aa r X i v : . [ m a t h . G T ] S e p On Raymond -Williams’ example
Michael Levin ∗ Abstract
Raymond and Wiliams in “Examples of p-adic transformation groups”.
Ann. ofMath. (2) 78 (1963) 92-106 constructed an action of the p -adic integers A p on an n -dimensional compactum X , n ≥
2, with the orbit space
X/A p of dimension n + 2.We present a simpler construction of such an example. Keywords:
Dimension Theory, Transformation Groups
Math. Subj. Class.:
The Hilbert-Smith conjecture asserts that a compact group acting effectively on a manifoldmust be a Lie group. This conjecture is equivalent to the following one: the group A p ofthe p -adic integers cannot act effectively on a manifold. Yang [2] showed that if A p actseffectively on a manifold M then dim M/A p = dim M + 2. In order to verify if the latterdimensional relation ever occurs in a more general setting, Raymond and Williams [1]constructed an action of A p on an n -dimensional compactum (=compact metric space) X, n ≥
2, with dim
X/A p = n + 2. We present a simpler approach for constructing suchan example. Let first note that it is sufficient to consider only the case n = 2 because theinduced action of A p on X × [0 , k defined by g ( x, t ) = ( gx, t ) , g ∈ A p , x ∈ X and t ∈ [0 , k provides the corresponding example with dim X × [0 , k = k + 2 and dim( X × [0 , k ) /A p =dim( X/A p ) × [0 , k = dim X/A p + k . Thus without loss of generality we restrict ourselves tothe dimension n = 2 (although the construction below can be easily adjusted to any finitedimension n ≥
2, it is slightly easier to visualize it for n = 2). Our example is constructedin the next section and a few related problems are posed in the last section. Consider the unit 3-sphere S in R . Represent R as the product R × R ⊥ of a coordinateplane R and its orthogonal complement R ⊥ and let S = S ∩ R and S ⊥ = S ∩ R ⊥ . Takethe closed ǫ -neighborhood F of S . Then F and F ⊥ = S \ int F can be represented as theproducts F = S × D of S and F ⊥ = D ⊥ × S ⊥ with D and D ⊥ being 2-disks. Thus we ∗ This research was supported by THE ISRAEL SCIENCE FOUNDATION (grant No. 522/14) ∂F = ∂F ⊥ = S × ∂D = ∂D ⊥ × S ⊥ . Let T = R / Z freely act on the circle ∂D by rotations. Then this action induces the corresponding free action on ∂F = ∂F ⊥ andthe later action obviously extends over F ⊥ as a free action and it extends over F by therotations of D induced by the rotations of ∂D . This way we have defined an action of T on S whose fixed point set is S and T acts freely on S \ S . Proposition 2.1
Let ∆ be a -simplex. There is an unknotted circle S in S = ∂ ∆ suchthat for each -simplex ∆ ′ the intersection S ∩ ∆ ′ is an interval whose end-points are in ∂ ∆ ′ and whose interior is in the interior of ∆ ′ . Proof.
Fix a 3-simplex ∆ ′ of ∆. First observe that S can be embedded into ∂ ∆ ′ so that S does not contain the vertices of ∆ ′ and the intersection of S with every 2-simplex ∆ ′′ of ∆ ′ is an interval whose end-points are in ∂ ∆ ′′ and whose interior is in the interior of ∆ ′′ . Takeany two 3-simplexes ∆ ′− and ∆ ′ + different from ∆ ′ and take any points x − ∈ S ∩ ∆ ′− ∩ ∆ ′ and x + ∈ S ∩ ∆ ′ + ∩ ∆ ′ not lying in ∂ (∆ ′− ∩ ∆ ′ ) and ∂ (∆ ′ + ∩ ∆ ′ ) respectively. Now for each3-simplex ∆ ′∗ of ∆ different from ∆ ′ , ∆ ′− and ∆ ′ + push the interior of the interval S ∩ ∆ ′∗ inside the interior ∆ ′∗ not moving the end-pints of the interval. The interior of the intervalof S connecting x − and x + inside ∆ ′− ∪ ∆ ′ + push into the interior of ∆ ′ not moving theend-points x − and x + . And finally push the intervals S ∩ ∆ ′− and S ∩ ∆ ′ + into the interiorof ∆ ′− and ∆ ′ + respectively not moving the end-points of the intervals. Thus S ⊂ ∂ ∆ hasthe required properties. (cid:4) From now we