aa r X i v : . [ m a t h . L O ] N ov MORE ON TIE-POINTS AND HOMEOMORPHISM 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. We picture (and denote) this as X = A ⊲⊳ x B where A, B are the closed sets which have a unique common accumulationpoint x . Tie-points have appeared in the construction of non-trivial autohomeomorphisms of β N \ N = N ∗ (e.g. [10, 7]) andin the recent study [4, 2] of (precisely) 2-to-1 maps on N ∗ . Inthese cases the tie-points have been the unique fixed point of aninvolution on N ∗ . One application of the results in this paper is theconsistency of there being a 2-to-1 continuous image of N ∗ whichis not a homeomorph of N ∗ . 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 both A and B .We let X = A ⊲⊳ x B denote this relation and say that x is a tie-point aswitnessed by A, B . Let A ≡ x B mean that there is a homeomorphismfrom A to B with x as a fixed point. If X = A ⊲⊳ x B and A ≡ x B , thenthere is an involution F of X (i.e. F = F ) such that { x } = fix( F ). Inthis case we will say that x is a symmetric tie-point of X .An autohomeomorphism F of N ∗ is said to be trivial if there is abijection f between cofinite subsets of N such that F = βf ↾ N ∗ .Since the fixed point set of a trivial autohomeomorphism is clopen, asymmetric tie-point gives rise to a non-trivial autohomeomorphism.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 : November 4, 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 917 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.In this paper we establish the following theorem. Theorem 1.1.
It is consistent that N ∗ has symmetric tie-points x, y as witnessed by A, B and A ′ , B ′ respectively such that N ∗ is not home-omorphic to the space A ⊲⊳ x = y A ′ Corollary 1.1.
It is consistent that there is a 2-to-1 image of N ∗ whichis not a homeomorph of N ∗ . One can generalize the notion of tie-point and, for a point x ∈ N ∗ ,consider how many disjoint clopen subsets of N ∗ \ { x } (each accumu-lating to x ) can be found. Let us say that a tie-point x of N ∗ satisfies τ ( x ) ≥ n if N ∗ \ { x } can be partitioned into n many disjoint clopensubsets each accumulating to x . Naturally, we will let τ ( x ) = n denotethat τ ( x ) ≥ n and τ ( x ) n +1. Each point x of character ω in N ∗ is a symmetric tie-point and satisfies that τ ( x ) ≥ n for all n . We listseveral open questions in the final section.More generally one could study the symmetry group of a point x ∈ N ∗ : e.g. set G x to be the set of autohomeomorphisms F of N ∗ thatsatisfy fix( F ) = { x } and two are identified if they are the same on someclopen neighborhood of x . Theorem 1.2.
It is consistent that N ∗ has a tie-point x such that τ ( x ) = 2 and such that with N ∗ = A ⊲⊳ x B , neither A nor B is ahomeomorph of N ∗ . In addition, there are no symmetric tie-points. The following partial order P , was introduced by Velickovic in [10]to add a non-trivial automorphism of P ( N ) / [ N ] < ℵ while doing as littleelse as possible — at least assuming PFA. Definition 1.1.
The partial order P is defined to consist of all 1-to-1functions f : A → B where • A ⊆ ω and B ⊆ ω • for all i ∈ ω and n ∈ ω , f ( i ) ∈ (2 n +1 \ n ) if and only if i ∈ (2 n +1 \ n ) • lim sup n → ω | (2 n +1 \ n ) \ A | = ω and hence, by the previouscondition, lim sup n → ω | (2 n +1 \ n ) \ B | = ω The ordering on P is ⊆ ∗ .We define some trivial generalizations of P . We use the notation P to signify that this poset introduces an involution of N ∗ becausethe conditions g = f ∪ f − satisfy that g = g . In the definitionof P it is possible to suppress mention of A, B (which we do) and
