aa r X i v : . [ m a t h . G T ] S e p Resolving compacta by free p -adic actions Michael Levin ∗ Abstract
We say that a compactum (compact metric space) Y is resolvable by a p -adicaction on a compactum X if there is a continuous action of the p -adic integers A p on X such that Y = X/A p . In this paper we study compacta Y that are resolvableby a free p -adic action on a compactum of a lower dimension and focus on compacta Y with dim Z [ p ] Y = 1. This is mainly motivated by p -adic actions on 1-dimensionalcompacta, the case that turns out to be highly non-trivial. More motivation forconsidering orbit spaces with dim Z [ p ] = 1 comes from Theorem (A). If A p acts on a finite dimensional compactum X so that Y = X/A p is infinite dimensional then there exists an invariant compactum X ′ ⊂ X on whichthe action of A p is free and whose orbit space Y ′ = X ′ /A p is infinite dimensional withdim Z [ p ] Y ′ = 1. (cid:4) We show
Theorem (B).
Let Y be a finite dimensional compactum with dim Z [ p ] Y = 1. Then(i) Y is resolvable by a free p -adic action on a compactum of dim ≤ dim Y − Y ≥ Y is not resolvable by a free p -adic action on a compactum of dim ≤ dim Y − Y ≥ (cid:4) The author was initially inclined to believe that for a 3-dimensional compactum Y with dim Z [ p ] Y = 1 (the case not covered by (ii) of Theorem (B)) the additionalassumption dim Z p Y = 2 imposed by Yang’s relations would imply that Y is resolvableby a free p -adic action on a 1-dimensional compactum and was surprised to find outthat Theorem (C).
There is a 3-dimensional compactum Y with dim Z [ p ] Y = 1 anddim Z p Y = 2 that cannot be resolved by a free p -adic action on a 1-dimensionalcompactum. (cid:4) Moreover
Theorem (D).
There is an infinite dimensional compactum Y with dim Z [ p ] Y = 1and dim Z Y = 2 that cannot be resolved by a free p -adic action on a finite dimensionalcompactum. (cid:4) Theorems (A), (C) and (D) are based on
Theorem (E).
If a compactum Y with dim ≥ n + 2 can be resolved by a free p -adicaction on an n -dimensional compactum then the second Cech cohomology H ( Y ; Z )of Y contains a subgroup isomorphic to Z p ∞ . Moreover for every closed A ⊂ Y withdim A ≥ n + 2 the image of this subgroup in H ( A ; Z ) under the homomorphisminduced by the inclusion is non-trivial. (cid:4) ∗ This research was supported by THE ISRAEL SCIENCE FOUNDATION (grant No. 522/14) Introduction
The Hilbert-Smith conjecture asserts that a compact group effectively (and continuously)acting on a manifold must be a Lie group. This assertion is equivalent to the following one:there is no effective action of A p (the group of p -adic integers) on a manifold. The Hilbert-Smith conjecture is proved for manifolds of dim ≤ A p in dim >
3. Yang [15] showed that if A p effectively acts on an n -manifold M then eitherdim M/A p = ∞ or dim M/A p = n + 2. This naturally suggests to examine if the latterdimensional relations may occur in a more general setting, mainly when M is just a finitedimensional compactum (=compact metric space). One of these relations was confirmedby Raymond and Williams [13] who constructed an action of A p on an n -dimensionalcompactum X , n ≥
2, with dim
X/A p = n + 2. However it remains open for more than 50years whether there exists a free action of A p on a finite dimensional compactum X suchthat dim X/A p = dim X + 2 or dim X/A p = ∞ .An important collection of dimensional properties of the orbit spaces under actions of A p is provided by Cohomological Dimension. Recall the cohomological dimension dim G X of a compactum X with respect to an abelian group G is the least integer n (or ∞ if such n does not exist) such that the Cech cohomology H n +1 ( X, A ; G ) vanishes for every closed A ⊂ X . Clearly dim G X ≤ dim X for every group G and, by the Alexandrov theorem,dim X = dim Z X if X is finite dimensional. A first example of an infinite dimensionalcompactum X with dim Z X < ∞ was constructed by A. Dranishnikov.Yang [15] showed that a free action of A p on a compactum X imposes the followingdimensional relations between X and Y = X/A p : dim Z [ p ] Y = dim Z [ p ] X , dim Z p ∞ X ≤ dim Z p ∞ Y ≤ dim Z p ∞ X + 1, dim Z p X ≤ dim Z p Y ≤ dim Z p X + 1 and dim X ≤ dim Z Y ≤ dim Z X + 2. Moreover if dim Z Y = dim Z X + 2 then dim Z p ∞ Y = dim Z p Y = dim Z X + 1.We will refer to these dimensional relations as the Yang relations.There is a nice characterization of cohomological dimension in terms of extensions ofmaps. A CW-complex K is said to be an absolute extensor for a space X , written ext-dim X ≤ K , if every map from a closed subset A of X to K continuously extends over X .It turns out that dim G X ≤ n if and only if the Eilenberg-MacLane complex K ( G, n ) isan absolute extensor for X . Using this characterization and representing K ( Z [ p ] ,
1) as theinfinite telescope of the p -fold covering S → S of the circle S one can easily show Proposition 1.1
Let X be a compactum. Then dim Z [ p ] X ≤ if and only if for every map f : A → S from a closed subset A of X to the circle S there is a natural number k suchthat f followed by the p k -fold covering map S → S of S extends over X . Let us say that a compactum Y can be resolved by a free p -adic action on a compactum X if there is a continuous free action of the p -adic integers A p on X such that Y = X/A p We will shorten this lengthy expression to a shorter one: Y is p -resolvable by X (keepingin mind that we consider only free actions of A p ).In this paper we study compacta Y that are p -resolvable by compacta of a lower di-mension and mainly focus on compacta with dim Z [ p ] Y = 1 and, in particular on compactawhich are p -resolvable by 1-dimensional compacta (and hence, by Yang’s relations, have2im Z [ p ] = 1), the case that turns out to be highly non-trivial. More motivation for consid-ering orbit spaces Y with dim Z [ p ] Y = 1 comes from Theorem 1.2 If A p acts on a finite dimensional compactum X so that Y = X/A p isinfinite dimensional then there exists an invariant compactum X ′ ⊂ X on which the actionof A p is free and whose orbit space Y ′ = X ′ /A p is infinite dimensional with dim Z [ p ] Y ′ = 1 . The most obvious obstruction to the p -resolvability of a compactum Y by a compactumof dim < dim Y is H ( Y ; Z p ) = 0. Indeed, A p is the inverse limit of Z p n and hence everycompactum that p -resolves Y can be represented as the inverse limit of Z p n -coverings of Y . Recall that H ( Y ; Z p ) represents the Z p -coverings (bundles) of Y . Then H ( Y ; Z p ) = 0implies that every Z p n -covering over Y is trivial. Hence any compactum that p -resolves Y must contain a copy of every component of Y , and therefore cannot be of a lower dimension.The conditions dim Z [ p ] Y = 1 and dim Y > f : A → S from a closed subset A of Y that does not extend over Y . By Proposition 1.1 there is a Z p k -covering map φ : S → S such that f followed by φ extends to g : Y → S . Take a component Y ′ ⊂ Y × S of the pull-back space of g and φ with the induced maps (projections) g ′ : Y ′ → S and φ ′ : Y ′ → Y . Notice that φ ′ is not1-to-1 since otherwise g ′ would provide an extension of g . Thus φ ′ is a Z p t -covering of Y with 0 < t ≤ k . The existence of a non-trivial Z p t -covering of Y implies H ( Y ; Z p ) = 0. Asimilar reasoning also shows that H ( Y ; Z p ) is infinite.It turns out that for a finite dimensional compactum Y with dim Y > Z [ p ] Y = 1 is already sufficient to p -resolve Y by a compactum of dim < dim Y but notof dim < dim Y − Theorem 1.3
Let Y be a finite dimensional compactum with dim Z [ p ] Y = 1 . Then(i) Y is p -resolvable by a compactum of dim ≤ dim Y − if dim Y ≥ and(ii) Y is not p -resolvable by a compactum of dim ≤ dim Y − if dim Y ≥ . The author was initially inclined to believe that for a 3-dimensional compactum Y withdim Z [ p ] Y = 1 (the case not covered by (ii) of Theorem 1.4) the additional assumptiondim Z p Y = 2 imposed by Yang’s relations would imply that Y is p -resolvable by a 1-dimensional compactum and was surprised to find out that Theorem 1.4
There is a -dimensional compactum Y with dim Z [ p ] Y = 1 and dim Z p Y = 2 that is not p -resolvable by a -dimensional compactum. Moreover
Theorem 1.5
There is an infinite dimensional compactum Y with dim Z [ p ] Y = 1 and dim Z Y = 2 that is not p -resolvable by a finite dimensional compactum. The following property is of crucial importance for proving Theorems 1.2, 1.4 and 1.5.
Theorem 1.6
Let a compactum Y be p -resolvable by an n -dimensional compactum and dim Y ≥ n + 2 . Then H ( Y ; Z ) contains a subgroup isomorphic to Z p ∞ . Moreover forevery closed A ⊂ Y with dim A ≥ n + 2 , the image of this subgroup in H ( A ; Z ) under thehomomorphism induced by the inclusion is non-trivial.