identify S with the boundary of a 4-smplex ∆ and assume that S ⊂ S satisfy the conclusions of Proposition 2.1.Recall that F is represented as the product F = S × D , take 5 disjoint closed intervals I ,..., I in S lying in the interior of different 3-simplexes of ∆ and denote B i = I i × D ′ where D ′ ⊂ D is a concentric disk in D . Taking D ′ to be small enough we may assume that B i lie in the interior of different 3-simplexes of ∆. Note that each 3-ball B i is invariant underthe action of T and hence M = S \ int( B ∪ · · · ∪ B ) is invariant as well. Proposition 2.2 (i) For every g ∈ T the map x → gx, x ∈ M, is ambiently isotopic to a homeomorphism φ : M → M by an isotopy of M that does not move the points of S ∪ ∂M and suchthat φ restricted to the -skeleton ∆ (2) coincides with the inclusion of ∆ (2) into M and φ (∆ ′ ∩ M ) = ∆ ′ ∩ M for every -simplex ∆ ′ of ∆ .(ii) There is a retraction of r : M → ∆ (2) such that for every -simplex ∆ ′ of ∆ wehave that r (∆ ′ ∩ M ) ⊂ ∂ ∆ ′ . Proof. (i) Since T is path-connected g : M → M is isotopic to the identity map of M by anisoptopy that does not move the points of M ∩ S (the fixed point set of T ). For each ∂B i ⊂ ∂M we can change this isotopy on a neighborhood of ∂B i in M to get in additionthat the points of ∂B i are not moved.(ii) For each 3-simplex ∆ ′ of ∆ define r on ∆ ′ ∩ M as the restriction of a radial projectionto ∂ ∆ ′ from a point of ∆ ′ that that does not belong to M . (cid:4) p consider the subgroup Z p = Z /p Z of T = R / Z and let Γ M = M/ Z p .Denote by M + the space which is the union of p copies M a , a ∈ Z p , of M with ∂M beingidentified in all the copies by the identity map. Thus all M a ⊂ M + intersect each other at ∂M . Define the action of Z p on M + by sending x ∈ M a to gx ∈ M g + a . Note that ∂M ⊂ M + is invariant under the action of Z p on M + and the natural projection α + : M + → M sendingeach M a to M by the identity map is equivariant.Let Γ + be the mapping cylinder of α + . The actions of Z p on M + and M induce thecorresponding action of Z p on Γ + . The spaces M + and M can be considered as naturalsubsets of the mapping cylinder Γ + , we will refer to M + and M as the bottom and the topof Γ + respectively.Let Γ = Γ + / Z p , γ + : Γ + → Γ the projection and Γ ∂M and Γ M the images in Γ of thebottom set M + and the top set M of Γ + under the map γ + . Note that α + : M + → M induces the corresponding map α : Γ ∂M → Γ M for which Γ is the mapping cylinder of α . Also note that Γ M = M/ Z p and Γ ∂M is the space obtained from M by collapsingto singletons the orbits of the action of Z p on M lying in ∂M . Thus there is a naturalprojection µ : M → Γ ∂M and ∆ (2) ⊂ M can be considered as a subset of Γ ∂M as well(we can identify M with one of the spaces M a ⊂ M + and regard µ as the projection M + → Γ ∂M = M + / Z p restricted to M a ).Let us define a map δ : Γ → ∆ by sending Γ ∂M to ∂ ∆ and Γ M to the barycenter of∆ such that δ is the identity map on ∆ (2) , δ ( x ) ∈ ∆ ′ for every 3-simplex ∆ ′ of ∆ and x ∈ µ ( M ∩ ∆ ′ ) and δ is linearly extended along the intervals of the mapping cylinder Γ(in our descriptions we abuse notations regarding simplexes of ∆ also as subsets of otherspaces involved). Denote δ + = δ ◦ γ + : Γ + → ∆.In the notation below an element g ∈ Z p that appears in a composition with other mapsis regarded as a homeomorphism of the space on which Z p acts. Proposition 2.3
For every -simplex ∆ i of ∆ there is a map r i + : δ − (∆ i ) → ∂ ∆ i suchthat(i) r i + coincides with δ + on δ − ( ∂ ∆ i )) ;(ii) for every triangulation of Γ + and every g , . . . , g ∈ Z p there is a map r + : Γ + → ∂ ∆ such that r + (Γ (3)+ ) ⊂ ∆ (2) and r + restricted to δ − (∆ i ) coincides with r i + ◦ g i for each i . Proof.