ORE ON TIE-POINTS AND HOMEOMORPHISM IN N ∗ to have the poset P consist simply of the functions g (and to treat A = min( g ) = { i ∈ dom( g ) : i < g ( i ) } and to treat B as max( g ) = { i ∈ dom( g ) : g ( i ) < i } ).Let P denote the poset we get if we omit mention of f but consistingonly of disjoint pairs ( A, B ), satisying the growth condition in Defini-tion 1.1, and extension is coordinatewise mod finite containment. Tobe consistent with the other two posets, we may instead represent theelements of P as partial functions into 2.More generally, let P ℓ be similar to P except that we assume thatconditions consist of functions g satisfying that { i, g ( i ) , g ( i ) , . . . , g ℓ ( i ) } has precisely ℓ elements for all i ∈ dom( g ) (and replace the intervals2 n +1 \ n by ℓ n +1 \ ℓ n in the definition).The basic properties of P as defined by Velickovic and treated byShelah and Steprans are also true of P ℓ for all ℓ ∈ N .In particular, for example, it is easily seen that Proposition 1.1. If L ⊂ N and P ∗ = Π ℓ ∈ L P ℓ (with full supports)and G is a P ∗ -generic filter, then in V [ G ] , for each ℓ ∈ L , there is atie-point x ℓ ∈ N ∗ with τ ( x ℓ ) ≥ ℓ . For the proof of Theorem 1.1 we use P × P and for the proof ofTheorem 1.2 we use P .An ideal I on N is said to be ccc over fin [3], if for each uncountablealmost disjoint family, all but countably many of them are in I . Anideal is a P -ideal if it is countably directed closed mod finite.The following main result is extracted from [6] and [8] which werecord without proof. Lemma 1.1 (PFA) . If P ∗ is a finite or countable product (repetitionsallowed) of posets from the set { P ℓ : ℓ ∈ N } and if G is a P ∗ -genericfilter, then in V [ G ] every autohomeomorphism of N ∗ has the propertythat the ideal of sets on which it is trivial is a P -ideal which is ccc overfin. Corollary 1.2 (PFA) . If P ∗ is a finite or countable product of posetsfrom the set { P ℓ : ℓ ∈ N } , and if G is a P ∗ -generic filter, then in V [ G ] if F is an autohomeomorphism of N ∗ and { Z α : α ∈ ω } is an increasingmod finite chain of infinite subsets of N , there is an α ∈ ω and acollection { h α : α ∈ ω } of 1-to-1 functions such that dom( h α ) = Z α and for all β ∈ ω and a ⊂ Z β \ Z α , F [ a ] = ∗ h β [ a ] . Each poset P ∗ as above is ℵ -closed and ℵ -distributive (see [8,p.4226]). In this paper we will restrict out study to finite products.The following partial order can be used to show that these productsare ℵ -distributive. A. DOW AND S. SHELAH
Definition 1.2.
Let P ∗ be a finite product of posets from { P ℓ : ℓ ∈ N } .Given { ~f ξ : ξ ∈ µ } = F ⊂ P ∗ (decreasing in the ordering on P ∗ ), define P ( F ) to be the partial order consisting of all g ∈ P ∗ such that there issome ξ ∈ µ such that ~g ≡ ∗ ~f ξ . The ordering on P ( F ) is coordinatewise ⊇ as opposed to ∗ ⊇ in P ∗ . Corollary 1.3 (PFA) . If P ∗ is a finite or countable product of posetsfrom the set { P ℓ : ℓ ∈ N } , and if G is a P ∗ -generic filter, then in V [ G ] if F is an involution of N ∗ with a unique fixed point x , then x is a P ω -point and N ∗ = A ⊲⊳ x B for some A, B such that F [ A ] = B .Proof. We may assume that F also denotes an arbitrary lifting of F to [ N ] ω in the sense that for each Y ⊂ N , ( F [ Y ]) ∗ = F [ Y ∗ ]. Let Z x = [ N ] ω \ x (the dual ideal to x ). For each Z ∈ Z x , F [ Z ] is alsoin Z and F [ Z ∪ F [ Z ]] = ∗ Z ∪ F [ Z ]. So let us now assume that Z denotes those Z ∈ Z x such that Z = ∗ F [ Z ]. Given Z ∈ Z , sincefix( F ) ∩ Z ∗ = ∅ , there is a collection Y ⊂ [ Z ] ω such that F [ Y ] ∩ Y = ∗ ∅ for each Y ∈ Y , and such that Z ∗ is covered by { Y ∗ : Y ∈ Y } . Bycompactness, we may assume that Y = { Y , . . . , Y n } is finite. Set Z = Y ∪ F [ Y ]. By induction, replace Y k by Y k \ S j Let P ∗ be a finite product of posets from { P ℓ : ℓ ∈ N } .In the forcing extension, V [ H ] , by ω <ω , there is a descending sequence F from P ∗ which is P ∗ -generic over V and, for which, P ( F ) is ccc and ω ω -bounding. It follows also that P ( F ) preserves that R ∩ V is of second category.This was crucial in the proof of Lemma 1.1. We can manage with the ω ω -bounding property because we are going to use Lemma 1.1. A posetis said to be ω ω -bounding if every new function in ω ω is bounded bysome ground model function.The following proposition is probably well-known but we do not havea reference. Proposition 1.2. Assume that Q is a ccc ω ω -bounding poset and that x is an ultrafilter on N . If G is a Q -generic filter then there is no set A ⊂ N such that A \ Y is finite for all Y ∈ x .Proof. Assume that { ˙ a n : n ∈ ω } are Q -names of integers such that1 (cid:13) Q “ ˙ a n ≥ n ”. Let A denote the Q -name so that (cid:13) Q “ A = { ˙ a n : n ∈ ω } ”. Since Q is ω ω -bounding, there is some q ∈ Q and a sequence { n k : k ∈ ω } in V such that q (cid:13) Q “ n k ≤ ˙ a i ≤ n k +2 ∀ i ∈ [ n k , n k +1 )”.There is some ℓ ∈ Y = S k [ n k + ℓ , n k + ℓ +1 ) is a member of x . On the other hand, q (cid:13) Q “ A ∩ [ n k + ℓ +1 , n k + ℓ +3 )” is not empty foreach k . Therefore q (cid:13) Q “ A \ Y is finite”. (cid:3) Another interesting and useful general lemma is the following. A. DOW AND S. SHELAH Lemma 1.3. Let F ⊂ P ℓ (for any ℓ ∈ N ) be generic over V , then foreach P ( F ) -name ˙ h ∈ N N , either there is an f ∈ F such that f (cid:13) P ( F ) “ ˙ h ↾ dom( f ) / ∈ V ”, or there is an f ∈ F and an increasing sequence n < n < · · · of integers such that for each i ∈ [ n k , n k +1 ) and each g < f such that g forces a value on ˙ h ( i ) , f ∪ ( g ↾ [ n k , n k +1 )) also forcesa value on ˙ h ( i ) .Proof. Given any f , perform a standard fusion (see [6, 2.4] or [8, 3.4]) f k , n k by picking L k ⊂ [ n k +1 , n k +2 ) (absorbed into dom( f k +1 )) so thatfor each partial function s on n k which extends f k ↾ n k , if there is someinteger i ≥ n k +1 for which no Theorem 2.1 (PFA) . If G is a generic filter for P ∗ = P × P , thenthere are symmetric tie-points x, y as witnessed by A, B and C, D re-spectively such that N ∗ is not homeomorphic to the space A ⊲⊳ x = y C Assume that N ∗ is homeomorphic to A ⊲⊳ x = y C and that z is the P ∗ -name of the ultrafilter that is sent (by the assumed homeomorphism)to the point ( x, y ) in the quotient space A ⊲⊳ x = y C .Further notation: let { a α : α ∈ ω } be the P -names of the infinitesubsets of N which form the mod-finite increasing chain whose remain-ders in N ∗ cover A \{ x } and, similarly let { c α : α ∈ ω } be the P -names(second coordinates though) which form the chain in C \ { y } .If we represent A ⊲⊳ x = y C as a quotient of ( N × ∗ , we may assume that F is a P ∗ -name of a function from [ N ] ω into [ N × ω such that letting Z α = F − ( a α × { } ∪ c α × { } ) for each α ∈ ω , then { Z α : α ∈ ω } forms the dual ideal to z , and F : [ Z α ] ω → ( a α ×{ }∪ c α ×{ } ) ω inducesthe above homeomorphism from Z ∗ α onto ( a ∗ α × { } ) ∪ ( c ∗ α × { } ).By Corollary 1.2, we may assume that for each β ∈ ω , there isa bijection h β between some cofinite subset of Z β and some cofinitesubset of ( a β × { } ) ∪ ( c β × { } ) which induces F ↾ [ Z β ] ω (since wecan just ignore Z α for some fixed α ). We will use F ↾ [ Z β ] ω = h β to mean that h β induces F ↾ [ Z β ] ω . Note that by the assumptions,for each β ∈ ω , there is a γ ∈ ω such that each of h − γ ( a γ ) \ Z β and h − γ ( c γ ) \ Z β are infinite. ORE ON TIE-POINTS AND HOMEOMORPHISM IN N ∗ Let H be a generic filter for ω <ω , and assume that F ⊂ P ∗ is chosenas in Lemma 1.2. In this model, let us use λ to denote the ω from V . Using the fact that F is P ∗ -generic