3n the proofs of Theorems 1.6, 1.5 and 1.4 we interpret the elements of H ( Y ; Z ) ascircle bundles over Y (like before we interpreted the elements of H ( Y ; Z p ) as Z p -bundlesover Y ). A few comments regarding Theorem 1.6 are given in Remark 3.2.The author knows no example of a compactum Y with dim Y > Z [ p ] Y = 1that satisfies the conclusions of Theorem 1.6. This together with the fact that the examplesin Theorems 1.4 and 1.5 are constructed in a certain generic way motivates Conjecture 1.7
No compactum Y with dim Z [ p ] Y = 1 and dim Y ≥ n + 2 is p -resolvableby an n -dimensional compactum. By (ii) of Theorem 1.4 the conjecture holds for 3 < dim Y < ∞ . Thus the meaningful casesof the conjecture are dim Y = 3 and dim Y = ∞ . Kolmogorov-Pontrjagin surfaces.
We will describe 2-dimensional compacta to whichwe will refer in the sequel as Kolmogorov-Pontjagin surfaces.Let p be a prime number and k ≥ p k ) the mapping cylinder of a p k -fold covering map ∂ ∆ → S and refer to the domain ∂ ∆ and the range S of this map as the bottom and the top of Ω( p k ) respectively.Let k = 0 , k , k , ... an increasing sequence of natural numbers. We will construct aKolmogorov-Pontrjagin surface Y determined by p and the sequence k n as the inverse limitof 2-dimensional finite simplicial complexes Ω n . Set Ω to be a 2-simplex ∆. Assume thatΩ n is constructed. Take a sufficiently fine triangulation of Ω n and in every 2-simplex ∆ ofΩ n remove the interior of ∆ and attach to ∂ ∆ the mapping cylinder Ω( p k n +1 ) by identifyingthe bottom of Ω( p k n +1 ) with ∂ ∆. Define Ω n +1 to be the space that obtained this way fromΩ n and define the bonding map ω n +1 : Ω n +1 → Ω n to be a map that sends each mappingcylinder Ω( p k n +1 ) to the corresponding simplex ∆ that identifies the bottom of Ω( p k n +1 )with ∂ ∆ and sends the top of Ω( p k n +1 ) to the barycenter of ∆. Denote Ω = lim ← (Ω n , ω n )and call Ω a Kolmogorov-Pontrjagin surface determined by the prime p and the sequence k n . We additionally assume that in the construction of a Kolmogorov-Pontrjagin surfaceΩ the triangulations of Ω n are so fine that the diameter of the images of the simplexes ofΩ n in Ω i , i < n under the map ω in = ω n ◦ · · · ◦ ω i +1 : Ω n → Ω i is < / n − i . Proposition 2.1
A compactum Y with dim Y ≤ m + 2 , m ≥ , and dim Z [ p ] Y = 1 admitsan m -dimensional map into a Kolmogorov-Pontrjagin surface. In the proof of Proposition 2.1 we will use the following facts.
Proposition 2.2 [2]
Let f : X → Y be an m -dimensional map (= a map whose fibers areof dim ≤ m ). Then dim X ≤ dim Y + m and dim G X ≤ dim G Y + m for every abeliangroup G . Lemma 2.3
Let T and T be finite trees, X a compactum and X ′ a σ -compact subset of X with dim X ′ ≤ . Then every map f : X → T × T can be arbitrarily closely approximated y a map f ′ : X → T × T such that f ′ coincides with f on A = f − ( ∂ ( T × T )) and f ′ is -dimensional on X ′ \ A where ∂ ( T × T ) = (( ∂T ) × T ) ∪ ( T × ∂T ) and ∂T and ∂T stand for the sets of the end points of T and T respectively. Proof of Lemma 2.3.
Represent f as f = ( f , f ) with f : X → T and f : X → T being the coordinate maps. Fix metrics on X , T and T , consider the induced supremummetric on Y = T × T and let ǫ >
0. Since T and T are finite trees there is δ > F ⊂ X that does not interesect A and every map f Fi : F → T i , i = 1 , δ -close to f i on F , f Fi extends over X to a map that ǫ -close to f i and coincides with f i on A .Consider a compact subset F ǫ ⊂ X ′ such that F ǫ ⊂ { x ∈ X : d ( x, A ) ≥ ǫ } and take afinite closed cover F of F ǫ by subsets of F ǫ of diameter < ǫ such that the images of the setsin F under both f and f are of diameter < δ and F splits into the union F = F ∪ F ∪ F of collections F i of disjoint sets.Denote by F i , i = 1 , F i . Then f i restricted to F i can be δ -approximated by a map f Fi : F i → T i that sends the sets in F i to distinct points in T i .Now extend f Fi to a map f ǫi : X → T i that is ǫ -close to f i and coincides with f i on A , andset f ǫ = ( f ǫ , f ǫ ) : X → Y = T × T .Then every fiber of f ǫ restricted to F ǫ intersects at most one set of F and at mostone set of F and hence every fiber of f ǫ restricted to F ǫ can be covered by finitely manydisjoint compact sets of diameter < ǫ . Recall that X ′ is σ -compact, f ǫ is ǫ -close to f andcoincides with f on A , and apply the standard Baire category argument to the functionspace of the maps from X to Y that coincide with f on A to get the desired result. (cid:4) Proof of Proposition 2.1.
Recall that a Kolmogorov-Pontrjagin surface is the inverselimit of finie simplicial complexes Ω n with the bonding maps ω n +1 : Ω n +1 → Ω n and Ω is a2-simplex. Take a σ -compact subset Y ′ of Y such that dim Y ′ ≤ Y \ Y ′ ≤ m − m -dimensional map f : Y → Ω . We will construct by induction on n spacesΩ n and m -dimensional maps f n : Y → Ω n . Assume the construction is completed for n and proceed to n + 1 as follows. Fix a triangulation of Ω n . Recall that dim Z [ p ] Y ≤ k n such that for every simplex ∆ of Ω n f n restricted to f − n ( ∂ ∆) and followed by a p k n -covering map ∂ ∆ → S extends over f − n (∆).This determines a simlicial complex Ω n +1 as described in the construction of Kolmogorov-Pontrjagin surfaces and a natural map f n +1 : Y → Ω n +1 . Clearly taking a sufficiently finetriangulation of Ω n we may assume that f n and ω n +1 ◦ f n +1 are as close as we wish. Notethat every point of every Ω i has a closed neighborhood homeomorphic to a product of afinite tree with a closed interval. Then, by Proposition 2.3, we can replace f n +1 by a mapwhich is 0-dimensional on Y ′ . Recall that dim Y \ Y ′ ≤ m − f n +1 is m -dimensional. Then it is easy to see that the whole construction can be carriedout so that the maps f n will determine an m -dimensional map from Y to Ω = lim ← Ω n . (cid:4) Lemma 2.4
Let f : ˜Ω( p k ) → Ω( p k ) be the Z p k -covering of Ω( p k ) induced by the Z p k -covering of the top of Ω( p k ) . Then f restricted to the preimage of the bottom of Ω( p k ) extends over ˜Ω( p k ) as a map to the bottom of Ω( p k ) . roof. Represent ˜Ω( p k ) as the union of p k copies of S × [0 ,
1] being glued along S × { } such that under f the set S × { } goes to the top of Ω( p k ) and the sets S × { } go to thebottom of Ω( p k ). Consider a map f from S × { } to the bottom of Ω( p k ) such that f followed by the natural projection of Ω( p k ) to its top coincides with f on S × { } . Thenfor each set S × [0 ,
1] the maps f = f restricted to S × { } and f are homotopic asmaps to the bottom of Ω( p k ) and hence extend over S × [0 ,
1] to a map to the bottom ofΩ( p k ). This defines a map required in the proposition. (cid:4) Proposition 2.5
Every Kolmogorov-Pontrjagin surface is p -resolvable by a -dimensionalcompactum. Proof.
