the name ˙ Y γ as in the proof of Lemma 3.2. Let p ∈ P β +1 andassume that P ∗ β is dense in P β . By density, we may assume that p ↾ β ∈ P ∗ β , H p ( β ) ⊂ dom( p ), T p ( β ) ⊂ T p (0) , and that p ↾ β has decidedthe members of the set D ˙ Y β ∩ T p ( β ) . We can assume further that foreach t ∈ D ˙ Y β ∩ T p ( β ) , p ↾ β has forced a value y t ∈ ˙ Y β ∩ x t \ S { x s : s ∈ T p and s t } such that y t > n p ( β ) . We are using that T isnot finitely branching to deduce that if t ∈ D ˙ Y β , then p ↾ β (cid:13) P β “ ˙ Y β ∩ x t \ S { x s : s ∈ T p and s t } is non-empty” (which followssince ˙ Y β must meet x s for each immediate successor s of t ). Chooseany m larger than y t for each t ∈ T p ( β ) . Without loss of generality,we may assume that the integer n ∗ witnessing that p ↾ β ∈ P ∗ β is atleast as large as m and that n ∗ ∈ T ξ ∈ H p ( β ) A ξ . Construct ¯ f just as inLemma 3.2, except that this time there is no requirement to actuallyhave fixed points so one member of ˙ Y β in each appropriate [ i ] T p ( β ) is allthat is required. Let ζ = max(dom( p ) ∩ β ). No new forcing decisionsare required of p ↾ β in order to construct a suitable ¯ f , hence this showsthat p ↾ β ∪ { ( β, q ) } (where q is constructed below p ( β ) as in Corollary3.2 in which H p ( ζ ) ∪ { ζ } is add to H q ) is the desired extension of p which is a member of P ∗ β +1 .Now to show that P γ +1 is ccc, let { p α : α ∈ ω } ⊂ P ∗ γ +1 . Clearlywe may assume that the family { p α (0) : α ∈ ω } are pairwise compat-ible and that there is a single integer n such that, for each α ∈ ω ,dom( p α (0)) = n × T α for some T α ∈ [ T ] <ω . Also, we may assume thatthere is some ( a, h ) such that, for each α , p α ↾ γ (cid:13) P γ “ p ( γ ) = ( a, h, T α , H α )”where H α = dom( p α ) ∩ C ∩ γ .The family { dom( p α ) ∩ γ : α ∈ ω } may be assumed to form a ∆-system with root R . For each ξ ∈ R , we may assume that, if ξ / ∈ C ,there is a single k ξ ∈ ω such that, for all α , p α ↾ ξ (cid:13) P ξ “ p α ( ξ ) ∈ ˙ Q ( ξ, k ξ )”, and if ξ ∈ C , then there is a single ( a ξ , h ξ ) such that p α ↾ ξ (cid:13) P ξ “ p α ( ξ ) = ( a ξ , h ξ , T α , H α ∩ ξ )”. For convenience, for each ξ / ∈ C let ˙ r ξ be a P ξ -name of a function from ω × ˙ Q ξ such that, for each k ∈ ω ,1 (cid:13) P ξ “ ˙ r ξ ( k, q, q ′ ) ≤ q, q ′ ( ∀ q, q ′ ∈ ˙ Q ( ξ, k ))” . Fix any α < β < ω and let H = H α ∪ H β . Recall that p α (0)and p β (0) are compatible. Recursively define a P ξ -name q ( ξ ) for ξ ∈ IE-POINTS AND FIXED-POINTS IN N ∗ dom( p α ) ∪ dom( p β ) so that q ↾ ξ (cid:13) P ξ “ q ( ξ ) = ( n, T α ∪ T β , f p α (0) ∪ f p β (0) ) ξ = 0˙ r ξ ( k ξ , p α ( ξ ) , p β ( ξ )) ξ ∈ R \ Cp α ( ξ ) ξ ∈ dom( p α ) \ ( R ∪ C ) p β ( ξ ) ξ ∈ dom( p β ) \ ( R ∪ C )( a ξ , h ξ , T α ∪ T β , H ∩ ξ ) ξ ∈ C. ”.Now we check that q ∈ P ξ by induction on ξ ∈ γ + 1.The first thing to note is that not only is this true for ξ = 1, but alsothat q (0) (cid:13) Q “ fin( T α ∪ T β ) ⊂ n ”. Since p α and p β are each in P ∗ γ +1 ,this show that condition (4) of Definition 3.2 will hold in all coordinatesin C .We also prove, by induction on ξ , that q ↾ ξ forces that for η < δ bothin H ∩ ξ and t ∈ T α ∪ T β , f δ [ x t \ n ] = x t \ n , f η ↾ ( N \ (fix( f η ) ∪ n )) ⊂ f δ and A δ \ n ⊂ A η .Given ξ ∈ H and the assumption that q ↾ ξ ∈ P ξ , and α = α q ( ξ ) =max( H ∩ ξ ), condition (3), (5), and (6) of Definition 3.2 hold by theinductive hypothesis above. It follows then that q ↾ ξ (cid:13) P ξ “ q ( ξ ) ∈ ˙ Q ξ ”.By the definition of the ordering on ˙ Q ξ , given that H ∩ ξ = H