A Proof Of The Hilbert-Smith Conjecture by Louis F. McAuley Dedicated to the memory of Deane Montgomery Abstract The Hilbert-Smith Conjecture states that if G is a locally compact group whi* *ch acts effectively on a connected manifold as a topological transformation group, then* * G is a Lie group. A rather straightforward proof of this conjecture is given. The moti* *vation is work of Chernavsky ("Finite-to-one mappings of manifolds", Trans. of Math. Sk. * *65 (107), 1964.). His work is generalized to the orbit map of an effective action of a p-* *adic group on compact connected n-manifolds with the aid of some new ideas. There is no at* *tempt to use Smith Theory even though there may be similarities. Smith's exact sequen* *ces are not used. 1. Introduction. In 1900, Hilbert proposed twenty-three problems [8]. For an excellent discus* *sion con- cerning these problems, see the Proceedings of Symposia In Pure Mathematics con* *cerning äM thematical Developments Arising From Hilbert Problems" [3]. The abstract by * *C.T. Yang [22] gives a review of Hilbert's Fifth Problem öH w is Lie's concept of co* *ntinu- ous groups of transformations of manifolds approachable in our investigation wi* *thout the assumption of differentiability?" Work of von Neumann [40] in 1933 showed that* * differ- entiability is not completely dispensable. This with results of Pontryagin [35* *] in 1939 suggested the specialized version of Hilbert's problem: If G is a topological g* *roup and a topological manifold, then is G topologically isomorphic to a Lie group? This i* *s generally 1 regarded as Hilbert's Fifth Problem. The first partial result was given by Brou* *wer [26] in 1909-1910 for locally euclidean groups of dimension 2. The best known partial* * results were given for compact locally euclidean groups and for commutative locally euc* *lidean groups by von Neumann [40] and Pontryagin [35], respectively. In 1952, the work of Gleason [5] and Montgomery-Zippin [13] proved: Every l* *ocally euclidean group is a Lie group. This solved Hilbert's Fifth Problem. A more general version of Hilbert's Fifth Problem is the following: If G is a locally compact group which acts effectively on a connected manifo* *ld as a topological transformation group, then is G a Lie group? The Hilbert-Smith Con* *jecture states that the answer is yes. Papers of Montgomery [34] in 1945 and Bochner-Montgomery [1] in 1946 establi* *shed the partial result: Let G be a locally compact group which acts effectively on a di* *fferentiable manifold M such that for any g 2 G, x 7! gx is a differentiable transformation * *of M . Then G is a Lie group and (G, M) is a differentiable transformation group. Ano* *ther partial result was given by a theorem of Yamabe [43] and a theorem of Newman [1* *5] as follows: If G is a compact group which acts effectively on a manifold and every* * element of G is of finite order, then G is a finite group. It has been shown [14] that an affirmative answer to the generalized version* * of Hilbert's Fifth Problem is equivalent to a negative answer to the following: Does there * *exist an effective action of a p-adic group on a manifold? It is proved here that the answer to this question is No. Thus, the Hilbert-* *Smith Con- jecture is true, i.e., A locally compact group acting effectively on a connecte* *d n-manifold must be a Lie group. A brief review of some of the consequences of efforts to solve this problem * *is given below. There are examples in the literature of effective actions of an infinit* *e compact 0-dimensional topological group G (each g 2 G-{identity} moves some point) on l* *ocally 2 connected continua. The classic example of Kolmogoroff [29] in 1937, is one wh* *ere G operates effectively but not strongly effectively [24] on a 1-dimensional local* *ly connected continuum (Peano continuum) such that the orbit space is 2-dimensional. In 1957* *, R.D. Anderson [24] proved that any compact 0-dimensional topological group G can act* * strongly effectively as a transformation group on the (Menger) universal 1-dimensional c* *urve M such that either (1) the orbit space is homeomorphic to M or (2) the orbit spa* *ce is homeomorphic to a regular curve. A related result is the example of Keldys [55* *] of a one-dimensional continuum with a zero-dimension and open mapping onto the squar* *e. The motivation for the proof of The Hilbert-Smith Conjecture is the remarkab* *le work of A.V. Chernavsky [27]. He proved that if f is a finite-to-one open and closed* * mapping on a connected metric manifold Mn onto a Hausdorff space Y , then (1) there is a natural number k so that for each x 2 Mn , the cardinality o* *f f-1 f(x) k (bounded multiplicity). (2) the elements of maximal multiplicity form a dense open set in Mn , and (3) for each open set U of Mn , there is ffl > 0 such that if f is any fi* *nite-to-one open and closed mapping of Mn onto some Hausdorff space Y and f is n* *ot a homeomorphism, then for some x 2 U , diam f-1 f(x) ffl (Newman's Prop* *erty). In 1960, C.T. Yang [44] proved that if a p-adic group, Ap, acts effectively * *as a transfor- mation group on X (a locally compact Hausdorff space of homology dimension not * *greater than n), then the homology dimension of the orbit space X=Ap is not greater tha* *n n + 3. If X is an n-manifold, then the homology dimension of X=Ap is n+2. If Ap acts s* *trongly effectively (freely) on an n-manifold X , then the dimension of X=Ap is either * *n + 2 or infinity. At about the same time (1961), Bredon, Raymond, and Williams [25] pro* *ved the same results using different methods. There are, of course, actions by p-adic * *groups on p-adic solenoids and actions by p-adic solenoids on certain spaces. See [25] f* *or some of these results. 