Let Ω = lim ← Ω n be a Kolmogorov-Pontrjagin surface determined by a sequence k n .Consider the first homology H (Ω n ; Z ) and let C i , i ≥ i , i ≤ n . Then the circles in C ∪ · · · ∪ C n considered as elements of H (Ω n ; Z ) form a collection of free generators of H (Ω n ; Z ).Take any sequence s n such that s = k and s n +1 − k n +1 ≥ s n for every n ≥ G n of H (Ω n ; Z ) by G n = { n X i =1 X α ∈ C i t α α : n X i =1 X α ∈ C i p s i − k i t α is divisible by p s n } . Then H (Ω n ; Z ) /G n = Z p sn and for the bonding map ω n +1 : Ω n +1 → Ω n we have that thatthe induced homomorphism ( ω n +1 ) ∗ : H (Ω n +1 ; Z ) → H (Ω n ; Z ) is onto and sends G n +1 into G n . Indeed ( ω n +1 ) ∗ ( P n +1 i =1 P α ∈ C i t α α ) = P ni =1 P α ∈ C i t α α and if P n +1 i =1 P α ∈ C i p s i − k i t α = p s n +1 − k n +1 ( P α ∈ C n +1 t α ) + P ni =1 P α ∈ C i p s i − k i t α is divisible by p s n +1 with s n +1 − k n +1 ≥ s n then we have that P ni =1 P α ∈ C i p s i t α is divisible by s n and therefore ( ω n +1 ) ∗ ( G n +1 ) ⊂ G n .Let h n : π (Ω n ) → H (Ω n ; Z ) be the Hurewicz homomorphism. Then˜ G n = π (Ω n ) /h − n ( G n ) = H (Ω n ; Z ) /G n = Z p sn . Consider the corresponding ˜ G n -covering f n : ˜Ω n → Ω n and lift the map ω n +1 : Ω n +1 → Ω n to ˜ ω n +1 : ˜Ω n +1 → ˜Ω n . The homomorphism ( ω n +1 ) ∗ induces the natural ephimorpism˜ g n +1 : ˜ G n +1 → ˜ G n so that the actions of ˜ G n +1 and ˜ G n agree with ˜ g n +1 and ˜ ω n +1 . Thisdefines the compactum X = lim ← ( ˜Ω n , ˜ ω n ), the action of A p = lim ← ( ˜ G n , ˜ g n ) on X and the map f : X → Ω determined by the maps f n .Note that if we consider α ∈ C n as a circle in Ω n then the preimage of α under themap f n splits into p s n − k n components (circles) and f n restricted to each component is a Z p kn -covering of α .Let us show that dim X ≤
1. Consider the triangulation of Ω n used for constructingΩ n +1 and a simplex ∆ of this triangulation. Let Ω( p k n +1 ) = ω − n +1 (∆) and ˜Ω n +1 ( p k n +1 ) acomponent of f − n +1 (Ω( p k n +1 )). Note that f n +1 restricted to ˜Ω( p k n +1 ) is a Z p kn +1 -covering ofΩ( p k n +1 ). Apply Lemma 2.4 to extend the map f n +1 restricted to the preimage in ˜Ω( p k n +1 )of the bottom ∂ ∆ of Ω( p k n +1 ) to a map φ : ˜Ω( p k n +1 ) → ∂ ∆. Now consider the triangulationof ˜Ω n induced by the triangulation of Ω n . The map φ lifts to ˜ φ : ˜Ω n +1 ( p k n +1 ) → ˜Ω n so that˜ φ ( ˜Ω n +1 ( p k n +1 )) ⊂ ˜ ω n +1 ( ˜Ω n +1 ( p k n +1 )) . Doing that for every simplex of Ω n we get a mapfrom ˜Ω n +1 to the 1-skeleton of ˜Ω n and this shows that dim X ≤ (cid:4) roof of Theorem 1.3. (i) Let dim Y = m + 2. By Proposition 2.1 there is an m -dimensional map f : Y → Ωto a Kolmogorov-Pontrjagin surface Ω. By Proposition 2.5 there is a free action of A p ona 1-dimensional compactum ˜Ω such that ˜Ω /A p = Ω. Let X be the pull-back of f and theprojection of ˜Ω to Ω. Then the projection of X to ˜Ω is an m -dimensional map and hence,by Proposition 2.2 dim X ≤ m + 1 and for the pull-back action of A p on X we have that X/A p = Y .(ii) Assume that Y is p -resolvable by a finite dimensional compactum X . Then, byYang’s relations dim Z [ p ] X = 1 and dim Z p ∞ X ≥ dim Z p ∞ Y −
1. Bokstein inequalities anddim Y ≥ Q X = dim Q Y = 1 and dim Z p ∞ Y = dim Y − ≥ Z p ∞ X ≥ X = dim Z p ∞ X + 1 ≥ dim Y − (cid:4) A bundle will always mean a locally trivial principal bundle. The circle S is consideredas the group S = U (1) = R / Z . Recall that the classifying space for circle bundles overcompacta is BS = CP ∞ = K ( Z ,
2) and therefore every circle bundle Z → Y over acompactum Y is represented by an element α of the second Cech cohomology H ( Y ; Z )that being considered as a map α : Y → K ( Z ,
2) allows to obtain the bundle X → Y asthe pull-back of the universal circle bundle E → K ( Z ,
2) and the map α .Let Y be a compactum and f : Z → Y a circle bundle over Y . Consider the subgroup Z m of S and in each fiber of f collapse the orbits under the action of Z m in S to singletons.This way we obtain the circle bundle f ′ : Z ′ → Y determined by the action of S = S / Z m on Z ′ . We will refer to f ′ : Z ′ → Y as the circle bundle induced by f : Z → Y and Z m . Lemma 3.1
Let f : Z → Y be a circle bundle over a compactum Y represented by α ∈ H ( Y ; Z ) and let f ′ : Z ′ → Y be the circle bundle induced by f and Z m . Then f ′ isrepresented by α ′ = mα ∈ H ( Y ; Z ) . Proof.
Consider the short exact sequence0 → Z m → S → S → . It defines a fiber sequence B Z m → BS → BS with the long exact sequence of the fibration . . . → π ( B Z m ) → π ( BS ) → π ( BS ) → π ( B Z m ) → π ( BS ) → . . . Recall that B Z m = K ( Z m ,
1) and BS = K ( Z ,
2) and get0 → Z → Z → Z m → . Thus the homomorphism S → S induces the map h : BS → BS which acts onthe second homotopy group of BS as the multiplication by m . Represent α as a map7 : Y → BS = K ( Z , α ′ is represented by h ◦ α that translates in H ( Y ; Z ) to α ′ = mα . (cid:4) Let A p freely act on a finite dimensional compactum X . Consider A p as a subgroup the p -adic solenoid Σ p and the induced action of A p on X × Σ p . Then Σ p naturally acts on X × A p Σ p = ( X × Σ p ) /A p with ( X × A p Σ p ) / Σ p = X/A p and there is a natural projectionof X × A p Σ p to S = Σ p /A p induced by the projections X × Σ p → Σ p → Σ p /A p . Note thatthe fibers of the projection X × A p Σ p → S = Σ p /A p are homeomorphic to X and hence,by Theorem 2.2, dim X × A p Σ p ≤ dim X + 1.Consider a decreasing sequence of subgroups A kp of A p such that A kp is isomorphic to A p , A p = A p , A kp /A k +1 p = Z p and the intersection of all A kp contains only 0. Denote Y = X/A p and Z n = ( X × A p Σ p ) /A kp . Note that the circle S = Σ p /A kp acts on Z k with Y = Z k /S .This turns each Z k into a circle bundle f k : Z k → Y over Y and the inclusion of A k +1 p into A kp defines the natural bundle map g k +1 : Z k +1 → Z k that witnesses that the bundle f k : Z k → Y is induced by the bundle f k +1 : Z k +1 → Y and Z p . Moreover, the bundle f : Z → Y is trivial since ( X × A p Σ p ) /A p = ( X/A p ) × (Σ p /A p ) = Y × S . Also note that Z = X × A p Σ p = lim ← ( Z k , g k ) and the projection f : Z = X × A p Σ p → Y = ( X × A p Σ p ) / Σ p coincides with the projection of Z to Z k followed by f k . Proof of Theorem 1.6 . Let A p acts freely on an n -dimensional compactum X with Y = X/A p of dim ≥ n + 2. Consider the circle bundles f k : Z k → Y described above andlet α k ∈ H ( Y ; Z ) represent f k . By Lemma 3.1, α k = pα k +1 . Recall that f : Z → Y is a trivial bundle and hence α = 0. Denote by H the subgroup of H ( Y ; Z ) generatedby α k , k = 0 , , . . . . We are going to show that H is isomorphic to Z p ∞ and satisfy theconclusions of the theorem.Let a compactum A ⊂ Y be of dim ≥ n +2. Aiming at a contradiction assume the imageof H under the inclusion of A into Y is trivial in H ( A ; Z ). Then every bundle f k : Z k → Y is trivial over A . Let ǫ > A does not admit an open cover of order ≤ n + 2by sets of diameter ≤ ǫ . Recall that dim Z ≤ n + 1. Then, since Z = lim ← ( Z k , g k ), there is asufficiently large k so that Z k admits an open cover U of order ≤ n + 2 so that the imagesof the sets in U under f k are of diameter ≤ ǫ . Take a section φ : A → Z k over A . Then U restricted to φ ( A ) and mapped by f k to A provides an open cover of A of order ≤ n + 2 bysets of diameter ≤ ǫ . Contradiction.Now assuming that A = Y we get that H is non-trivial and, since α k = pα k +1 and α = 0, we deduce that H is isomorphic to Z p ∞ . (cid:4) Remark 3.2