over V , we may treat all thefunctions h α ( α ∈ λ ) as members of V since we can take the valuationof all the P ∗ -names using F . Assume that ˙ h is a P ( F )-name of a finite-to-1 function from N into N × h α ⊂ ∗ h for all α ∈ λ .We show there is no such ˙ h .Since P ( F ) is ω ω -bounding, there is a increasing sequence of integers { n k : k ∈ ω } and an ~f = ( g , g ) ∈ F such that(1) for each i ∈ [ n k , n k +1 ), ~f (cid:13) P ( F ) “ ˙ h ( i ) ∈ ([0 , n k +2 ) × i ∈ [ n k , n k +1 ), ~f (cid:13) P ( F ) “ ˙ h − ( { i } × ⊂ [0 , n k +2 )”(3) for each k and each j ∈ { , } there is an m such that n k < m < m +1 < n k +1 , and [2 m , m +1 ) \ dom g j has at least k elements.Choose any ( g ′ , g ′ ) = ~f < ~f such that N \ dom( g ′ ) ⊂ S k [ n k +1 , n k +2 )and N \ dom( g ′ ) ⊂ S k [ n k +4 , n k +5 ). Next, choose any ~f < ~f andsome α ∈ λ such that ~f (cid:13) P ( F ) “ dom( g ′ ) ⊂ ∗ a α ∪ g ′ [ a α ] and dom( g ′ ) ⊂ ∗ c α ∪ g ′ [ c α ]”. For each γ ∈ λ , note that ~f (cid:13) P ( F ) “ a γ \ a α ⊂ ∗ N \ dom( g ′ )”and similarly ~f (cid:13) P ( F ) “ c γ \ c α ⊂ ∗ N \ dom( g ′ )”.Now consider the two disjoint sets: Y = S k [ n k , n k +3 ) and Y = S k [ n k +3 , n k +6 ). Since z is an ultrafilter in this extension, by possiblyextending ~f even more, we may assume that there is some j ∈ { , } and some β > α such that ~f (cid:13) P ( F ) “ Y j ⊂ ∗ Z β ”. Without loss ofgenerality (by symmetry) we may assume that j = 0. Consider any γ ∈ λ . Since we are assuming that h γ ⊂ ∗ ˙ h , we have that ~f forces that h γ [ Z γ \ Z α ] = ∗ ˙ h [ Z γ \ Z α ]. We also have that ~f (cid:13) P ( F ) “ ˙ h [ Y ] ∗ ⊃ ( a γ \ a α ) × { } = ∗ h γ [ Z γ \ Z α ] ∩ N × { } ”. Putting this all together, we nowhave that ~f forces that ˙ h [ Z β ] almost contains ( a γ \ a α ) × { } for all γ ∈ λ ; which clearly contradicts that ˙ h [ Z β ] is supposed to be almostequal to h β [ Z β ].So now what? Well, let H be a generic filter for P ( F ) and considerthe family of functions H λ = { h α : α ∈ λ } which we know does nothave a common finite-to-1 extension.Before proceeding, we need to show that H λ does not have any ex-tension h . If ˙ h is any P ( F )-name of a function for which it is forced that h α ⊂ ∗ ˙ h for all α ∈ λ , then there is some ℓ ∈ N such that ˙ Y = h − ( h ( ℓ ))is (forced to be) infinite. It follows easily that ˙ Y is forced to be almostcontained in every member of z . By Lemma 1.2 this cannot happen.Therefore the family H λ does not have any common extension. A. DOW AND S. SHELAH Given such a family as H λ , there is a well-known proper poset Q (see[1, 3.1], [3, 2.2.1], and [10, p9]) which will force an uncountable cofinal I ⊂ λ and a collection of integers { k α,β : α < β ∈ I } satisfying that h α ( k α,β ) = h β ( k α,β ) (and both are defined) for α < β ∈ I . So, let ˙ Q bethe ω <ω ∗ P ( F )-name of the above mentioned poset. In addition, let ˙ ϕ be the ω <ω ∗ P ( F ) ∗ ˙ Q -name of the enumerating function from ω onto I ,and let ˙ k α,β (for α < β ∈ ω ) be the name of the integer k ˙ ϕ ( α ) , ˙ ϕ ( β ) . Thusfor each α < β ∈ ω , there is a dense set D ( α, β ) ⊂ ω <ω ∗ P ( F ) ∗ ˙ Q suchthat for each member p of D ( α, β ), there are functions h α , h β in V andsets Z α = dom( h α ) , Z β = dom( h β ) and integers k = k ( α, β ) ∈ Z α ∩ Z β such that p (cid:13) ω <ω ∗ P ( F ) ∗ ˙ Q “ F ↾ [ Z α ] ω = h α , F ↾ [ Z β ] ω = h β , h α ( k ) = h β ( k )” . Finally, let ˙ Q be the ω <ω ∗ P ( F ) ∗ ˙ Q -name of the σ -centered posetwhich forces an element ~f ∈ P ∗ which is below every member of F .Again, there is a countable collection of dense subsets of the properposet ω <ω ∗ P ( F ) ∗ ˙ Q ∗ ˙ Q which determine the values of ~f .Applying PFA to the above proper poset and the family of ω