q ( ξ ) and T α ∪ T β = T q ( ξ ) , it follows that the inductive hypothesis then holds for ξ + 1.It is trivial for ξ ∈ dom( q ) \ C , that q ↾ ξ ∈ P ξ implies that q ↾ ξ (cid:13) P ξ “ q ( ξ ) ∈ ˙ Q ξ ”. This completes the proof that q ∈ P γ +1 , and it is trivialthat q is below each of p α and p β . (cid:3) Remark . If we add a trivial tree T to the collection { T λ : λ ∈ Λ } (i.e. T has only a root), then the root of T has a single extension which isa maximal node t , and with no change to the proof of Theorem 3.1,one obtains that F induces an automorphism on x ∗ t with a single fixedpoint. Therefore, it is consistent (and likely as constructed) that β N \ N will have symmetric tie-points of type ( c , c ) in the model V [ G ∩ P ] and V [ G ]. Remark . In the proof of Theorem 2.1, it is easy to arrange thateach K λ ( λ ∈ Λ) is also K T λ for a ( T λ -generic) full tower, T λ , of N -involutions. However the generic sets added by the forcing P will pre-vent this tower of involutions from extending to a full involution.4. Questions
In this section we list all the questions with their original numbering.
Question 1.1.
Can there be a tie-point in β N \ N with δ -type ( κ, λ )with κ ≤ λ less than the character of the point? Question 1.2.
Can β N \ N have tie-points of δ -type ( ω , ω ) and( ω , ω )? Question 1.3.
Does p > ω imply there are no tie-points of b -type( ω , ω )? Question 1.4. If F is an involution on β N \ N such that K = fix( F )has empty interior, is K a (symmetric) tie-set? Question 1.5.
Is there some natural restriction on which compactspaces can (or can not) be homeomorphic to the fixed point set ofsome involution of β N \ N ? Question 1.6. If F is an involution of N ∗ , is the quotient space N ∗ /F (in which each { x, F ( x ) } is collapsed to a single point) a homeomorphiccopy of β N \ N ? Question 3.1.
Can the tower T in Theorem 3.1 be constructed so that F T extends to an involution of β N \ N with fix( F ) = K T ? References [1] J¨org Brendle and Saharon Shelah,
Ultrafilters on ω —their ideals and their car-dinal characteristics , Trans. Amer. Math. Soc. (1999), no. 7, 2643–2674.MR 1686797 (2000m:03111)[2] Alan Dow, Two to one images and PFA , Israel J. Math. (2006), 221–241.MR 2282377[3] Alan Dow and Geta Techanie,
Two-to-one continuous images of N ∗ , Fund.Math. (2005), no. 2, 177–192. MR 2162384 (2006f:54003)[4] Thomas Jech, Set theory , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, The third millennium edition, revised and expanded.MR 1940513 (2004g:03071)[5] Ronnie Levy,
The weight of certain images of ω , Topology Appl. (2006),no. 13, 2272–2277. MR MR2238730 (2007e:54034)[6] S. Shelah and J. Stepr¯ans. Non-trivial homeomorphisms of βN \ N without theContinuum Hypothesis. Fund. Math. , 132:135–141, 1989.[7] S. Shelah and J. Stepr¯ans. Somewhere trivial autohomeomorphisms.
J. LondonMath. Soc. (2) , 49:569–580, 1994.
IE-POINTS AND FIXED-POINTS IN N ∗ [8] Saharon Shelah and Juris Stepr¯ans, Martin’s axiom is consistent with the ex-istence of nowhere trivial automorphisms , Proc. Amer. Math. Soc. (2002),no. 7, 2097–2106 (electronic). MR 1896046 (2003k:03063)[9] B. Veliˇckovi´c. Definable automorphisms of P ( ω ) /f in . Proc. Amer. Math. Soc. ,96:130–135, 1986.[10] Boban Veliˇckovi´c. OCA and automorphisms of P ( ω ) / fin. Topology Appl. ,49(1):1–13, 1993.
Department of Mathematics, Rutgers University, Hill Center, Pis-cataway, New Jersey, U.S.A. 08854-8019
Current address : Institute of Mathematics, Hebrew University, Givat Ram,Jerusalem 91904, Israel
E-mail address ::