In general the assumption dim Y ≥ n + 2 in Theorem 1.6 cannot be weakened todim Y ≥ n + 1. Indeed, it is easy to see that a Kolmogorov-Pontrjagin surface Ω has H (Ω; Z ) = 0 and, as it was shown in the previous section, Ω is p -resolvable by a 1-dimensional compactum. In this connection we would like to point out an iteresting phe-nomenon that occurs in the proof of Theorem 1.6 for Y = Ω: all the compacta Z k arehomeomorphic to Y × S and the bonding maps g k +1 : Z k +1 → Z k look the same, andstill we get that Z = lim ← Z k is 2-dimensional despite that each Z k is 3-dimensional. This8appens because we cannot fix a trivialization Z k = Y × S for each Z k so that with respectto these trivializations each g k +1 will have the form g k +1 ( y, s ) = ( y, ps ) , ( y, s ) ∈ Y × S .However, there is a special case of Theorem 1.6 that admits a strengthening. Namely,one can show that if the compactum Y in Theorem 1.6 is p -resolvable by an n -dimensionalmanifold then the assumption dim A ≥ n + 2 can be weakened to dim A ≥ n + 1. Let P denote the set of all primes. The Bockstein basis is the collection of groups σ = { Q , Z p , Z p ∞ , Z ( p ) | p ∈ P} where Z p = Z /p Z is the p -cyclic group, Z p ∞ = dirlim Z p k isthe p -adic circle, and Z ( p ) = { m/n | n is not divisible by p } ⊂ Q is the p -localization ofintegers.By a Moore space M ( G, n ) for a group in the Bockstein basis σ we always mean thestandard Moore space. All CW-complexes are assumed to be countable and all the spacesare assumed to be separable metrizable. A compactum X is said to be hereditarily infinitedimensional if every closed subset of X is either 0-dimensional or infinite dimensional.(1) (Dranishnikov’s first extension criterion [3]) Let X be a compactum and K a CW-complex such that ext-dim X ≤ K . Then dim H n ( K ) X ≤ n for every n ≥ H n ( K ) is the reduced homology of K with the coefficients in Z .(2) (Dranishnikov’s second extension criterion [3]) Let K be a simply connected CW-complex, X a finite dimensional compactum with dim H n ( K ) X ≤ n for every n thenext-dim X ≤ K .(3) (Dranishnikov’s splitting theorem [4, 5]) Let X be a space and K and K CW-complexes such that ext-dim X ≤ K ∗ K . Then X splits into X = A ∪ A withext-dim A ≤ K , ext-dim A ≤ K and A being F σ in X .(4) [9] Let X be a compactum with dim Q X ≤ n . Then ext-dim X ≤ M ( Q , n ).(5) (A factorization theorem that was actually proved in [10]) Let σ ′ be a subcollection ofthe Bockstein basis σ and X a compactum (not necessarily finite dimensional) suchthat ext-dim X ≤ M ( G, n G ) for every G ∈ σ ′ . Then every map f : X → K to afinite CW-complex K can be arbitrarily closely approximated by a map f ′ : X → K such that f ′ factors through a compactum Z with dim Z ≤ dim K and ext-dim Z ≤ M ( G, n G ) for every G ∈ σ ′ .(6) It follows from the results of Ancel [1] and Pol [12] (see also [8]) on C -spaces that everyinfinite dimensional compactum of finite integral cohomological dimension containsa hereditarily infinite dimensional compactum.(7) [15] Let A p act on an n -dimensional compactum X . Then dim Z X/A p ≤ n + 3. Proposition 4.1
A compactum X is finite dimensional if and only if there is naturalnumber n such that ext-dim X ≤ M ( Q , n ) and ext-dim X ≤ M ( Z p , n ) for every prime p . roof. If dim X = k > G ≤ k for every group G and, by (2), ext-dim X ≤ M ( G, n ) for n = k + 1 and every G ∈ σ .Now assume that ext-dim X ≤ M ( Q , n ) and ext-dim M ( Z p , n ) for every prime p . Fix ǫ > ǫ -map (a map with fibers of diameter < ǫ ) f : X → K to a finitedimensional cube K . By (5) f factors through a (2 ǫ )-map g : X → Z with Z being finitedimensional with ext-dim Z ≤ M ( Z p , n ) for every prime p and ext-dim Z ≤ M ( Q , n ). By(1) dim Z p Z ≤ n for every prime p and dim Q Z ≤ n and hence, by Bockstein inequali-ties, dim Z Z ≤ n + 1 and since Z is finite dimensional dim Z = dim Z Z ≤ n + 1. Thus forevery ǫ > X admits a (2 ǫ )-map a compactum of dim ≤ n +1 and hence dim X ≤ n +1. (cid:4) Proof of Theorem 1.2.
Let dim X = n and let A ⊂ Y be the subset of Y corre-sponding to all the finite orbits. Then dim A ≤ n . Replace A by a larger a G δ -subset of Y with dim A ≤ n . Since Y is infinite dimensional, we have that Y \ A is infinite dimensional,and since it is also σ -compact, there is an infinite dimensional compactum in Y \ A . Thusreplacing Y and X by this compactum and its preimage we may assume that the action of A p is free.By (7) and (6) we may also assume that Y is hereditarily infinite dimensional. Letus show that Y contains an infinite dimensional compactum of dim Z [ p ] = 1. Aiming at acontradiction assume that every compactum Y ′ in Y with dim Y ′ > Z [ p ] > Z [ q ] Y ′ > q = p because otherwise H ( Y ′ ; Z [ q ]) = 0 and hence,by the universal coefficients theorem, H ( Y ′ ; Z ) is q -torsion and cannot contain a copy of Z p ∞ that violets Theorem 1.6.Note that for every prime q we have that the join M ( Z [ q ] , ∗ M ( Z q , n +3) is contractibleand hence ext-dim Y ≤ M ( Z [ q ] , ∗ M ( Z q , n + 3). Then, by (3), Y = Y q ∪ Y q withext-dim Y q ≤ M ( Z [ q ] , Y q ≤ M ( Z q , n + 3) and Y q being σ -compact. By (1),dim Z [ q ] Y q ≤ Z q Y q ≤ n + 3. Since any compactum in Y of positive dimension isof dim Z [ q ] > Y q ≤ Y the union of Y q for all prime q and let Y = Y \ Y . Then dim Y ≤ Y ≤ M ( Z q , n + 3) for every prime q . Enlarging Y to a 0-dimensional G δ -subsetof Y we get that Y \ Y ⊂ Y is σ -compact and infinite dimensional and hence contains aninfinite dimensional compactum Y ′ in Y . By (7), dim Z Y ≤ n + 3 and hence dim Q Y ′ ≤ n + 3. Then, by (4), ext-dim Y ′ ≤ M ( Q , n + 3). Recall that ext-dim Y ′ ≤ M ( Z q , n + 3)for every prime q . Thus, by 4.1, Y ′ is finite dimensional, and we arrive at a contradiction.This shows that Y contains an infinite dimensional compactum with dim Z [ p ] ≤ (cid:4) A map between simplicial or CW-complexes is said to be combinatorial if the preimage ofevery subcomplex of the range is a subcomplex of the domain.10 .1 Extending partial maps
Let M be a finite simplicial complex, K a connected CW-complex and f : A → K a cellularmap from a subcomplex A of M . We will show how to construct a CW-complex M ′ anda map µ : M ′ → M such that µ restricted to A ′ = µ − ( A ) and followed by f extends to amap f ′ : M ′ → K .Let A i = A ∪ M ( i ) be the union of A with the i -skeleton M ( i ) of M . Extend f over A to a cellular map f : A → K and set M = A with µ being the inclusion. Assume thatwe already constructed a CW-complex M i , cellular maps µ i : M i → A i and f i : M i → K so that f i +1 extends f i , M i ⊂ M i +1 and µ i +1 | M i = µ i for all the relevant indices up to n .The CW-complex M n +1 is obtained from M n by the following procedure. If A n alreadycontains M ( n +1) set M n +1 = M n , f n +1 = f n and µ n +1 = µ n . Otherwise, for every ( n + 1)-simplex ∆ not contained in A n attach to M ∆ n = µ − n ( ∂ ∆) the mapping cylinder of f n | M ∆ n .The map f n naturally extends over each mapping cylinder defining f n +1 : M n +1 → K .Extend the map µ n to µ n +1 by sending each mapping cylinder to the corresponding simplex∆ so that the top ( K -level) of the mapping goes to the barycenter of ∆ and µ n +1 is linearon the intervals of the mapping cylinder.Finally for i = dim M denote M ′ = M i , µ = µ i and f ′ = f i . Note that M ′ admits atriangulation for which µ is combinatorial provided there are simplicial structures on M and K for which f is simplicial. Proposition 5.1
Assume that in the above construction K = S , the subcomplex A con-tains the -skeleton of M and the map f : A → S if of degree p k (that is f lifts to a Z p k -covering of S ).Let N be a subcomplex of M , m = dim N and N ′ = µ − ( N ) . Consider the homomor-phisms ( ∗ ) H m ( N ′ ; Z p t ) → H m ( N ; Z p t )( ∗∗ ) H m ( N ; Z p ) → H m ( N ; Z p t ) → H m ( M ; Z p t )( ∗ ∗ ∗ ) H m ( N ′ ; Z p ) → H m ( N ′ ; Z p t ) → H m ( M ′ ; Z p t ) induced by the map µ , the mononorphism Z p → Z p t and the inclusions of N into M and N ′ into M ′ respectively. Then for t ≤ k we have that(1) the homomorphism in (*) is an isomorphism and(2) if the composition of the homomorphisms in (**) is trivial then the composition ofthe homomorphisms in (***) is trivial as well. Proof.