men-tioned dense sets, we find there is a sequence { h ′ α , Z ′ α : α ∈ ω } , integers { k α,β : α < β ∈ ω } , and a condition ~f ∈ P ∗ such that, for all α < β and k = k ( α, β ), ~f (cid:13) P ∗ “ F ↾ [ Z ′ α ] ω = h ′ α , F ↾ [ Z ′ β ] ω = h ′ β , h ′ α ( k ) = h ′ β ( k )” . But, we also know that we can choose ~f so that there is some λ ∈ ω ,and some h λ , Z λ such that, for all α ∈ ω , Z ′ α ⊂ ∗ Z λ and F ↾ [ Z λ ] ω = h λ .It follows of course that for all α ∈ ω , there is some n α such that h ′ α ↾ [ n α , ω ) ⊂ h λ . Let J ∈ [ ω ] ω , n ∈ ω , and h ′ a function withdom( h ′ ) ⊂ n such that n α = n and h ′ α ↾ n = h ′ for all α ∈ J . We nowhave a contradiction since if α < β ∈ J then clearly k = k ( α, β ) ≥ n and this contradicts that h ′ α ( k ) and h ′ β ( k ) are both supposed to equal h λ ( k ). 3. proof of Theorem 1.2 Theorem 3.1 (PFA) . If G is a generic filter for P , then a tie-point x is introduced such that τ ( x ) = 2 and with N ∗ = A ⊲⊳ x B , neither A nor B is a homeomorph of N ∗ . In addition, there is no involution F on N ∗ which has a unique fixed point, and so, no tie-point is symmetric. Assume that V is a model of PFA and that P = P . The ele-ments of P are partial functions f from N into 2 which also satisfythat lim sup n ∈ N | n +1 \ (2 n ∪ dom( f )) | = ∞ . The ordering on P is that ORE ON TIE-POINTS AND HOMEOMORPHISM IN N ∗ f < g ( f, g ∈ P ) if g ⊂ ∗ f . For each f ∈ P , let a f = f − (0) and b f = f − (1).Again we assume that { a α : α ∈ ω } is the sequence of P -namessatisfying that N ∗ = A ⊲⊳ x B and A \ { x } = S { a ∗ α : α ∈ ω } . Of courseby this we mean that for each f ∈ G , there are α ∈ ω , a ∈ [ N ] ω , and f ∈ G such that a f ⊂ ∗ a ⊂ a f and f (cid:13) P “ˇ a = a α ”.Next we assume that, if A is homeomorphic to N ∗ , then F is a P -name of a homeomorphism from N ∗ to A and let z denote the pointin N ∗ which F sends to x . Also, let Z α be the P -name of F − [ a α ]and recall that N ∗ \ { z } = S { Z ∗ α : α ∈ ω } . As above, we may alsoassume that for each α ∈ ω , there is a P -name of a function h α withdom( h α ) = Z α such that F ↾ [ Z α ] ω is induced by h α .Furthermore if τ ( x ) > 2, then one of A \ { x } or B \ { x } can bepartitioned into disjoint clopen non-compact sets. We may assumethat it is A \ { x } which can be so partitioned. Therefore there is somesequence { c α : α ∈ ω } of P -names such that for each α < β ∈ ω , c β ⊂ a β and c β ∩ a α = ∗ c α . In addition, for each α < ω there must bea β ∈ ω such that c β \ a α and a β \ ( c β ∪ a α ) are both infinite.Now assume that H is ω <ω -generic and again choose a sequence F ⊂ P which is V -generic for P and which forces that P ( F ) is ccc and ω ω -bounding. For the rest of the proof we work in the model V [ H ] andwe again let λ denote the ordinal ω V .In the case of P we are able to prove a significant strengthening ofLemma 1.3. Lemma 3.1. Assume that ˙ h is a P ( F ) -name of a function from N to N . Either there is an f ∈ F and such that f (cid:13) P ( F ) “ ˙ h ↾ dom( f ) / ∈ V ”,or there is an f ∈ F and an increasing sequence m < m < · · · ofintegers such that N \ dom( f ) = S k S k where S k ⊂ m k +1 \ m k and foreach i ∈ S k the condition f ∪ { ( i, } forces a value on ˙ h ( i ) .Proof. First we choose f ∈ F and some increasing sequence n < n < · · · n k < · · · as in Lemma 1.3. We may choose, for each k , an m k such that n k ≤ m k < m k +1 ≤ n k +1 such that lim sup k | m k +1 \ (2 m k ∪ dom( f )) | = ∞ . For each k , let S k = 2 m k +1 \ (2 m k ∪ dom( f )). Byre-indexing we may assume that | S k | ≥ k , and we may arrange that N \ dom( f ) is equal to S k S k and set L = N . For each k ∈ L , let i k = min S k and choose any f ′ < f such that (by definition of P ) I = { i