Recall that Γ + is the mapping cylinder of α + : M + → M and let β + : Γ + → M bethe projection to the top set M of Γ + . Also recall that M + is the union of p copies M a , a ∈ Z p , of M intersecting each other at ∂M .Denote ∆ (2) a = δ − (∆ (2) ) ∩ M a , (∆ i ∩ M ) a = δ − (∆ i ) ∩ M a , ( ∂ ∆ i ) a = δ − ( ∂ ∆ i ) ∩ M a and( S ∩ M ) a = α − ( S ∩ M ) ∩ M a . Note that ( S ∩ M ) a ∩ (∆ i ∩ M ) a splits into two intervalseach of them connects in M a the sets ∂B i and ( ∂ ∆ i ) a . One of this intervals we will denoteby ( S i ) a .By (i) of Proposition 2.2 the map β + can be isotoped relative to β − (( S ∩ M ) ∪ ∂M )into a map ω + : Γ + → M such that ω + and δ + restricted to each ∆ (2) a ⊂ M + coincide.Consider the map r from (ii) of Proposition 2.2 and note that ( r ◦ ω + )(Γ + ) ⊂ ∆ (2) . Definethe map r i + as the map r ◦ ω + restricted to δ − (∆ i ).3i) follows from (ii) of Proposition 2.2.(ii) For a subset A ⊂ M + by the mapping cylinder of α + over A we mean the map-ping cylinder of α + : A → α + ( A ) which is a subset of Γ + . Let S ⊂ M + be the union of( S i ) a for all i and a , and B ⊂ M + the union of all the balls B i ⊂ M + Consider a CW-structure of Γ + for which the interiors of the 4-cells are the interiors in Γ + of the mappingcylinders of α + over (∆ i ∩ M ) a \ ( S i ) a without the points belonging to M + and M . Thusthe 3-skeleton of this CW-structure is the union of M + ∪ M and the mapping cylinder of α + over δ − (∆ (2) ) ∪ S ∪ B .Let be the map φ : M + → ∂ ∆ be defined by the maps r i + ◦ g i on each δ − (∆ i ). Notethat the maps φ and r ◦ ω + coincide on δ − (∆ (2) ) ∪ S . Also note φ and r ◦ ω + are homotopicon B by a homotopy relative to B ∩ S . Thus r ◦ ω + restricted to the union of M ⊂ Γ + withthe mapping cylinder of α + over δ − (∆ (2) ) ∪ B ∪ S can be homotoped to a map Φ suchthat φ and Φ restricted to δ − (∆ (2) ) ∪ B ∪ S coincide. Thus we can extend φ to a map φ + from the 3-skeleton of the CW-structure of Γ + to ∆ (2) .Then, since ∆ (2) is contractible inside ∂ ∆, we may extend φ + to a map r + : Γ + → ∂ ∆.For any triangulation of Γ + , the 3-skeleton of the triangulation can be pushed off the in-teriors of the 4-cells of Γ + relative to the 3-skeleton of the CW-structure of Γ + and (ii)follows. (cid:4) Denote by Γ ∗ the space obtained from Γ + by collapsing the fibers of γ + to singletonsover the set δ − (∆ (2) ) and let the maps γ ∗ : Γ ∗ → Γ, δ ∗ : Γ ∗ → ∆, r i ∗ : δ − (∆ i ) → ∂ ∆ i and r ∗ : Γ ∗ → ∂ ∆ be induced by γ + , δ + , r i + and r + respectively and consider Γ ∗ with the actionof Z p induced by the action of Z p on Γ + . Proposition 2.4 (i) The conclusions of Proposition 2.3 hold with the subscript “ + ” being replaced every-where by the subscript “ ∗ ”.(ii) The fixed point set of the action of Z p on Γ ∗ is -dimensional and there are trian-gulations of Γ ∗ and Γ and a subdivision of ∆ for which the action of Z p on Γ ∗ is simplicialand the maps γ ∗ and δ are simplicial. Proof. (i) is obvious and (ii) can be derived from the construction of the spaces and themaps involved. (cid:4)
Let L be a finite 4-dimensional simplicial complex and λ : L → ∆ a simplicial map suchthat λ is 1-to-1 on each simplex of L . Denote by L ′ the pull-back space of the maps λ and δ : Γ → ∆ and by Ω : L ′ → L the pull-back of δ . Proposition 2.5
The map
Ω : L ′ → L induces an isomorphism of H ( L ′ ; Z p ) and H ( L ; Z p ) . Proof.