Note that for µ is 1-to-1 over the 1-skeleton of M and for very 2-simplex ∆ of M ,we have that µ is either 1-to-1 over ∆ or µ − (∆) is the mapping cylinder of f restricted ∂ ∆ attached to ∂ ∆. Then, since f is of degree p k , one can easily see that (1) holds fordim N ≤ N > L be the union of all the simplexes of dim ≤ m − M . Note that dim µ − ( L ) ≤ m −
2. Collapse thefibers of µ over L ∩ N and denote by N ∗ the CW-complex obtained this way from N ′ andlet µ ′ : N ′ → N ∗ and µ ∗ : N ∗ → N be the induced natural maps.11et N − = N ( m − and N ′− = µ − ( N − ). Note that µ ∗ is 1-to-1 over N − and hencewe can identify ( µ ∗ ) − ( N ′− ) with N − . Then, since dim µ − ( L ) ≤ m −
2, we get thatthe map µ ′ induces isomorphisms H m ( N ′ ; Z p t ) → H m ( N ∗ ; Z p t ) and H m ( N ′ , N ′− ; Z p t ) → H m ( N ∗ , N − ; Z p t ). Moreover, it follows from Construction 5.1 that, since t ≤ k , for ev-ery m -simplex ∆ of N (with respect to the origimal triangulation of M ), we have that H m (( µ ∗ ) − (∆) , ( µ ∗ ) − ( ∂ ∆); Z p t ) = Z p t and µ ∗ induces an isomorphism H m (( µ ∗ ) − (∆) , ( µ ∗ ) − ( ∂ ∆); Z p t ) → H m (∆ , ∂ ∆; Z p t )(everything here can be easily visualized for m = 3). Then µ ∗ also induces an isomorphism H m ( N ∗ , N − ; Z p t ) → H m ( N, N − ; Z p t ). Now apply the long exact sequences for the pairs( N, N − ) and ( N ∗ , N − ) and the 5-lemma to show, by induction om m , that (1) holds. Notethat we also showed that( † ) µ induces an isomorphism H m ( N ′ , N ′− ; Z p t ) → H m ( N, N − ; Z p t ).Let us turn to (2). Clearly we can replace M by its ( m + 1)-skeleton and assume thatdim M = m + 1. Let M − be the m -skeleton of M and M ′− = µ − ( M − ). By (1) and ( † ) weget that µ induces isomorphisms H m ( M ′− ; Z p t ) → H m ( M − ; Z p t ) and H m +1 ( M ′ , M ′− ; Z p t ) → H m +1 ( M, M − ; Z p t ). Then applying the long exact sequences of the pairs ( M ′ , M ′− ) and( M, M − ) and the 5-lemma we get that the homomorphism H m ( M ′ ; Z p t ) → H m ( M ; Z p t ) isinjective on the image of H m ( M ′− ; Z p t ) in H m ( M ′ ; Z p t ) under the homomorphism inducedby the inclusion of M ′− into M ′ . Since dim M ′− = dim N ′ = m we have that the homomor-phism H m ( N ′ ; Z p t ) → H m ( M ′− ; Z p t ) induced by the inclusion of N ′ into M ′− is injective.Combining all this together we get that (2) holds. (cid:4) Proposition 5.2
Assume that in Construction 5.1 the subcomplex A contains the m -skeleton of M and K is a Moore space M ( Z p , m ) .Let N be a subcomplex of M with dim N ≤ m and N ′ = µ − ( N ) . Consider the homo-morphisms ( ∗ ) H m ( N ′ ; Z p t ) → H m ( N ; Z p t )( ∗∗ ) H m ( N ; Z p ) → H m ( N ; Z p t ) → H m ( M ; Z p t )( ∗ ∗ ∗ ) H m ( N ′ ; Z p ) → H m ( N ′ ; Z p t +1 ) → H m ( M ′ ; Z p t +1 ) induced by the mononorphisms Z p → Z p t and Z p → Z p t +1 and the inclusions of N into M and N ′ into M ′ respectively. Then for every t > we have that(1) the homomorphism in (*) is an isomorphism and(2) if the composition of the homomorphisms in (**) is trivial then the composition ofthe homomorphisms in (***) is trivial as well. Proof.
Note that µ is 1-to-1 over the m -skeleton of M and hence we can identify the m -skeleton M ( m ) of M with µ − ( M ( m ) ) and assume that N = N ′ . In particular it implies(1). We also can replace M by the ( m + 1)-skeleton of M and assume that dim M ≤ m + 1.12onsider α ∈ H m ( N ; Z p ) and let β ∈ H m ( N ; Z p t ) be the image of α and γ ∈ H m ( N ; Z p t +1 )the image of β under the homomorphisms H m ( N ; Z p ) → H m ( N ; Z p t ) and H m ( N ; Z p t ) → H m ( N ; Z p t +1 ) induced by the monomorphisms Z p → Z p t and Z p t → Z p t +1 respectively.Consider β as an m -cycle in the chain complex C ( N ; Z p t ). Then γ is represented bythe cycle pβ in the chain complex C ( N ; Z p t +1 ) Since the composition (**) is trivial, thecycle β is homologous to 0 in C ( M ; Z p t ) and hence β = ∂θ where θ = c ∆ i + · · · + c k ∆ k , c i ∈ Z p t and ∆ i are ( m + 1)-simplexes in M . Thus γ = p∂θ = c ( p∂ ∆ ) + · · · + c k ( p∂ ∆ k )in C ( N ; Z p t +1 ). Recall that for every ( m + 1)-simplex ∆, µ − (∆) is either contractibleor homotopy equivalent to the mapping cylinder of f | ∂ ∆ → M ( Z p , m ) and hence p∂ ∆ ishomologous to 0 in C ( M ′ ; Z p t +1 ). Thus we get that γ is homologous to 0 in C ( M ′ ; Z p t +1 )and hence the composition (***) is trivial. (cid:4) Let M be a finite simplicial complex and g : L → M a circle bundle. We will say that g is p -flexible if for every k > f : M (1) → L over the 1-skeleton of M such that for a every 2-simplex ∆ of M and a trivialization g − (∆) = ∆ × S we havethat the map f restricted to ∂ ∆ and followed by the projection ∆ × S → S lifts to the Z p k -covering of S . We will refer to p k as a degree of f . Clearly this definition does notdepend on the trivializations over 2-simplexes of M .We will describe two versions of killing non-trivial p -flexible bundles that will be usedin different contexts. Let g : L → M be a p -flexible circle bundle over a finite simplicial complex M . Let f : M (1) → L be a section of degree p k over the 1-skeleton of M . We will construct aCW-complex M ′ and a map µ : M ′ → M so that the pull-back bundle g ′ : L ′ → M ′ of f via µ admits a section f ′ : M ′ → L ′ .Set M = M (1) , µ : M → M to be the inclusion and f = f : M → L . We willconstruct by induction finite CW-complexes M i , maps µ i : M i → M ( i ) and f i : M i → L sothat g ◦ f i = µ i .Assume that the construction is completed for i ≤ n and proceed to n + 1 as fol-lows. Consider ( n + 1)-simplex ∆ of M consider a trivialization g − (∆) = ∆ × S and let f ∆ : µ − n ( ∂ ∆) → S be the map f n restricted to µ − n ( ∂ ∆) and followed by the projection g − (∆) = ∆ × S → S .Consider the mapping cylinder M ∆ of f ∆ and let µ ∆ : M ∆ → ∆ be the map sending thetop of M ∆ to the barycenter of ∆ and linearly extending µ n restricted to µ − n ( ∂ ∆). Thenthe map f n restricted µ − n ( ∂ ∆) naturally extends over M ∆ to a map f ∆ : M ∆ → g − (∆)such that µ ∆ = g ◦ f ∆ . Now attach M ∆ to µ − i ( ∂ ∆) and doing that for every ( n + 1)-simplex ∆ of M we obtain the CW-complex M n +1 and the maps µ n +1 : M n +1 → M ( n +1) and f n +1 : M n +1 → L induced by the maps µ ∆ and f ∆ respectively. Clearly g ◦ f n +1 = µ n +1 .Finally for i = dim M set M ′ = M i , µ ′ = µ i , L ′ = the pull-back of L under µ ′ , g and g ′ : L ′ → M ′ the pull-back of g . Then f i induces a section f ′ : M ′ → L ′ . Note that M ′ admits a triangulation for which µ is combinatorial.13 roposition 5.3 The conclusions of Proposition 5.1 hold for the construction above.
The proof of Proposition 5.1 applies to prove Proposition 5.3 as well.