3 In 1961, Frank Raymond published the results of his study of the orbit space* * M=Ap assuming an effective action by Ap (as a transformation group) on an n-manifold* * M . Later (1967), Raymond [38] published work on two problems in the theory of gene* *ralized manifolds which are related to the (generalized) Hilbert Fifth Problem. In 1963, Raymond and Williams [39] gave examples of compact metric spaces Xn* * of dimension n and an action by a p-adic group, Ap, on Xn such that dim Xn =Ap = n* * + 2. Work related to and used in [39] is the paper [41] by Williams. In [41], Willia* *ms answers a question of Anderson [24; p. 799] by giving a free action by a compact 0-dime* *nsional group G on a 1-dimensional Peano continuum P with dim P=G = 2. In 1976, I described [32; 33] what I called p-adic polyhedra which admit per* *iodic home- omorphisms of period p. Proper inverse systems {Pi, OEi} of p-adic ni-polyhedra* * have the property that the inverse limit X = lim-Pi admits a free action by a p-adic gro* *up. In 1980, one of my students, Alan J. Coppola [28] generalized results of C.T* *. Yang [44] which involve homologically analyzing p-adic actions. Coppola formalized these* * so that homological calculations could be done in a more algorithmic manner. He define* *d a p- adic transfer homomorphism and used it to produce all of the relevant Smith-Yan* *g exact sequences which are used to homologically analyze Zpr-actions on compact metric* * spaces. Coppola studied p-adic actions on homologically uncomplicated spaces. In partic* *ular, he proved that if X is a compact metric Ap-space of homological dimension no great* *er than n and X is homologically locally connected, then the (n + 3)-homology of any c* *losed subset A X=Ap vanishes. In 1983, Robinson and I proved Newman's Theorem for finite-to-one open and c* *losed mappings on manifolds [10]. We formalized Newman's Property (and variations) a* *nd studied this property for discrete open and closed mappings on generalized cont* *inua in 1984 [11]. In 1985, H-T Ku, M-C Ku, and Larry Mann investigated in [30] the connections* * between 4 Newman's Theorem involving the size of orbits of group actions on manifolds and* * the Hilbert-Smith Conjecture. They establish Newman's Theorem (Newman's Property [1* *1]) for actions of compact connected non-Lie groups such as the p-adic solenoid. In 1997, D. Repov~s and E.V. ~S~cepin [51] gave a proof of the Hilbert-Smith* * Conjecture for actions by Lipschitz maps. See also related work by Shchepin [52]. In the s* *ame year, Iozke Maleshick [53] proved the Hilbert-Smith conjecture for Hölder actions. In 1999, Gaven J. Martin [54] announced a proof of The Hilbert-Smith Conject* *ure for quasiconformal actions on Riemannian manifolds and related spaces. The crucial idea that works here is M.H.A. Newman's idea used in his proof t* *hat for a given compact connected n-manifold M , there is an ffl > 0 such that if h is an* *y periodic homeomorphism of period p, a prime > 1, of M onto itself, then there is some x * *2 M such that the orbit of x, {x, h(x), . .,.hp-1(x)}, has diameter ffl. It is well kn* *own that the collection of orbits under the action of a transformation group G on a compact * *Hausdorff space X is a continuous decomposition of X . The works [20; 21] of David Wilson and John Walsh [18] show that there exist* * continuous decompositions of n-manifolds Mn , n 3, into Cantor sets. This paper shows t* *hat such decompositions can not be equivalent to those induced by any action of a p* *-adic transformation group Ap acting on Mn . I owe a special debt of gratitude to Patricia Tulley McAuley who has been ex* *tremely helpful in reading drafts of numerous attempts to solve this problem and who ha* *s provided helpful insights with regard to ~Cech homology. I am not only most deeply inde* *bted to the work of A.V. Chernavsky [27] but also extremely grateful to him for the num* *erous important and helpful suggestions which he made after a careful reading of a ma* *nuscript and in stimulating conversations with him on a visit to Istanbul Bilgi Universi* *ty in January, 2002. I also wish to thank Eric Robinson, David C. Wilson, and John J. Walsh fo* *r having read earlier attempts some of which worked for n = 3. 5 OUTLINE OF THE PROOF It is well known that if a locally compact group G acts effectively on a con* *nected n- manifold M and G is not a Lie group, then there is a subgroup H of G isomorphic* * to a p-adic group Ap which acts effectively on M . Thus, the Hilbert-Smith Conject* *ure can be established by proving that there is no effective action by a p-adic group A* *p on a connected n-manifold M . The conjecture is proved by the following theorem. A* *s seen later, there is no loss of generality, in assuming that M is compact, orientabl* *e, and without boundary. The proof given below can be adopted to the situation where M is a l* *ocally compact, orientable, and without boundary by replacing finite open coverings of* * M and M=Ap with locally finite open coverings with the same properties since the orbi* *t map OE : M ! M=Ap is open, closed, and proper. Definition. An