k : k ∈ L } ⊂ ( f ′ ) − (0) and (by assumption on ˙ h ) f ′ forces avalue on ˙ h ( i k ) for each k ∈ L . Set f = f ′ ↾ ( N \ I ) and for each k ∈ L , let S k = S k \ ( { i k } ∪ dom( f )). By further extending f wemay also assume that f ∪ { ( i k , } also forces a value on ˙ h ( i k ). Choose L ⊂ L such that lim k ∈ L | S k | = ∞ . Notice that each member of i k isthe minimum element of S k . Again, we may extend f and assume that N \ dom( f ) is equal to S k ∈ L S k . Suppose now we have some infinite L j ,some f j , and for k ∈ L j , an increasing sequence { i k , i k , . . . , i j − k } ⊂ S k .Assume further that S jk ∪ { i ℓk : ℓ < j } = S k \ dom( f j )and that lim k ∈ L j | S jk | = ∞ . For each k ∈ L j , let i jk = min( S jk \ { i ℓk : ℓ < j } ). By a simple recursion of length 2 j , there is an f j +1 < f j suchthat, for each k ∈ L j , { i ℓk : ℓ ≤ j } ⊂ S k \ dom( f j +1 ) and for eachfunction s from { i ℓk : ℓ ≤ j } into 2, the condition f j +1 ∪ s forces a valueon ˙ h ( i jk ). Again find L j +1 ⊂ L j so that lim k ∈ L j +1 | S j +1 k | = ∞ (where S j +1 k = S k \ dom( f j +1 )) and extend f j +1 so that N \ dom( f j +1 ) is equalto S k ∈ L j +1 S j +1 k .We are half-way there. At the end of this fusion, the function ¯ f = S j f j is a member of P because for each j and k ∈ L j +1 , 2 m k +1 \ (2 m k ∪ dom( ¯ f )) ⊃ { i k , . . . , i jk } . For each k , let ¯ S k = S k \ dom( ¯ f ) and, bypossibly extending ¯ f , we may again assume that there is some L suchthat lim k ∈ L | ¯ S k | = ∞ and that, for k ∈ L , ¯ S k = { i , i k , . . . , i j k k } forsome j k . What we have proven about ¯ f is that it satisfies that for each k ∈ L and each j < j k and each function s from { i k , . . . , i j − k } to 2,¯ f ∪ s ∪ ( i jk , 0) forces a value on ˙ h ( i jk ).To finish, simply repeat the process except this time choose maximalvalues and work down the values in ¯ S k . Again, by genericity of F , theremust be such a condition as ¯ f in F . (cid:3) Returning to the proof of Theorem 3.1, we are ready to use Lemma3.1 to show that forcing with P ( F ) will not introduce undesirable func-tions h analogous to the argument in Theorem 1.1. Indeed, assumethat we are in the case that F is a homeomorphism from N ∗ to A asabove, and that { h α : α ∈ λ } is the family of functions as above. Ifwe show that ˙ h does not satisfy that h α ⊂ ∗ ˙ h for each α ∈ λ , then weproceed just as in Theorem 1.1. By Lemma 3.1, we have the condition f ∈ F and the sequence S k ( k ∈ N ) such that N \ dom( f ) = S k S k and that for each i ∈ S k S k , f ∪ { ( i, } forces a value (call it ¯ h ( i )) on˙ h ( i ). Therefore, ¯ h is a function with domain S k S k in V . It suffices tofind a condition in P below f which forces that there is some α suchthat h α is not extended by ˙ h . It is useful to note that if Y ⊂ S k S k is such that lim sup | S k \ Y | is infinite, then for any function g ∈ Y , f ∪ g ∈ P . ORE ON TIE-POINTS AND HOMEOMORPHISM IN N ∗ We first check that ¯ h is 1-to-1 on a cofinite subset. If not, there isan infinite set of pairs E j ⊂ S k ¯ S k , ¯ h [ E j ] is a singleton and such thatfor each k , ¯ S k ∩ S j E j has at most two elements. If g is the functionwith dom( g ) = S j E j which is constantly 0, then f ∪ g forces that ˙ h agrees with ¯ h on dom( g ) and so is not 1-to-1. On the other hand, thiscontradicts that there is f < f ∪ g such that for some α ∈ ω , a α almost contains ( f ∪ g ) − (0) and the 1-to-1 function h α with domain a α is supposed to also agree with ˙ h on dom( g ).But now that we know that ¯ h is 1-to-1 we may choose any f ∈ F such that f < f and such that there is an α ∈ ω with f − (0) ⊂ a α ⊂ f − (0), and f has decided the function h α . Let Y be any infinitesubset of N \ dom( f ) which meets each ¯ S k in at most a single point. If¯ h [ Y ] meets Z α in an infinite set, then choose f < f so that f [ Y ] = 0and there is a β > α such that Y ⊂ a β . In this case we will have that f forces that Y ⊂ a β \ a α , ˙ h ↾ Y ⊂ ∗ h β , and h β [ Y ] ∩ h β [ a α ] is infinite(contradicting that h β is 1-to-1). Therefore we must have that ¯ h [ Y ] isalmost disjoint from Z α . Instead consider f < f so that f [ Y ] = 1.By extending f we may assume that there is a β < ω such that f (cid:13) P ( F ) “ Z β ∩ ¯ h [ Y ] is infinite”. However, since f (cid:13) P ( F ) “ h β ⊂ ∗ ˙ h ”, wealso have that f (cid:13) P ( F ) “ h β ↾ ( a β \ a α ) ⊂ ∗ ¯ h and ( a β \ a α ) ∩ Y = ∗ ∅ ”contradicting that ¯ h is 1-to-1 on dom( f ) \ dom( f ). This finishes theproof that there is no P ( F ) name of a function extending all the h α ’s( α ∈ λ ) and the proof that F can not exist continues as in Theorem1.1.Next assume that we have a family { c α : α ∈ λ } as described aboveand suppose that C = ˙ h − (0) satisfies that (it is forced) C ∩ a α = ∗ c α for all α ∈ λ . If we can show there is no such ˙ h , then we will knowthat in the extension obtained by forcing with P ( F ), the collection { ( c α , ( a α \ c α )) : α ∈ λ } forms an ( ω , ω )-gap and we can use a properposet Q to “freeze” the gap. Again, meeting ω dense subsets of theiteration ω <ω ∗ P ( F ) ∗ Q ∗ Q (where Q is the σ -centered poset asin Theorem 1.1) introduces a condition f ∈ P which forces that c λ will not exist. So, given our name ˙ h , we repeat the steps above upto the point where we have f and the sequence { S k : k ∈ N } sothat f ∪ { ( i, } forces a value ¯ h ( i ) on ˙ h ( i ) for each i ∈ S k S k and N \ dom( f ) = S k S k . Let Y = ¯ h − (0) and Z = ¯ h − (1) (of coursewe may assume that ¯ h ( i ) ∈ i ). Since x is forced to be anultrafilter, there is an f < f such that dom( f ) contains one of Y or Z . If dom( f ) contains Y , then f forces that ˙ h [ a β \ dom( f )] = 1 andso ( a β \ dom( f )) ⊂ ∗ ( N \ C ) for all β ∈ ω . While if dom( f ) contains Z , then f forces that ˙ h [ a β \ dom( f )] = 0, and so ( a β \ dom( f )) ⊂ ∗ C for all β ∈ ω . However, taking β so large that each of c β \ dom( f )and ( a β \ ( c β ∪ dom( f )) are infinite shows that no such ˙ h exists.Finally we show that there are no involutions on N ∗ which have aunique fixed point. Assume that F is such an involution and that z is the unique fixed point of F . Applying Corollaries 1.2 and 1.3, wemay assume that N ∗ \ { z } = S α ∈ ω Z ∗ α and that for each α , F ↾ Z ∗ α isinduced by an involution h α .Again let H be ω <ω -generic, λ = ω V , and F ⊂ P be P -generic over V . Assume that ˙ h is a P ( F )-name of a function from N into N . Itsuffices to show that no f ∈ F forces that ˙ h mod finite extends each h α ( α ∈ λ ).At the risk of being too incomplete, we leave to the reader the factthat Lemma 1.3 can be generalized to show that there is an f ∈ F suchthat either f (cid:13) P “ ˙ h ↾ Z α / ∈ V ”, or there is a sequence { n k : k ∈ N } asbefore. This is simply due to the fact that the P -name of the ultrafilter x can be replaced by any P -name of an ultrafilter on N . Similarly,Lemma 3.1 can be generalized in this setting to establish that theremust be an f ∈ F and a sequence of sets { m k , S k , T k : k ∈ K ∈ [ N ] ω } with bijections ψ : S k → T k such that S k ⊂ (2 m k +1 \ m k ) ⊂ [ n k , n k +1 ), T k ⊂ [ n k , n k +1 ), N \ dom( f ) ⊂ S k S K , and for each k and i ∈ S k and¯ f < f ¯ f forces a value on ˙ h ( ψ ( i )) iff i ∈ dom( ¯ f ). The difference hereis that we may have that f (cid:13) P “ dom( f ) ⊂ Z α ”, but there will besome values of ˙ h not yet decided since V [ H ] does not have a functionextending all the h α ’s. Set Ψ = S ψ which is a 1-to-1 function.The contradiction now is that there will be some f ′ < f such that f ′ (cid:13) P “Ψ ∗ ( x ) = z ” (because we know that x is not a tie-point). There-fore we may assume that Ψ(dom( f ′ ) ∩ dom(Ψ)) is a member of z and sothat Ψ(dom(Ψ) \ dom( f ′ )) is not a member of z . By assumption, thereis some ¯ f < f ′ and an α ∈ λ such that ¯ f (cid:13) P “Ψ(dom(Ψ) \ dom( f ′ )) ⊂ Z α ”. However this implies ¯ f forces that ˙ h (Ψ( i )) = h α (Ψ( i )) for almostall i ∈ S k S k \ dom( ¯ f ), contradicting that ¯ f does not force a value on¯ h (Ψ( i )) for all i / ∈ ¯ f . 4. questions Question 4.1. Assume PFA. If G is P -generic, and N ∗ = A ⊲⊳ x B isthe generic tie-point introduced by P , is it true that A is not homeo-morphic to N ∗ ? Is it true that τ ( x ) = 2? Is it true that each tie-pointis a symmetric tie-point? ORE ON TIE-POINTS AND HOMEOMORPHISM IN N ∗ Remark . The tie-point x introduced by P does not satisfy that τ ( x ) = 3. This can be seen as follows. For each f ∈ P , we canpartition min( f ) into { i ∈ dom( f ) : i < f ( i ) < f ( i ) } and { i ∈ dom( f ) : i < f ( i ) < f ( i ) } .It seems then that the tie-points x ℓ introduced by P ℓ might be bettercharacterized by the property that there is an autohomeomorphism F ℓ of N ∗ satisfying that fix( F ℓ ) = { x ℓ } , and each y ∈ N ∗ \ { x } has an orbitof size ℓ . Remark . A small modification to the poset P will result in a tie-point N ∗ = A ⊲⊳ x B such that A (hence the quotient space by the associatedinvolution) is homeomorphic to N ∗ . The modification is to build intothe conditions a map from the pairs { i, f ( i ) } into N . A natural way todo this is the poset f ∈ P +2 if f is a 2-to-1 function such that for each n , f maps dom( f ) ∩ (2 n +1 \ n ) into 2 n \ n − , and again lim sup n | n +1 \ (dom( f ) ∪ n ) | = ∞ . P +2 is ordered by almost containment. The genericfilter introduces an ω -sequence { f α : α ∈ ω } and two ultrafilters: x ⊃ { N \ dom( f α ) : α ∈ ω } and z ⊃ { N \ range( f α ) : α ∈ ω } . Foreach α and a α = min( f α ) = { i ∈ dom( f α ) : i = min( f − α ( f α ( i )) } , weset A = { x } ∪ S α a ∗ α and B = { x } ∪ S α (dom( f α ) \ a α ) ∗ , and we havethat N ∗ = A ⊲⊳ x B is a symmetric tie-point. Finally, we have that F : A → N ∗ defined by F ( x ) = z and F ↾ A \ { x } = S α ( f α ) ∗ is ahomeomorphism. Question 4.2. Assume PFA. If L is a finite subset of N and P L =Π { P ℓ : ℓ ∈ L } , is it true that in V [ G ] that if x is tie-point, then τ ( x ) ∈ L ; and if 1 / ∈ L , then every tie-point is a symmetric tie-point? References [1] Alan Dow, Petr Simon, and Jerry E. Vaughan, Strong homology and theproper forcing axiom , Proc. Amer. Math. Soc. (1989), no. 3, 821–828.MR MR961403 (90a:55019)[2] Alan Dow and Geta Techanie, Two-to-one continuous images of N ∗ , Fund.Math. (2005), no. 2, 177–192. MR MR2162384 (2006f:54003)[3] Ilijas Farah, Analytic quotients: theory of liftings for quotients over analyticideals on the integers , Mem. Amer. Math. Soc. (2000), no. 702, xvi+177.MR MR1711328 (2001c:03076)[4] Ronnie Levy, The weight of certain images of ω , Topology Appl. (2006),no. 13, 2272–2277. MR MR2238730 (2007e:54034)[5] S. Shelah and J. Stepr¯ans. Non-trivial homeomorphisms of βN \ N without theContinuum Hypothesis. Fund. Math. , 132:135–141, 1989.[6] S. Shelah and J. Stepr¯ans. Somewhere trivial autohomeomorphisms. J. LondonMath. Soc. (2) , 49:569–580, 1994. [7] 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)[8] Juris Stepr¯ans, The autohomeomorphism group of the ˇCech-Stone compactifi-cation of the integers , Trans. Amer. Math. Soc. (2003), no. 10, 4223–4240(electronic). MR 1990584 (2004e:03087)[9] B. Velickovic. 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 ::