Let g : L → M be a p -flexible circle bundle over a finite simplicial complex M . Let m = dim M and k = tm , and let f : M (1) → L be a section of degree p km over the 1-skeleton of M . We will construct a CW-complex M ′ and a map µ : M ′ → M so that thepull-back bundle g ′ : L ′ → M ′ of f via µ admits a section f ′ : M ′ → L ′ .Set M = M (1) , µ : M → M to be the inclusion and f = f : M → L . We willconstruct by induction finite CW-complexes M i , maps µ i : M i → M ( i ) and f i : M i → L sothat g ◦ f i = µ i and g i is a map of degree p ( m − i ) k . By this we mean that for every ( i + 1)-simplex ∆ of M and a trivialization f − (∆) = ∆ × S of g over ∆ the map f i restricted to µ − i ( ∂ ∆) and followed by the projection ∆ × S → S lifts to the Z p ( m − i ) k -covering of S .Assume that the construction is completed for i ≤ n and proceed to n + 1 as follows.Consider ( n + 1)-simplex ∆ of M consider a trivialization g − (∆) = ∆ × S and a map f ∆ : µ − n ( ∂ ∆) → S witnessing that f n restricted to µ − i ( ∂ ∆) is a map of degree p ( m − n ) k andconsider the finite telescope T of m maps S → S each of them is the Z p t -covering of S .the mapping cylinder M ∆ of f ∆ followed by the embedding of S in T as the first circle of T .Let µ ∆ : M ∆ → ∆ be the map sending the top ( T -level) of M ∆ to the barycenter of ∆ andlinearly extending µ n restricted to µ − n ( ∂ ∆). Then the map f n restricted µ − n ( ∂ ∆) naturallyextends over M ∆ to a map f ∆ : M ∆ → g − (∆) of degree p ( m − n − k such that µ ∆ = g ◦ f ∆ .Now attach M ∆ to µ − i ( ∂ ∆) and doing that for every ( n + 1)-simplex ∆ of M we obtain theCW-complex M n +1 and the maps µ n +1 : M n +1 → M ( n +1) and f n +1 : M n +1 → L induced bythe maps µ ∆ and f ∆ respectively. Clearly µ n +1 = g ◦ f n +1 .The only thing that we need to check is that that f n +1 is of degree p ( m − n − k on µ − n +1 (∆)for every ( n + 2)-simplex ∆ of M .The only thing that we need to check is that that f n +1 restricted to µ − n +1 ( ∂ ∆) is of degree p ( m − n − k for every ( n + 2)-simplex ∆ of M . Consider a trivialization g − (∆) = ∆ × S of g − (∆) and denote by ∆ , . . . , ∆ n +1 the ( n + 1)-simplexes contained in ∆. Note thatfor µ − n +1 ((∆ ∪ · · · ∪ ∆ i ) ∩ ∆ i +1 ) is connected for every 0 ≤ i ≤ n . Then the cycles of H ( µ − n +1 (∆ i )), 0 ≤ i ≤ n + 1 generate H ( µ − n +1 ( ∂ ∆)). Since f n +1 restricted to each ∆ i andfollowed by the projection α : ∆ × S → S is of degree p k we get that the homomorphisminduced by α ◦ f n +1 sends H ( µ − n +1 ( ∂ ∆)) into the elements of H ( S ) = Z divisible by p ( m − n − k . This means that f n +1 restricted to µ − n +1 ( ∂ ∆) is of degree p ( m − n − k .Finally set m = dim M , M ′ = M m , µ ′ = µ m , L ′ = the pull-back of L under µ ′ , g and g ′ : L ′ → M ′ the pull-back of g . Then f m induces a section f ′ : M ′ → L ′ . Note that M ′ admits a triangulation for which µ is combinatorial.Let T j , ≤ j ≤ m be the subtelescope of T consisting of the first j maps S → S . Thus T m = T and in the construction above we can consider the subcomplexes M j ⊂ M ′ = M m obtained from M ′ by leaving only the subtelescope T j from T in all the mapping cylinders M ∆ . Proposition 5.4
Let N be a connected CW-complex whose homotopy groups are finite p -torsion groups. Consider Construction 5.2.2 with t being such such that p t π n ( N ) = 0 for very ≤ n ≤ m and assume that f N : A N → N is a map from a subcomplex A N of M such that f N does not extend over M . Then µ restricted to µ − ( A N ) and followed by f N does not extend over M ′ . Proof.
Let M ∗ by the space obtained from M ′ by collapsing the fibers of µ over N tosingletons, µ ∗ : M ∗ → M the map induced by µ and M j ∗ = µ ∗ ( M j ) , ≤ j ≤ m Recallthat M m = M ′ and hence M m ∗ = M ∗ . Thus we can consider N as a subcomplex of M ∗ and aiming at a contradiction assume that f N extends to f ∗ : M ∗ → N . Recall that inConstruction 5.2.2 we denote by T the telescope of m copies of Z p t -coverings S → S andby T j , 1 ≤ j ≤ m the subtelescope of the first j maps of T ( T m = T ).Consider the first barycentric subdivision βM of the triangulation of M . Note that forever 0-dimensional simplex (vertex) ∆ of βM , ( µ ∗ ) − (∆) is either a singleton or homeo-morphic to T = T m . Then ( µ ∗ ) − (∆) ∩ M m − ∗ is either a singleton or homeomorphic to thesubtelescope T m − of T and hence f ∗ is null-homotopic on ( µ ∗ ) − (∆) ∩ M m − ∗ . Thus we canreplace M m − ∗ by the space M m − obtained from M m − ∗ by collapsing ( µ ∗ ) − (∆) ∩ M m − ∗ to singletons for every 0-simplex ∆ of βM , and µ ∗ and f ∗ by the induced maps µ m − and f m − to M and N respectively and assume that µ m − is 1-to-1 over the 0-simplexes of βM .Now consider a 1-simplex ∆ of βM . Then ( µ m − ) − (∆) is either contractible or homo-topy equivalent to Σ T m − . Then ( µ m − ) − (∆) ∩ M m − is either a singleton or home-omorphic to the subtelescope Σ T m − of Σ T m − and hence f m − is null-homotopic on( µ m − ) − (∆) ∩ M m − . Thus f m − restricted to M m − factors up to homotopy throughthe space obtained from M m − by collapsing the fibers of µ m − | M m − over the simplex ∆.Doing that consecutively for all the 1-simplexes of βM we obtain the space M m − and themaps µ m − : M m − → M and f m − : M m − → N induced by µ m − and f m − respectivelysuch that µ m − is 1-to-1 over the 1-simplexes of βM .Procced by induction and construct for every i ≤ m − M m − i − i and the maps µ m − i − i : M m − i − i → M and f m − i − i : M m − i − i → N and finally get for i = m − M − M m − = M and f m − extends f N that contradicts the assumptions of the proposition. (cid:4) We will describe how to construct an inverse limit Y = lim ← ( M i , µ i ) of finite simplicialcomplexes M i performing countably many procedures on certain objects determined byeach M i . We will mainly deal with the objects described in Section 5 like partial maps andand circle bundles and the procedures like extending partial partial maps and killing somenon-trivial circle bundles. We assume that any object on M j can be transferred to any M i with i ≥ j via the map µ ji = µ i ◦ · · · ◦ µ j +1 : M i → M j ( µ ii is the identity map of M i ). Bytransferring a partial map on M j to M i we just mean that a partial map f : A → K from aclosed subset A of M j to to CW-complex K moves to the partial map f ◦ µ ji : ( µ ji ) − ( A ) → K on M i . And by transferring a circle bundle over M j to M i we mean the pull-back of thebundle to M i via the map µ ji . We also assume for each finite simplicial complex we can fixcountably many objects on which we want to perform appropriate procedures.The inverse system ( Y i , µ i ) is constructed as follows. Consider a bijection β : N × N → N such that β ( j, n ) ≥ j . Assume that Y j and µ j are already constructed for j ≤ i . Moreover15or each j ≤ i we also fixed countably many objects Q j for Y j that we need to take careof and the objects in Q j are indexed by natural numbers. Proceed to i + 1 as follows.Let ( j, n ) = i . Transfer the object indexed by n in Q j to M i , construct M i +1 and µ i +1 inorder to perform the procedure appropriate for this object. And, finally, pick out countablymany objects Q i +1 for Y i +1 needed to be taken care of and index the objects in Q i +1 bythe natural numbers.Thus we get in the inverse limit determining Y we took care of all the objects pickedout for each M i . Theorem 1.4 follows form Theorem 1.6 and the case n = 1 of the following proposition. Proposition 6.1
For every n ≥ there is a compactum Y with dim Z [ p ] Y = 1 , dim Z p Y = n + 1 and dim Y = n + 2 such that H ( Y ; Z ) does not contain a subgroup isomorphic to Z p ∞ . Proof.
We will construct Y as the inverse limit of ( n + 2)-dimensional finite simplicialcomplexes M i and combinatorial bonding maps µ i +1 : M i +1 → M i . In order to show that Y has the required properties we consider for each i a subcomplex A i of M i such that A i +1 = µ − i +1 ( A i ) and two natural numbers k i and t i such that k i +1 ≥ k i , t i +1 ≥ t i and k i ≥ t i . Set M to be an ( n + 2)-ball, A = S n +1 the boundary of this ball, and k = t = 1.We require that(1) dim A i = n + 1, H n +1 ( A i ; Z p ) = Z p and the homomorphism H n +1 ( A i +1 ; Z p ) → H n +1 ( A i ; Z p ) induced by µ i +1 is an isomorphism;(2) Let H n +1 ( A i ; Z p ) → H n +1 ( A i ; Z p ti ) and H n +1 ( A i ; Z p ti ) → H n +1 ( M i ; Z p ti ) be the ho-momorphisms induced by the monomorphism Z p → Z p t and the inclusion of A i into M i respectively. Then the following composition is trivial H n +1 ( A i ; Z p ) → H n +1 ( A i ; Z p ti ) → H n +1 ( M i ; Z p t ) . Clearly the relevant parts of (1) and (2) hold for i = 0. Assuming that the construction iscompleted for i and we proceed to i + 1 performing one the following procedures. Procedure I (taking care of non-flexible bundles).
Let g : L → M i be a circlebundle which is not p -flexible. Then take any natural number k such that there is nosection over M (1) i of degree p k . Set M i +1 = M i , µ i +1 =the identity map, k i +1 = max { k, k i } and t i +1 = t i . Procedure II (taking care of flexible bundles).