n-manifold (M, d) is said to have Newman's Property w.r.t. the c* *lass L(M, p) (as stated below) iff there is ffl > 0 such that for any OE 2 L(M; p), * *there is some x 2 M such that diam OE-1OE(x) ffl using the metric d on M . Theorem. If L(M, p) is the class of all orbit mappings OE : M ! M=Ap where Ap a* *cts effectively on a compact, connected, and orientable n-manifold M and each h 2 * *Ap is homotopic to the identity (hence, h has degree one and preserves the orientatio* *n of M ), then M has Newman's Property w.r.t. L(M, p). It is well known that M does not have Newman's Property w.r.t. L(M, p). H* *ence, L(M, p) = ; and the Hilbert-Smith Conjecture is true. Lemma 2. (A consequence of a Theorem of Floyd [4].) Suppose that M is a compa* *ct connected orientable n-manifold. There is a finite open covering W1 of M such t* *hat (1) order W1 = n+1 and (2) there is a finite open refinement W2 of W1 which covers * *M such that if W is any finite open covering of M refining W2, then ßW1 : ~Hn(M) ! Hn* *(W1) maps H~n(M) isomorphically onto the image of the projection ßWW1 : Hn(W ) ! Hn* *(W1). 6 [Here, if U is either a finite or locally finite open covering of M , then H* *n(U) is the nth simplicial homology group of the nerve N(U) of U . The coefficient group is* * always Zp and H~n(M) denotes the nth ~Cech homology of M .] Now, choose U = W1, and a finite open covering W of M which star refines W2 * *where W1 and W2 satisfy Lemma 2. Let ffl be the Lebesque number of W2. Choose OE 2 L(M, p) such that diam OE-* *1OE(x) < ffl for each x 2 M . Construct the sequences of coverings {V m} and {Um } as in Lem* *ma 5 below where V 1 star refines W2 and order Um = n + 1 along with projections ff* *m , fim , and ßm yielding the following commutative diagram: * ff* fi*m+1 Hn(V m) -fim Hn(Um ) - m Hn(V m+1) - Hn(Um+1 ) m " fi*m. - ff*m m+1 " fi*m+1. - ff*m* *+1 * ß*m+1 Hn(Vnm) ßm- Hn(Vnm+1) - Hn(Vnm+* *2) Here m is the natural map of an n-cycle in Hn(Vnm), the nth simplicial homolog* *y group of the n-skeleton of the nerve N(V m) of V m, into its homology class in Hn(V m). * *The other maps are those induced by the projections ffm , fim and ßm . The upper sequence* *, of course, yields the ~Cech homology group H~n(M) as its inverse limit. Furthermore, it ca* *n be easily shown, using the diagram, that ~Hn(M) is isomorphic to the inverse limit G = li* *m-Hn(Vnm), of the lower sequence. Specifically, fl : ~Hn(M) ! G defined by fl( ) = {fi*m(ß* *Um ( ))} is an isomorphism of H~n(M) onto G. We shall use the isomorphism in what follows a* *nd for convenience we shall let fl( ) = {znm( )}, i.e. znm( ) = fi*m(ßUm ( )) 2 Hn(Vn* *m). The group lim-Hn(Vnm) is used because the operator oe (introduced later) is applied* * to actual n-cycles rather than to elements of a homology class. This is the reason for th* *e sequence {Um }. An operator oem is defined on the n chains of N(V m+1) for each m. The oper* *ator oem maps n-cycles to n-cycles and commutes with the projections ß*m: Hn(Vnm+1) ! Hn* *(Vnm) 7 and, hence, induces an automorphism on ßV m+1(H~n(M)) Hn(Vnm+1). See Lemmas 6 and 7. Distinguished families of n-simplices in N(V m) are defined. Now, let zm =* * znm( ) where is the generator of G ~= Zp. For each n-simplex ffin in zm ( ), there* * is a unique distinguished family Smj of n-simplices in N(V m) which contains ffin . * *If Cj is the collection of all n-simplices in zm ( ) which are in Smj, then the sum of the c* *oefficients of those members of Cj (as they appear in zm ( )) is 0 mod p. Take the projection * *ßV mU from V m to U = W1. Hence, ßV mU has the property that all members of a disting* *uished family Smj of n-simplices in N(V m) project to the same simplex in N(U1). Thus* *, the projection of those members of zm ( ) which are in Smj project to the same simp* *lex ffij in N(U1) and the coefficient of ffij is 0 mod p. Thus, ßV mU : Hn(Vnm) ! Hn(U) tak* *es the nontrivial n-cycle zm ( ) to the 0-n cycle mod p. This violates the conclusion * *of Lemma 2. Thus, M has Newman's Property w.r.t. the class L(M, p). Hence, ffl is a Ne* *wman's number and the Theorem is proved. It is well known that if Ap acts effectively on a compact connected n-manifo* *ld M , then given any ffl > 0, there is an effective action of Ap on M such that diam OE-1O* *E(x) < ffl for each x 2 M . That is, M fails to have Newman's property w.r.t. the class L(M, p* *). It fol- lows that Ap can not act effectively on a compact connected n-manifold M . Cons* *equently, the Hilbert-Smith Conjecture is true. Details of the proof follow. 2. Some properties of the orbit mapping of an effective action by Ap on a compa* *ct connected orientable n-manifold M . Suppose that OE is the orbit mapping of a p-adic group Ap acting effectively* * as a transformation group on an orientable n-manifold Mn = M where p is a prime lar* *ger than 1. By [12; 21], there is a sequence Ap = H0 H1 H2 . .o.f open (and c* *losed) 8 subgroups of Ap which closes down on the identity e of Ap such that when j > i,* * Hi=Hj is a cyclic group of order pj-i. Let hij : Ap=Hj ! Ap=Hi and hi : Ap ! Ap=Hi be homomorphisms induced by the inclusion homomorphisms (quotient homomorphisms) on Ap and Ap=Hj. Then {Ap=Hi; hij} is an inverse system and {hi} gives an isomorph* *ism of Ap onto lim-Ap=Hi. Now, let a 2 Ap - Hi. For each natural number i, let ai b* *e the coset aHi in Ap=Hi. Then ai is a periodic homeomorphism of M=Hi onto M=Hi with aqibeing the