Recall that for every 3-simplex ∆ ′ of ∆, δ − (∆ ′ ) is the mapping cylinder a map ofdegree p from ∂ ∆ ′ to a 2-sphere S (the boundary of one of the 3-balls B i ).Also recall that Γ = δ − (∆) is the mapping cylinder of the map α and from the definitionof α one can also observe that H ( δ − (∆) , δ − ( ∂ ∆); Z p ) = Z p and δ induces an isomorphismbetween H ( δ − (∆) , δ − ( ∂ ∆); Z p ) and H (∆ , ∂ ∆; Z p ) = Z p .4nd finally recall that δ is 1-to-1 over the 2-skeleton of ∆. Consider the long ex-act sequences of the pairs ( L ′ , L ′ (3) ) and ( L, L (3) ) for the homology with coefficients in Z p . The facts above imply that Ω induces isomorphisms H ( L ′ (3) ; Z p ) → H ( L (3) ; Z p ) and H ( L ′ , L ′ (3) ; Z p ) → H ( L, L (3) ; Z p ). Then, by the 5-lemma, Ω induces an isomorphism H ( L ′ ; Z p ) → H ( L ; Z p ) as well. (cid:4) Proposition 2.6
Let G = Z p k act simplicially on a finite -dimensional simplicial complex K such that the action of G is free on K \ K (2) . Then there is a finite -dimensionalsimplicial complex K ′ , a simplicial action of G ′ = Z p k +1 on K ′ and a map ω : K ′ → K such that the action of G ′ is free on K ′ \ K ′ (2) and(i) the actions of G and G ′ agree with ω and the natural epimorphism h : G ′ → G . Bythis we mean that ω ( g ′ x )) = h ( g ′ ) ω ( x ) for for every x ∈ K ′ and g ′ ∈ G ′ ;(ii) there is a map κ : K ′ → K (3) such that κ ( K ′ (3) ) ⊂ K (2) and κ ( ω − (∆ K )) ⊂ ∆ K forevery simplex ∆ K of K ;(iii) the map K ′ /G ′ → K/G determined by ω induces an isomorphism H ( K ′ /G ′ ; Z p ) → H ( K/G ; Z p ) . Proof.
Replacing the triangulation of K by a subdivision we may assume that L = K/G is a simplicial complex, the projection π : K → L is a simplicial map and L admits asimplicial map λ L : L → ∆ to a 4-simplex ∆ such that λ L is 1-to-1 on each simplex of L . For every 4-simplex ∆ L of L fix a 4-simplex ∆ K of K such that π (∆ K ) = ∆ L anddenote by K − the union of the 3-skeleton K (3) of K with all the 4-simplexes of K that wefixed. Let K ′− be the pull-back space of the maps λ L ◦ π | K − : K − → ∆ and δ ∗ : Γ ∗ → ∆, ω − : K ′− → K the pull-back map of δ ∗ and λ ∗ : K ′− → Γ ∗ the pull back of λ L ◦ π | K − .Let g be a generator of G ′ and l : G ′ → Z p an epimorphism. We will first define the ac-tion of G ′ on ω − − ( K (3) ). For each 3-simplex ∆ L of L define the action of G ′ on ω − − ( π − (∆ L ))as follows. Fix a 3-simplex ∆ K of π − (∆ L ) and let x ∈ ω − − (∆ K ). Define y = g t x for1 ≤ t ≤ p k − y ∈ K ′ such that ω − ( y ) ∈ h ( g t )(∆ K ) and λ ∗ ( y ) = λ ∗ ( x ),and for t = p k define y = g t x as the point y ∈ ∆ K such that λ ∗ ( y ) = l ( g ) λ ∗ ( x ). Wedo this independently for every 3-simplex ∆ L of L and this way define the action of g on ω − − ( K (3) ). It is easy to see that the action of g is well-defined, g t for t = p k +1 is the identitymap of ω − − ( K (3) ) and hence the action of g defines the action of G ′ on ω − − ( K (3) ). Note that(*) for g t ∈ G ′ , t = p k , and g ∗ = l ( g ) ∈ Z p we have that λ ∗ ◦ g t and g ∗ ◦ λ ∗ coincide on ω − − ( ∂ ∆ K ) for every 4-simplex ∆ K of K .Now we will enlarge K ′− to a space K ′ and extend the action of G ′ over K ′ . Let∆ L be a 4-simplex of L . Recall that we fixed a 4-simplex ∆ K in π − (∆ L ). For every g ′ = g t ∈ G ′ , ≤ p k − g ′ ( ω − − ( ∂ ∆ K )) a copy of the space ω − − (∆ K ) (which is isin its turn a copy of Γ ∗ ) by identifying g ′ ( ω − − ( ∂ ∆ K )) with ω − − ( ∂ ∆ K ) according to g ′ andfor x ∈ ω − − (∆ K ) define g ′ x as the as the point corresponding to x in the attached space.We