Let g : L → M i be a flexiblebundle. Apply 5.2.1 with M = M i and k = k i construct M i +1 = M ′ and µ i +1 = µ . Set k i +1 = k i and t i +1 = t i . 16 rocedure III (extending partial maps to a circle). Let A be a subcomplex of M i and f : A → S a map. Extend f over the 1-skeleton M (1) of M and replace f by acellular map homotopic to f followed by a map S → S of degree k i . Thus we assumethat A contains the 1-skeleton of M i and f : A → S is a cellur map of degree p k i (recallthat “of degree p k i ” that f lifts to a Z p ki -covering of S ). Apply 5.1 with M = M i and K = S to construct M i +1 = M ′ and µ i +1 = µ . Set k i +1 = k i and t i +1 = t i . Procedure IV (extending partial maps to M ( Z p , n + 1) ). Let A be a subcomplexof M i and f : A → M ( Z p , n + 1) a map. Extend f over the ( n + 1)-skeleton on M i and re-place f by a homotopic cellular map. Thus we assume that A contains the ( n + 1)-skeleton M ( n +1) of M i and f : A → M ( Z p , n + 1) is a cellular map. Apply 5.1 with M = M i and K = M ( Z p , n + 1) to construct M i +1 = M ′ and µ i +1 = µ . Set k i +1 = k i + 1 and t i +1 = t i + 1.Let us check that after preforming the above procedures the conditions (1) and (2) hold.It is obvious for Procedure I. Procedures II and III preserve (1) and (2) by Propositions5.1 and 5.3 because t i +1 = t i ≤ k i . Procedure IV preserves (1) and (2) by Proposition 5.2.Let us show that Y = lim ← M i is ( n + 2)-dimensional. Clearly dim Y ≤ n + 2. Considerthe map µ i | A i : A i → A = S n +1 . Since dim A i ≤ n + 1, the long exact sequence generatedby 0 → Z p → Z p ti → Z p ti − implies that H n +1 ( A i ; Z p ) → H n +1 ( A i ; Z p t ) is injective for every i and hence, by (1) and(2), µ i induces a non-trivial homomorphismker( H n +1 ( A i ; Z p ti ) → H n +1 ( M i , Z p ti )) → H n +1 ( A ; Z p ti ) . This implies that µ i restricted to A i does not extend over M i as a map to A = S n +1 andhence dim Y = n + 2.Assume that Y is constructed as described in 5.3. The objects that we pick out for each M i are all the (non-isomorphic) circle bundles over M i and countably many maps fromsubcomplexes of M i to S and M ( Z p , n + 1) representing all possible maps up to homotopy.The procedures that we perform are Procedures I-IV.Assuming that on each M i we fix a sufficiently finite triangulation, we get that: • Procedure III together with Proposition 1.1 implies that dim Z [ p ] Y ≤ • Procedure IV implies that ext-dim Y ≤ M ( Z p , n + 1) and hence, by Dranishnikov’sextension criterion, dim Z p Y ≤ n + 1.Then dim Y = n +2 and Bockstein inequalities imply that dim Z [ p ] Y = 1 and dim Z p Y = n + 1.Let us show that H ( Y ; Z ) does not contain a subgroup isomorphic to Z p ∞ . Sincedim Z [ p ] Y = 1 we have that H ( Y ; Z ) ⊗ Z [ p ] = 0 and hence H ( Y ; Z ) is p -torsion. Thuswe need to show H ( Y ; Z ) does not contain a non-trivial element α that is divisible by p k for every k . Aiming at a contradiction assume that such α does exist. Consider thecircle bundle g Y : Z → Y corresponding to α . Then Construction 5.3 guarantees that at acertain step i of the construction we will consider a circle bundle g : L → M i such that g Y is the pull-back of g under the projection µ i : Y → Y i .17f g is flexible then we apply Procedure II and, by Construction 5.2.1, the pull-backbundle g i +1 : L i +1 → M i +1 of g under the map µ i +1 admits a section and hence both g i +1 and g Y are trivial. Thus we arrive at a contradiction with α = 0.If g : L → M i is non flexible then we apply Procedure I. Denote by g j : L j → M j , j ≥ i ,the pull-back bundle of g to M j under µ ij = µ j ◦ · · · ◦ µ i +1 : M j → M i with µ ii being theidentity map and g i = g and L i = L . Denote by G ij : L j → L i the induced map. Recallthat α is divisible by p k for every k . Then for a sufficiently large j there is a circle bundle g ∗ j : L ∗ j → M j such that g j : L j → M j is induced by g ∗ j and Z p ki +1 , see Lemma 3.1. Let G ∗ j : L ∗ j → L j be a bundle map induced by the homomorphism S → S / Z p ki +1 , Take asection s ∗ j : M (1) j → L ∗ j .Consider a 2-simplex Ω i of M i and let g − i (Ω i ) = Ω i × S be a trivialization of L i overΩ i . Consider Ω j = ( µ ij ) − (Ω i ). Analyzing Procedures I-IV we deduce that dim Ω j = 2, forProcedures I and IV we have that Ω j +1 = Ω j , and for Procedures II and III we have thatΩ j +1 is obtained from Ω j by taking a triangulation of Ω j and replacing each 2-simplex ∆of Ω j by by a mapping cylinder of a map ∂ ∆ → S of degree k j attached to ∂ ∆. Notethat k j ≥ k i +1 for j > i . Also note that µ ij is 1-to-1 over the 1-skeleton M (1) i of M i andhence M (1) i can be considered as a subset M (1) i ⊂ M (1) j of the 1-skeleton of M j and then ∂ Ω i considered as a subset of Ω j is homologous to 0 in H (Ω j ; Z p ki +1 ), j > i .Consider the trivialization g − j (Ω j ) = Ω j × S of L j over Ω j induced by the trivializationof g − i (Ω i ) = Ω i × S and consider the section s j : M (1) j → L j which is s ∗ j followed by G ∗ j .Note that for every 2-simplex ∆ of Ω j , the map s j restricted to ∂ ∆ and followed by theprojection of g − j (Ω j ) = Ω j × S to S is of degree p k i +1 . Then, since ∂ Ω i is homologousto 0 in H (Ω j ; Z p ki +1 ), we get that s j restricted to ∂ Ω i and followed by the projectionof g − j (Ω j ) = Ω j × S to S is of degree p k i +1 as well. Thus we get that the section s i = µ ij ◦ s j | M (1) i : M (1) i → L i is of degree p k i +1 and this violates our choice of k i +1 inProcedure I. (cid:4) Theorem 1.5 follows from Theorem 1.6 and the following proposition.
Proposition 7.1
There is an infinite dimensional compactum Y with dim Z [ p ] Y = 1 and dim Z Y = 2 such that H ( Y ; Z ) does not contain a subgroup isomorphic to Z p ∞ . Let K be a CW-complex and N a connected CW-complex. By map( K, N ) we de-note the space of pointed maps from K to N with the compact-open topology. We willwrite map( K, n ) ∼ = 0 if map( K, N ) is weakly homotopy equivalent to a point. Note thatmap(
K, n ) ∼ = 0 if and only if for every n ≥
0, every map from Σ n K to N is null-homotopic.In the proof of Proposition 7.1 we will use the following facts. Theorem 7.2 (Millers theorem (The Sullivan conjecture)) [14] . Let G be a finite groupand N a connected finite CW-complex. Then map( K ( G, , N ) ∼ = 0 . roposition 7.3 [11] Let M be a countable CW-complex, N a connected CW-complexwhose homotopy groups are finite, A N a subcomplex of M , and f : A N → N a map thatcannot be continuously extended over M then there exists a finite subcomplex M N of M such that f N | A N ∩ M N : A N ∩ M N → N cannot be continuously extended over M N . Proposition 7.4
Let N be a CW-complex whose homotpy groups are finite p -torsion groups.Then map( K ( Z [ p ] , , N ) ∼ = 0 . Proof.
Note that Σ n K ( Z [ p ] ,
1) can be represented as the infinite telescope of a map S n → S n of degree p . Then π n (Σ n K ( Z [ p ] , Z [ p ] and every finite subtelesope of K ( Z [ p ] , S n . Since π n ( N ) is a p -torsion group we have that every map f : Σ n K ( Z [ p ] , → N sends π n (Σ n K ( Z [ p ] , π n ( N ) and hence f is null-homotopicon every finite subtelescope of Σ n K ( Z [ p ] , f extends over Σ(Σ n K ( Z [ p ] , f is null-homotopic. (cid:4) Proposition 7.5
Consider Construction 5.1. Let N be a connected CW-complex such that map( K, N ) ∼ = 0 and f N : A N → N is a map from a subcomplex A N of M such that f N doesnot extend over M . Then µ restricted to µ − ( A N ) and followed by f N does not extend over M ′ . Proof.