identity mapping where q = pi is the period of ai. Consequently, H* *i acts as a transformation group on M and Ap=Hi acts as a cyclic transformation group on * *M=Hi. As above, let {Hi} be a sequence of open (and closed) subgroups of Ap such t* *hat (a) Hi Hi+1 for each i, (b) if j i, then Hi=Hj is a cyclic group of order pj-i,* * and (c) Ap=Hi is a cyclic group of order pi. Since Ap acts effectively on M (a compact * *connected n-manifold), the cyclic group Ap=Hi acts effectively on M=Hi with orbit space M* *=Ap. It follows that if ffl > 0, then there is a natural number j such that Hj ~=* * Ap acts effectively on M such that if OEj : M ! M=Hj is the orbit map of the action, th* *en diam OE-1OE(x) < ffl for each x 2 M . Observe that if H is a non empty open and clos* *ed subgroup of Ap, then for some i, H = Hi. Observe that if Ap acts effectively as a transformation group on M where M * * is an orientable connected metric n-manifold, then some orbit is infinite and not dis* *crete. This follows from a Theorem of Chernavsky [27] that if OE : M ! Y (Hausdorff) is a d* *iscrete open and closed (continuous) mapping, then there is a natural number k such that car* *dinality of OE-1OE(x) k for each x 2 M (bounded multiplicity). Furthermore, the union of * *the orbits of maximal cardinality is a dense open set W in M . The stability group of all * *the points in W is a certain Hv and, thus, Hv acts as the identity on M . Consequently, th* *e action of Ap is not effective contrary to the hypothesis. Thus, the orbit mapping OE :* * M ! M=Ap where Ap acts effectively on M is not a discrete open and closed mapping. Hence* *, some orbit is infinite and not discrete. 9 The following lemma is crucial to defining certain coverings of M with disti* *nguished families of open sets. Lemma 1. Suppose that OE is the orbit mapping OE : M ! M=Ap where Ap acts effec* *tively on a compact orientable connected n-manifold M and each h 2 Ap is homotopic to the identity (hence, h has degree one and preserves the orientation of M ). F* *or each z 2 M=Ap - OE(FOE) and ffl > 0 where FOE= {x | OE-1OE(x) = x}, there is a conne* *cted open set U such that (1) diam U < ffl, (2) z 2 U , and ps[ (3) OE-1(U) = Ui where s is a natural number such that i=1 (a) Ui is a component of OE-1(U) for each i, 1 i ps, (b) ~Ui\ ~Uj= ; for i 6= j , (c) OE(Ui) = U for each i, and (d) U1 is homeomorphic to Uj for each j , 1 < j ps (by maps compatib* *le with the projections OE | Uj). [The homeomorphism taking U1 to Uj is a * *power of a fixed element g 2 Ap - H1 where {Hi} closes down on the ident* *ity in Ap and g preserves the orientation of M . This g is used in Lemma * *6.] Proof. Since z 2 M=Ap - OE(FOE), OE-1(z) is non degenerate. It follows from Wh* *yburn's theory of open mappings and light open mappings [19] that for ffl > 0, there is* * a connected open set U such that (1) diam U < ffl, (2) z 2 U , and (3) OE-1(U) consists of* * a finite number (larger than one) of components U1, U2, . .,.Um such that ~Ui\ ~Uj= ; f* *or i 6= j , and OE(Ui) = U for each i. For each Uj, a component of OE-1(U), there is an open and closed subgroup Gj* * of Ap which is the largest subgroup of Ap which leaves Uj invariant and the map induc* *ed by OE maps Uj=Gj onto U . Since Gj is a normal subgroup of Ap, Gi = Gj for each i and* * j . 10 Furthermore, Ap=Gj is a cyclic group of order ps where s is a natural number. T* *here are ps pairwise distinct components of OE-1(U). (See [36: Lemma 2]). It follows tha* *t G1 = Hi for some i where {Hi} is the sequence of open and closed subgroups of Ap which * *closes down on the identity e 2 Ap (mentioned above) and s = i. Let a 2 Ap - Hi such t* *hat aHi generates the cyclic group Ap=Hi. For each natural number i, let ai be the* * coset aHi in Ap=Hi. Thus, ai is a periodic homeomorphism of M=Hi onto M=Hi with aqi= e where q = pi is the period of ai. Let fi: M ! M=Hi be the orbit map of the action of Hi on M and gi: M=Hi! M=Ap be the orbit map of the action of the cyclic group Ap=Hi on M=Hi. That is, OE =* * gifi. pi There are pi cosets vm Hi m=1 where v1 = e (the identity) such that for each * *x 2 U1, [pi (1) OE-1OE(x) = vm Hi(x) where vm Hi(x) = {h(x) | h 2 vm Hi}, m=1 (2) vm Hi(x) 2 Um =Hi, (3) vm is an orientation preserving homeomorphism of M onto M , and i-1 (4) if Ap=Hi= {ai, a1i, a2i, . .,.api } (a cyclic group), then there are e* *lements k1, k2, . .,.kpi where k1 = e such that (a) km (ßi(x)) = ßi(vm Hi(x)) where ßi maps OE-1(U) onto OE-1(U)=Hi and (b) km maps U1=Hi homeomorphically onto Um =Hi with km = vm Hi2 Ap=Hi which is a homeomorphism of M=Hi onto M=Hi. Thus, vm (x) 2 Um . Let z 2 Um . Hence, ßi(z) 2 Um =Hi and (vm Hi)-1(ßi(z))* * = v-1mHi(ßi(z)) 2 U1=Hi. Consequently, v-1mHi(ßi(z)) = ßi(v-1m(z)) 2 U1=Hi which * *implies that v-1m(z) 2 U1. Finally, vm (v-1m(z)) = z and vm maps U1 homeomorphically o* *nto Um . Lemma 1 is proved. 3. Special coverings, distinguished families, and distinguished subfamilies. 