will define the action of g t , t = p k , on ω − − (∆ K ) by y = g t x, x ∈ ω − − (∆ K ) such that y ∈ ω − − (∆ K ) and λ ∗ ( y ) = l ( g ) λ ∗ ( x ). By (*) the action of g on ω − − (∆ K ) agrees with theaction of g on ω − − ( K (3) ). We do the above procedure independently for every 4-simplex ∆ L L and this way we define the space K ′ and the action of G ′ on K ′ . We extend ω − and λ ∗ to the maps ω : K ′ → K and λ ′ K : K ′ → Γ by ω ( g t x ) = ω − ( x ) and λ ′ K ( g t x ) = γ ∗ ( λ ∗ ( x ))for x in a fixed 4-simplex ∆ K and 1 ≤ t ≤ p k .It is easy to verify that the action of G ′ on K ′ and the maps ω and λ ′ K are well-definedand the conclusion (i) of the proposition holds. Moreover λ ′ K ◦ g ′ = λ ′ K for every g ′ ∈ G ′ andhence λ ′ K defines the corresponding map λ ′ L : L ′ = K ′ /G ′ → Γ. Then L ′ is the pull-backof the maps λ L : L → ∆ and δ : Γ → ∆ with λ ′ L being the pull-back of λ L and the mapΩ : L ′ → L induced by ω being the pull-back of δ . Thus, by Proposition 2.5, the conclusion(iii) of the proposition holds as well.Consider any triangulation of K ′ for which the preimages under ω of the simplexesof K ′ are subcomplexes of K ′ . Then the map r ∗ : Γ ∗ → ∆ (3) provided by Propositions2.4 and 2.3 for g = · · · = g = 0 ∈ Z p defines the corresponding map κ − : K ′− → K (3) such that κ ( ω − − ( K (3) )) ⊂ K (2) . The construction above and Propositions 2.4 and 2.3allow us to extend the map κ − to a map κ : K ′ → K (3) satisfying the conclusion (ii) of theproposition. Recall that the triangulation of K is a subdivision of the original triangulationof K . Replacing κ by its composition with the simplical approximation of the identity mapof K with respect to the new and original triangulations of K we get that the conclusion(ii) of the proposition holds.The rest of the conclusions of the proposition follows from (ii) of Proposition 2.4. (cid:4) Now we are ready to construct our example. Set K to be any finite 4-dimensional sim-plicial complex with H ( K ; Z p ) = 0 and let the trivial group Z p = 0 trivially act on K . Construct by induction on i finite 4-dimensional simplicial complexes K i , an actionof Z p i on K i and maps ω i +1 : K i +1 → K i so that K i +1 , the action of Z p i +1 on K i +1 and ω i +1 satisfy the conclusions of Proposition 2.6 with K , K ′ , k and ω being replaced by K i , K i +1 , i and ω i +1 respectively. Consider X = lim ← ( K i , ω i ). Replacing the triangulation of K i by a sufficiently fine barycentric subdivision we can achieve, by (ii) of Proposition 2.6,that dim X ≤
2. The conclusions (i) and (iii) of Proposition 2.6 imply that A p = lim ← Z p i acts on X so that for L i = K i / Z p i and the map Ω i +1 : L i +1 → L i induced by ω i we havethat Y = X/A p = lim ← ( L i , Ω i ) and H ( Y ; Z p ) = 0. Thus dim Y = 4. Note that from theconstruction of K ′ in Proposition 2.6 one can derive that X contains the 2-skeleton K (2) i of each K i and hence dim X = 2. It is very challenging to try to modify the construction presented in this paper to approachthe following problems.
Problem 3.1
Does there exist an action of A p on a -dimensional compactum with the -dimensional orbit space? Problem 3.2
Does there exist an action of A p on an n -dimensional compactum X withan invariant subset X ′ ⊂ X so that dim X ′ < n , dim X/A p = n + 2 and the action of A p isfree on X \ X ′ ? References [1] Raymond, Frank; Williams, R. F.
Examples of p-adic transformation groups.
Ann.of Math. (2) 78 (1963) 92-106.[2] Yang, Chung-Tao. p-adic transformation groups.p-adic transformation groups.