Let M ∗ by the space obtained from M ′ by collapsing the fibers of µ over N tosingletons and µ ∗ : M ∗ → M the induced map. Thus we can consider N as a subcomplexof M ∗ and aiming at a contradiction assume that f N extends to f ∗ : M ∗ → N .Consider the first barycentric subdivision βM of the triangulation of M . Note that forever 0-dimensional simplex (vertex) ∆ of βM , ( µ ∗ ) − (∆) is either a singleton or homeo-morphic to K . Then f ∗ is null-homotopic on ( µ ∗ ) − (∆). Thus we can replace M ∗ by thespace M ∗ obtained from M ∗ by collapsing ( µ ∗ ) − (∆) to singletons for every 0-simplex ∆ of βM and µ ∗ and f ∗ by the induced maps µ ∗ and f ∗ to M and N respectively and assumethat µ ∗ is 1-to-1 over the 0-simplexes of βM .Now consider a 1-simplex ∆ of βM . Then ( µ ∗ ) − (∆) is either contractible or homotopyequivalent to Σ K and hence f ∗ is null-homotopic on ( µ ∗ ) − (∆). Thus f ∗ factors up tohomotopy through the space obtained from M ∗ by collapsing the fibers of µ ∗ over thesimplex ∆. Doing that consecutively for all the 1-simplexes of βM we obtain the space M ∗ and the maps µ ∗ : M ∗ → M and f ∗ : M ∗ → N induced by µ ∗ and f ∗ respectively suchthat µ ∗ is 1-to-1 over the 1-simplexes of βM .Procced by induction and finally get for m = dim M that M ∗ m = M and f ∗ m extends f N that contradicts the assumptions of the proposition. (cid:4) Proof of Proposition 7.1.
We will construct Y as the inverse limit of ( n + 2)-dimensionalfinite simplicial complexes M i and combinatorial bonding maps µ i +1 : M i +1 → M i . In orderto show that Y has the required properties we consider for each i a subcomplex A i of M i suchthat A i +1 = µ − i +1 ( A i ) and a natural numbers k i such that k i +1 ≥ k i . Set A to be a Moorespace M ( Z p , M the cone over A and k = 1. We denote µ ij = µ j ◦ · · · ◦ µ i +1 : M j → M i with µ ii being the identity map and require that(1) µ i | A i : A i → A = M ( Z p ,
2) does not extend over M i as a map to M ( Z p , i = 0. Assuming that the construction is completed for i and weproceed to i + 1 performing one the following procedures. Procedure I (taking care of non-flexible bundles).
Let g : L → M i be a circlebundle which is not p -flexible. Then take any natural number k such that there is no sec-tion over M (1) i of degree p k . Set M i +1 = M i , µ i +1 =the identity map and k i +1 = max { k, k i } .Clearly (1) holds for i + 1. Procedure II (taking care of flexible bundles).
Note that the homotopy groups of M ( Z p ,
2) are finite p -torsion groups, denote by t any natural number such that t ≥ k i and p t π j ( M ( Z p , j ≤ m = dim M i . Let g : L → M i be a flexible bundle. Apply5.2.2 with M = M i , N = M ( Z p , , A N = A i , f N = µ i | A i : A N = A i → N = A = M ( Z p , t and m as above to construct M i +1 = M ′ and µ i +1 = µ ′ , and set k i +1 = k i . ByProposition 5.4, we get that (1) holds for i + 1. Procedure III (extending partial maps to a circle).
Let A be a subcomplex of M i and f : A → K ( Z [ p ] ,
1) a map. Recall that K ( Z [ p ] ,
1) can be represented as the infinitetelescope of the Z p -covering map S → S . Extend f over the 1-skeleton M (1) of M andreplace f by a cellular map homotopic to f such that f ( A ) = S ⊂ K ( Z [ p ] ,
1) and f asa map to S is of degree k i . Apply 5.1 with M = M i and K = K ( Z [ p ] ,
1) to construct M i +1 = M ′ and µ i +1 = µ . By Propositions 7.4 and 5.4 we get that (1) holds for i + 1. ByProposition 7.3 we can replace in Construction 5.1 the complex K = K ( Z [ p ] ,
1) by a finitesubcomplex of K ( Z [ p ] ,
1) containing f ( A ) and still preserve (1). Thus we assume that M ′ is a finite CW-complex and set k i +1 = k i . Procedure IV (extending partial maps to K ( Z p ∞ , ). Let A be a subcomplexof M i and f : A → K ( Z p ∞ ,
1) a map. Extend f over the 1-skeleton of M i . Repre-sent K ( Z p ∞ ,
1) as the infinite telescope of the maps K ( Z p n , → K ( Z p n +1 ,
1) induced bythe monomorphisms Z p n → Z p n +1 and replace f by a homotopic cellular map f : A → K ( Z p n , ⊂ K ( Z p ∞ ,
1) such that f ( M (1) i ) = S and f restricted to M (1) i and consideredas a map to S is of degree p k i . Apply 5.1 with M = M i and K = K ( Z p n ,
1) to construct M i +1 = M ′ and µ i +1 = µ . By Theorem 7.2 and Proposition 7.5 we get that (1) holds for i + 1. By Proposition 7.3 we can replace in Construction 5.1 the complex K = K ( Z p n , K ( Z p n ,
1) containing f ( A ) and still preserve (1). Thus we assumethat M ′ is a finite CW-complex and set k i +1 = k i .Let Y = lim ← M i . Clearly (2) implies that dim Y > Y is constructed as described in 5.3. The objects that we pick out foreach M i are all the (non-isomorphic) circle bundles over M i and countably many mapsfrom subcomplexes of M i to K ( Z [ p ] ,
1) and K ( Z p ∞ ,
1) representing all possible maps up tohomotopy. The procedures that we perform are Procedures I-IV.Assuming that on each M i we fix a sufficiently finite triangulation, we get that: • Procedure III implies that ext-dim ≤ K ( Z [ p ] ,
1) and hence dim Z [ p ] Y ≤ Procedure IV implies that ext-dim Y ≤ K ( Z p ∞ ,
1) and hence dim Z p Y ≤ n + 1.Then dim Y > Z Y = 2 and hence Y isinfinite dimensional.Let us show that H ( Y ; Z ) does not contain a subgroup isomorphic to Z p ∞ . Sincedim Z [ p ] Y = 1 we have that H ( Y ; Z ) ⊗ Z [ p ] = 0 and hence H ( Y ; Z ) is a p -torsion group.Thus we need to show H ( Y ; Z ) does not contain a non-trivial element α that is divisibleby p k for every k . Aiming at a contradiction assume that such α does exist. Consider thecircle bundle g Y : Z → Y corresponding to α . Then Construction 5.3 guarantees that at acertain step i of the construction we will consider a circle bundle g : L → M i such that g Y is the pull-back of g under the projection µ i : Y → Y i .If g is flexible then we apply Procedure II and, by Construction 5.2.1, the pull-backbundle g i +1 : L i +1 → M i +1 of g under the map µ i +1 admits a section and hence both g i +1 and g Y are trivial. Thus we arrive at a contradiction with α = 0.If g : L → M i is non flexible then we apply Procedure I. Denote by g j : L j → M j , j ≥ i ,the pull-back bundle of g to M j under µ ij and g i = g and L i = L . Denote by G ij : L j → L i the induced map. Recall that α is divisible by p k for every k . Then for a sufficiently large j there is a circle bundle g ∗ j : L ∗ j → M j such that g j : L j → M j is induced by g ∗ j and Z p ki +1 , see Lemma 3.1. Let G ∗ j : L ∗ j → L j be a bundle map induced by the homomorphism S → S / Z p ki +1 , Take a section s ∗ j : M (1) j → L ∗ j .Let Ω be a 2-dimensional subcomplex of M j . Denote by µ − j +1 (Ω) the subcomplex of µ − j +1 (Ω) which is the closure of the following set µ − j +1 (Ω) \ { the fibers of µ j +1 over the barycenters of the 2-simplexes of Ω } .Consider a 2-simplex Ω i of M i and let g − i (Ω i ) = Ω i × S be a trivialization of L i over Ω i .For every j > i define Ω j by Ω i +1 = µ − i (Ω i ) , Ω i +2 = µ − i +2 (Ω i +1 ) , . . . , Ω j = µ − j (Ω j − ).Analyzing Procedures I-IV we deduce that dim Ω j = 2, for Procedure I we have thatΩ j +1 = Ω j , and for Procedures II, III and IV we have that Ω j +1 is obtained from Ω j bytaking a triangulation of Ω j and replacing each 2-simplex ∆ of Ω j by a mapping cylinderof a map ∂ ∆ → S of degree k t attached to ∂ ∆. Note that k j ≥ k i +1 for j > i . Also notethat µ ij is 1-to-1 over the 1-skeleton M (1) i of M i and hence M (1) i can be considered as asubset M (1) i ⊂ M (1) j of the 1-skeleton of M j and then ∂ Ω i considered as a subset of Ω j ishomologous to 0 in H (Ω j ; Z p ki +1 ), j > i .Consider the trivialization g − j (Ω j ) = Ω j × S of L j over Ω j induced by the trivializationof g − i (Ω i ) = Ω i × S and consider the section s j : M (1) j → L j which is s ∗ j followed by G ∗ j .Note that for every 2-simplex ∆ of Ω j , the map s j restricted to ∂ ∆ and followed by theprojection of g − j (Ω j ) = Ω j × S to S is of degree p k i +1 . Then, since ∂ Ω i is homologousto 0 in H (Ω j ; Z p ki +1 ), we get that s j restricted to ∂ Ω i and followed by the projectionof g − j (Ω j ) = Ω j × S to S is of degree p k i +1 as well. Thus we get that the section s i = µ ij ◦ s j | M (1) i : M (1) i → L i is of degree p k i +1 and this violates our choice of k i +1 inProcedure I. (cid:4) eferences [1] Ancel, Fredric D. The role of countable dimensionality in the theory of cell-likerelations.
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