11 Let L(M, p) = {OE | OE is the orbit mapping of an effective action of a p-ad* *ic group Ap (p a prime with p > 1) on a compact connected metric orientable n-manifold with* *out boundary, OE : M ! M=Ap}. For each OE 2 L(M, p), let FOE= {x | x 2 M and OE-1OE* *(x) = x}, the fixed set of the action Ap. It would simplify the proof of lemmas which* * follow to know that M is triangulable. Without this knowledge, a theorem of E.E. Floyd is* * used. Notation. Throughout this paper, H~n(X) will denote the nth ~Cech homology grou* *p of X with coefficients in Zp, the integers mod p, p a fixed prime larger than 1. A* *lso, Hn(K) will denote the nth simplicial homology of a finite simplicial complex K , with* * coefficients in Zp. If U is a finite open covering of a space X , then N(U) denotes the ne* *rve of U , Hn(U) is the nth simplicial homology group of N(U), and ßU the usual proje* *ction homomorphism ßU : ~Hn(X) ! Hn(U). Definition. If f is a mapping of M onto Y , then an open covering U of M i* *s said to be a saturated open covering (more precisely, saturated w.r.t. f ) iff for e* *ach u 2 U , f-1 f(u) = u. That is, u is an open inverse set. The next lemma follows. Lemma 2. Suppose that M is a compact connected metric n-manifold. There is a fi* *nite open covering W1 of M such that (1) order W1 = n + 1 and (2) there is a finite open refinement W2 of W1 which covers M such that if* * W is any finite open covering of M refining W2, then ßW1 : H~n(M) ! Hn(* *W1) maps H~n(M) isomorphically onto the image of the projection ßWW1 : Hn(* *W ) ! Hn(W1). Proof. Adapt Theorem (3.3) of [4] to the situation here and use (2.5) of [4]. If M is triangulable, then there is a sufficiently fine triangulation T such* * that if U con- sists of the open stars of the vertices of T , then ßU : ~Hn(M) ! Hn(U) is an i* *somorphism onto (where, of course, H~n(M) ~=Zp). 12 Standing Hypothesis: In the following, M is a compact, connected, and orient* *able metric n-manifold. Also, L(M, p) is the class of all orbit mappings OE : M ! M=* *Ap where Ap acts effectively on M and each h 2 Ap is homotopic to the identity (hence, * *h has degree one and preserves the orientation of M ). The finite open coverings W1 * *and W2 which satisfy Lemma 2 will be used in certain lemmas and constructions which fo* *llow. 1 Suppose also that Y = M=Ap has a countable basis Q = Bi i=1 such that (a) for * *each i, Bi is connected and uniformly locally connected and (b) if H is any subcolle* *ction of Q " " and h 6= ;, then h is connected and uniformly locally connected (a cons* *equence of h2H h2H a theorem due to Bing and Floyd [50]). All of the open sets used in Y below to* * construct coverings Rn of Y not related to OE(F ) are in Q. Lemma 3. Suppose that OE 2 L(M, p) and F = FOE. Then there is a finite open irr* *educible covering WF which covers F such that (1) WF star refines W2 (as in Lemma 2) and if w 2 WF , then OE(w) 2 Q, (2) order WF n + 1, (3) WF is a saturated open covering of F , (4) if BdF = @F = F - interiorF and W@F = {w | w 2 WF and w \ B 6= ;}, then order W@F n, (5) for w 2 WF such that w \ intF 6= ;, then either w 2 W@F or int F ~w* *, and (6) if ffin is an n-simplex in the nerve, N(WF ), the nerve of WF , then th* *e nucleous, __* *____ N[ffin], of ffin lies in int F (the interior of F ), indeed, int F N[* *ffin]. "n NOTE: An n-simplex is {u0, u1, . .,.un} = ffin where ui2 WF and N[ffin] = * * uj. j=0 Proof. If int F = ;, then dim F n - 1 and there is a finite irreducible open * *covering WF0 of F such that order WF0 n and has properties (1), (4), (5), and (6). Fo* *r each w0 2 WF0, let w = union of all OE-1OE(x) such that x 2 w0 and w0 OE-1OE(x). I* *t follows 13 that w is open [19] since OE is open and closed. Hence, WF = {w | w02 WF0} sati* *sfies (1) - (6). Assume that int F 6= ;. Cover @F = BdF with a finite irreducible collect* *ion C of open saturated sets such that order C n and C star refines W2. Cover F - * *C* (C* denotes the union of the elements of C ) with a finite irreducible collecti* *on D of open saturated sets such that order D n + 1, D star refines W2, and if d 2 D * *, then int F ~d. Let C = {c1, c2, . .,.cs} and D = {d1, d2, . .,.dt}. By a Theorem* * [47, p. 158, use the corresponding theorem for compact metric spaces], there is an open* * covering V = {vi| 1 i s + t} such that order V n + 1, ci vi for 1 i s, and di* * vi for 1 i t. Let WF = {wi | wi is the union of all OE-1OE(x) such that x 2 vi, 1 i s * *+ t, and vi OE-1OE(x)}. Clearly, WF satisfies (1)-(6). The lemma is proved. A proof of the necessity of the Theorem in 47 for compact metric spaces. A c* *ompact metric space (X, d) has dimension n if and only if for each finite irreducibl* *e open covering U1, U2, . .,.Uk, there is a finite open covering V1, V2, . .,.Vk such * *that (1) Ui Vi for each i, 1 i k , and (2) order {V1, V2, . .,.Vk} n + 1. Proof of the necessity. Suppose that dim X n. Given U = {U1, U2, . .,.Uk},* * a finite irreducible covering of X . By a Theorem in 47, there is a finite irreducible o* *pen refinement G = {G1, G2, . .,.Gm } such that (1) for each i, there is some j such that Ui * * Gj and Ut 6 Gj for any t 6= i and (2) order G n + 1. Let V1 be the union of all Gi * *such that U1 Gi and let Vi be the union of all Gj such that Ui Gj and Ut 6 Gj for 1 * * t < i, that is, Vt 6 Gj. It follows that order V n + 1 where V = {V1, V2, . .,.Vk}.* * Clearly, Ui Vi for 1 i k . Next, we extend WF to a special covering V of M (defined below) by coverin* *g M - [ w in a special way. Recall The Standing Hypothesis. We use the following l* *emma. w2WF 14 Lemma 4. Suppose that OE 2 L(M, p). Then there exists a finite open covering R* * of Y = M=Ap such that (1) if y 2 OE(F ) where F = FOE, then there is r 2 R such that y 2 r = OE(w* *) for some w 2 WF , (as described in Lemma 3), [ (2) if OE(WF ) = OE(w) and y 2 Y - OE(WF ), then there is r 2 R such t* *hat w2WF y 2 r , r 2 Q where Q is the basis in The Standing Hypothesis, ~r\ OE(F* * ) = ;, OE-1(r) = r1 [ r2 [ . .[.rq, q = pt for some natural number t, such tha* *t for each i = 1, 2, . .,.q , ri is a component of OE-1(r), ri maps onto r * * under OE, __r __ \irj= ; for i 6= j , and ri is homeomorphic to rj for each i and j wi* *th a homeomorphism compatible with the projection OE (indeed, there is an or* *ientation preserving homeomorphism, an element of Ap, which takes ri onto rj), (3) R is irreducible, and (4) if rx 2 R and rx 6= OE(w) for any w 2 WF , ry 2 R, rx \ ry 6= ;, OE-1(r* *x) consists of exactly pmx components, OE-1(ry) consists of exactly pmy componen* *ts, and mx my, then each component of OE-1(ry) meets exactly pmx-my componen* *ts of OE-1(rx). Proof. Obtain WF using Lemma 3. Since Y - OE(WF ) is compact, use Lemma 1 to ob* *tain a finite irreducible covering R0 of Y - OE(WF ) of sets r satisfying the condit* *ions of the lemma such that R0 star refines {OE(u) | u 2 W2}. Property (4) of the conclusi* *on of Lemma 4 is satisfied by using the compactness of Y and choosing R0 such that e* *ach r 2 R0 has sufficiently small diameter and r 2 Q (the basis in The Standing Hyp* *othesis). Let R = R0[ {OE(w) | w 2 WF }. The lemma is established. Let V 0= {~c| c is a component of OE-1(r) for some r 2 R such that r 6= OE(w* *) for any w 2 WF } [ {w~| w 2 WF } star refines W2, and V = {v | ~v2 V 0}. The irreducibl* *e finite open covering V = V 1of M which contains WF = WF1generated by the irreducible o* *pen covering R of Y in Lemma 4 is just the first step in establishing Lemma 5 belo* *w. 15 [t1i 1 Definition. For each ri 2 R1, OE-1(ri) = f1ijand f1ijtij=1= Fi1is called a* * distin- j=1 guished family of open sets in V 1 where f1ijis a component of OE-1(ri). If r 6* *= OE(w) for any w 2 WF1, then t1i= psi where si is a natural number. If r = OE(w) for some * *w 2 WF1, then t1i= 1. Lemma 5. There are sequences {V m} and {Um } of finite open coverings of M cofi* *nal in the collection of all open coverings of M such that (1) V m+1 star refines Um , (2) V 1 star refines W2 of Lemma 2, (3) Um star refines V m, (4) order Um = n + 1, (5) {mesh V m} ! 0 (6) V m is generated by a finite open covering Rm of Y = M=Ap, and if WFm * *is the subcollection of V m which covers F = FOE, then V m, WFm, and Rm have * *the properties stated in Lemma 4 where Rm replaces R, V m replaces V , and* * WFm replaces WF , (7) there are projections ßm : V m+1 ! V m such that (a) ßm = fim ffm where ffm : V m+1 ! Um and fim : Um ! V m, m+1 tm+1 (b) ßm takes each distinguished family fij ji=1 in V m+1 (defined * *in a man- ner like those defined for V 1 and V 2 below) onto a distinguished* * family tm fmsjjs=1in V m, (c) ßm extends to a simplicial mapping (also, ßm ) of N(V m+1) into N* *(V m) such that if ffin is an n-simplex in N(V m+1) and if ßm (ffin) = o* *en , an n- simplex in N(V m), then N[ffin] \ F 6= ; if and only if N[oen] \ * *F 6= ;. (Also, ffm and fim denote the extensions of ffm and fim to sim* *plicial map- pings ffm : N(V m+1) ! N(Um ) and fim : N(Um ) ! N(V m)* * where 16 ßm = fim ffm .) (d) ßm : V m+1 ! V m is equivariant relative to the natural actions o* *f certain cyclic groups whose orders are powers of p and these groups are ge* *nerated by some g 2 Ap, g not the identity. Also, ßm induces ß*m : N(V m* *+1) ! N(V m) which is equivariant relative to some cyclic group Zps whic* *h acts on N(V m+1) and projects to Zpt which acts on N(V m). The proof of Lemma 5, although straightforward, is long and tedious. The exi* *stence of V = V 1 in Lemma 4 (which star refines W2) generated by R = R1 is an initial st* *ep of a proof using mathematical induction. Additional first steps are described belo* *w. These should help make it clear how the induction is completed to obtain a proof of L* *emma 5. The finite open covering V can be partitioned into either the distinguished* * families t1 Fi1= f1ij}ji=1the elements of which are the components of OE-1(ri) for some ri* * 2 R1 with ri6= OE(w) for each w 2 WF = WF1and t1i= psior distinguished families of s* *ingletons {w} where w 2 WF1. As defined above, V = V 1 is generated by R = R1. Clearly, V 1 star refines W2. Observe that VF1= {v | v 2 V 1 and v \ F 6= ;}* * = WF1 and that VB1= {v | v 2 VF1 and v \ BdF 6= ;} = WB1. Also, order VB1 n and ord* *er VF1 n + 1. Note that order V 1may be larger than n + 1 since if OE is the orbi* *t mapping of an effective action by a p-adic transformation group, then dim Y = n + 2 or * *1 [22]. The covering V 1 is defined to be a special covering of M w.r.t. OE generat* *ed by R. Of course, OE is fixed throughout this discussion as in the statements of Lemma* *s 3 and q s 4. Recall that it follows from Lemma 1, that if f1kjj=1 and f1mj j=1 are t* *wo non singleton families (those containing more than one element) in V 1 such that fo* *r some i s and t, f1ki\ f1mt6= ;, then for each j , the number of elements of f1mj j=1 wh* *ich have a non empty intersection with f1kjis a constant ck and for each j , the number of* * elements of q f1kjj=1 which have a non empty intersection with f1mjis a constant cm where c* *k = pb, 17 b 0, and cm = pd, d 0. A singleton family (which contains exactly one eleme* *nt) either meets each member of a non singleton family or meets no member of a non singlet* *on family. Construction Of U1 Of Order n + 1 Which Refines V 1 The reason that the sequence {Um } is constructed is to prove (using the def* *initions of ffm , fim , and ßm = fim ffm ) that the inverse limit of the nth simplicial ho* *mology of the n-skeleta of the nerve of V m is Zp which permits the application of oe (define* *d below) to actual n-cycles. The operator oe can not be applied (as defined) to elements of* * a homology group. The next step is to describe a special refinement U1 of V 1 which has order * *n + 1 and other crucial properties. First, construct an auxiliary covering ^U1. List the non degenerate distinguished families of V 1 as F11, F21, . .,.Fn11* *where Fi1= t1 f1ijji=1where t1i= pbi for some bi. Recall that f1ijis homeomorphic to f1itfor* * each i, j , and t that makes sense. Since R1 (which generates V 1) is irreducible, it f* *ollows that if f1ij2 Fi1and f1st2 Fs1where Fi1and Fs1are distinguished families in V 1 with* * i 6= s, then f1ijand f1stare independent, that is, f1ij6 f1stand f1st6 f1ij. For each i, 1 i n1, choose a closed and connected subset KYi in OE(f1i1)* * = r1i2 R1 t1 where Fi1= f1ijji=1such that (1) KMij= OE-1(KYi) \ f1ij, KMijis homeomorphic to KMisfor any s and j that* * makes sense, (2) KM = {intKMij| 1 i n1 and 1 j t1i} covers M - [WF1 and K~Y = [ {intKYi| 1 i n1} covers Y - OE(w), and w2W1F (3) KMijis connected, 1 j t1i. To see that this is possible, choose a closed subset Ai of r1i, 1 i n1, * *such that [ A = {Ai | 1 i n1} covers Y - OE(w) = Y 0[47; 49]. Note that there exi* *sts w2W1F a natural number k such that {Ai = r1i- N_1k(@r1i) | 1 i n1} covers Y 0. T* *o see 18 n1 [ this, suppose that for each k , there is xk 62 (r1i- N_1k(@r1i)). Since Y 0* *is compact, i=1 there is a subsequence {xn(k)} of {xk} which converges to x 2 r1qfor some q . T* *here is some m such that x 2 r1q- N_1_m(@r1q) and x 2 interior(r1q- N__1_m+1(@r1q)) whi* *ch leads to a contradiction. Choose pi 2 r1iand let Cm (pi) be the component of r1i- N_1_m(@r* *1i) which 1[ contains pi. It will be shown that Cm (pi) = r1i. Suppose that there is q 2* * r1isuch 1 m=1 [ that q 62 Cm (pi). Since r1iis uniformly locally connected and locally compa* *ct (r~1iis a m=1 Peano continuum), there is a simple arc piq from pi to q in r1i. Consequently, * *for some m, r1i-N_1_m(@r1i) piqwhich is in Cm (pi). This is contrary to the assumption ab* *ove. Hence, 1[ Cm (pi) = r1i. For pi 2 r1ifixed as above, choose pij2 OE-1(pi) \ f1ijfor 1* * j t1i. m=1 Let Cim(pij) be the component of OE-1(Cm (pi)) which is in f1ijand contains pj.* * It will 1[ * * [1 be shown that Cim(pij) = f1ij. If this is false, then there is a qj 2 f1ij-* * Cim(pij). m=1 * * m=1 Since f1ijis ulc and locally compact, there is a simple arc pijqj from pij to q* *j in f1ij. Now, r1i OE(pijqj). For m large enough, Cm (pi) OE(pijqj) and some componen* *t of OE-1OE(pijqj) contains pijqj and lies in Cim(pij). This is contrary to the assu* *mption above. 1[ Hence, Cim(pij) = f1ij. m=1 Let KYi = intCm (pi) where m is sufficiently large that (1) and (3) are sati* *sfied and also (4) int Cm (pi) Ai. Then (2) is also satisfied, that is, the components o* *f OE-1(Cm (pi)) may be taken as KMij, 1 i n1. Next, shrink the elements of OE(WF1) = {OE(w) | w 2 WF1} as follows: Order * *WF1 as w11, w12, . .,.w1n2and choose a natural number s such that {Kn1+i = OE(w1i)-N1_* *s(@OE(w1i)) | [ n1[ 1 i n2} has the property that {intKn1+i | 1 i n2} covers OE(w)- * *intKYi. w2W1F i=1 It is well known that a metric space X has dim X n if and only if X has a * *sequence {Gi} of open coverings of X such that 19 (1) Gi+1 refines Gi for each i, (2) order Gi n + 1 for each i, and (3) mesh Gi< 1_i. If X is a manifold, then the elements of Gi can be chosen to be connected and u* *niformly locally connected. If X is a triangulable n-manifold, then it is easy to see th* *is by using barycentric subdivisions of a triangulation of X . Choose such a sequence {Gi} * *of open coverings of M such that {OE(g) | g 2 G1} star refines KY = {intKYi| 1 i n1* * + n2}. The next step is to show how to choose some Gi from which U^1 will be chosen* * and later modified to give U1 with the desired properties. Let ffl be the Lebesque number of the covering KY . Choose Gt, mesh Gt < 1_t* *, such that diam OE(g) < ffl_8for g 2 Gt. Define G(y) = {g | g 2 Gt and y 2 OE(g)}. Statement 1. If y 2 Y , then there is s, 1 s n1 + n2, such that int KYs [ ____ OE(g). g2G(y) Proof. It follows from the choice of ffl, t. Choose ^U1 Gt such that ^U1 is an irreducible finite covering of M . Define U^(y) = {g | g 2 ^U1 and y 2 OE(g)}. Since Gt is an open covering of M with connected open sets, mesh Gt < 1_t, and * *order Gt n + 1, we obtain: Statement 2. U^1 has the following properties: (1) order ^U1 n + 1 and ^U1 star refines V 1, (2) if u 2 ^U1, then u is connected, and [ ____ (3) if y 2 Y then there is s, 1 s n1 + n2, such that int KYs OE(* *g). g2U^(y) 20 Construction Of U1 Which Refines ^U1 In A Special Way For any collection S of sets, let [S be the union of sets in S and \S be the* *ir inter- section. For each y 2 Y , let Q(y) = OE(U^(y)) = {OE(u) | u 2 ^U1and y 2 OE(u)}. Ther* *e are at most a finite number of such sets distinct from each other. Order these sets as Q1, * *Q2, . .,.Qm1 [ ____ such that for i < j , Qi 6= Qj and card Qi card Qj. Let Oi = \Qi- (\Qj ) wh* *ere j card Dj and Bs 2 Di for some i, 1 i j , and if h 2 H0j, then OE(h)\ Bs * *= ;. This yields Property (5). Furthermore, if h 2 H0i, 1 < i j , then OE(h) \ (\D* *t) = ; for [j 1 t < i (Property (6)). Consequently, H0ihas Properties (2), (3), and (5)-* *(8) where i=1 in (5), 1 t j . 22 j [ Use Nagata's Theorem as in the proof of Lemma 3 to shrink each element of * * H0i i=1 j[ j[ to obtain H00i, 1 i j , such that H00icovers OE-1(\Di) and has Proper* *ty (4) i=1 i=1 * * [j while keeping properties (2), (3), and (5)-(8). For convenience of notation, su* *ppose H0i * * i=1 [j has properties (2)-(8) and covers OE-1(\Di). It follows by mathematical indu* *ction that m i=1 [0 H00= H0icovers OE-1(B) and has properties (2)-(8). There is no loss of gene* *rality in i=1 assuming that if h 2 H00and h \ intF 6= ;, then either int F ~hor h \ @F 6= ;* * and that if h 2 H00and h \ (M - F ) 6= ;, then either M - F ~hor h \ @(F ) 6= ;. Note: A finite open covering C of a closed subset N of M such that dim N = n* * - 1 where the elements of C are open relative to N and order C = n can be extended* * to a collection C0 of open sets in M such that card C = cardC0, C0 covers M and or* *der C0 = n. See [48]. Let HF0= {h | h 2 H00and h \ F 6= ;}. Let HF0(@F ) = {h | HF0 and h \ @F 6= * *;}. Let W1 be a finite irreducible open covering of @F such that W1 HF0(@F ), sta* *r refines both U^1 and WF1, and if w 2 W1 - HF0(@F ), then w~\ OE-1(B) = ;. By the use of Nagata's Theorem as in the proof of Lemma 3, there is an open irreducible refin* *ement H000 of H00- ({h | h 2 H00, h \ F 6= ;, and h \ @F = ;} [ W1) such that (1) order H0* *00= n, (2) if H000(@F ) = {h | h 2 H000and h \ @F 6= ;}, then order of H000(@F ) = n, * *and (3) H000 covers OE-1(B) - intF and has Properties (2)-(8) above. Now, let W2 be an irred* *ucible [ open covering of F - H000(@F ) such that if w 2 W2 then int F w . Again, by* * the use of Nagata's Theorem as in the proof of Lemma 3, there is an open irreducible co* *vering Q x of OE-1(B) [ F such that (1) W 2F= {q | q 2 Q and q \ F 6= ;} has order n + 1, x x (2) W 2@F= {q(q 2 W 2F and q \ @F 6= ;} has order n, [ x [ x (3) H0 = {q | q 2 Q and q \ (OE-1(B) - W 2F) 6= ;} covers OE-1(B) - W 2F,* * (4) order 23 H0 = n, and (5) H0 has properties (1)-(8) above. x Cover M - ([H0 [ ([W 2F)) with a finite irreducible collection C of open se* *ts which star refines U^1. By the method used in the proof of Lemma 3, the elements of * *H0, x C , and W 2Fcan be shrunk to collections H00, C0, and W^2F, respectively, such* * that (9) H00[C0 = G covers M -[W^2Firreducibly, W^2Fcovers F , order G n+1, order H00 * * n, H00covers OE-1(B) - [W^2F, if g 2 G - H00, then ~g\ OE-1(B) = ; and ~g\ F = ;, * *order (G[W^2@F) n+1, order W^2@F n, and order (G[W^2F) n+1. For the sake of nota* *tion, x suppose that H0 = H00and W 2F= ^WF2(Here, the elements of W^2Fmay not be satur* *ated, x * * x and by the notation the elements of W 2F= ^WF2may not be saturated, i.e., if w* *02 W 2F, then w0 may not equal OE-1OE(w0).). x x Recall that if ffin is an n-simplex in N(W 2F), the nerve of W 2F, then N* *[ffin], the ______ nucleous of ffin has the property that int F N[ffin]. Suppose, without loss o* *f generality, ______ that if ffin is an n-simplex in N(G), then N[ffin]\ F = ;. This property may be* * obtained in the construction above mimicing the proof of Lemma 3. x Now, change H0 into a cover H00of OE-1(B) - [W 2Fadding to it some new open* * sets, as follows (but keeping Properties (1)-(9)). ________ Let n-1 = {N[ffin-1\]OE-1(B) | ffin-1 is an (n - 1)-simplex in N(H0)}. Co* *ver the elements of n-1 with a finite irreducible collection Hn-1 of open sets such th* *at (1) if h 2 Hn-1 , then diam h < 1_t0and H0[ Hn-1 has Property (5) of H0 abo* *ve, (2) if h 2 Hn-1 , then [H0 h, (3) if h 2 Hn-1 and g 2 G - H0, then ~g\ ~h= ;, (4) if d 2 n-1 , then at most one member of Hn-1 meets d (and, therefore, * *contains d), (5) the closures of the members of Hn-1 are pairwise disjoint, (6) order {h - [ n-1 | h 2 H0} n - 1, (7) order (H0[ Hn-1) n + 1, and 24 ______ (8) if ffin is an n-simplex in N(H0[ Hn-1), then N[ffin]\ F = ;. _______ [ For each i, 1 < i n, let n-i = {(N[ffin-i]- ([Hn-j)) \ OE-1(B) | ffin-i* * is an j card Dt. Hence, k < t. This contradicts the choice of t. By Property * *(5) of ____ H , OE(u)\ Bs = ;. The Lemma is proved. If (7) is false, then there exists u 2 U10, y 2 \Dt for the smallest t, and * *there exists h 2 ^U1such that y 2 OE(h) and OE(h) 6 OE(u). This implies that there exists B* *s such that @OE(h) Bs and OE(u)\Bs 6= ; by Property (10). Clearly, Bs 62 Dt since OE(h)\(* *\Dt) 6= ;. 26 This contradicts the Lemma which states that OE(u) \ Bs = ;. It is not difficul* *t to see that U10has properties (1)-(10). x x Let U1 = U10[ W 2F. This cover has order n + 1 and W 2@Fis the subcollectio* *n of U1 ___* *___ which covers BdF = @F . Also, if ffin is an n-simplex in N(U1), then either N[f* *fin]\F = ; ______ or int F N[ffin]. Observe that if u 2 U10and OE(u) \ B = ;, then for some i, Oi OE(u) by the * *definition of O ; otherwise, u 2 H . Construction of V 2 Which Refines U1 Next, construct V 2. Recall that for each y 2 Y we have defined U^(y) = {u * *| u 2 ^U1 [ ____ and y 2 OE(u)}. There is some s, 1 s n1 + n2, such that KYs OE(u). (* *And u2U^(y) every KYs is a slightly shrunken member of the covering of Y , which generates * *V 1). x Let WF2= {w | w0 2 W 2Fand w is the union of all OE-1OE(x) such that x 2 w0* * and w0 OE-1OE(x), that is, w is saturated and OE-1OE(w) = w}. For each y 2 B - OE* *([WF2), choose r2y2 Q (the basis for Y described above) such that (1) y 2 r2y, 0 1 0 * * 1 " " (2) if U(y) = {u | u 2 U1 (not ^U1) and y 2 OE(u)}, then @ OE(u)A\@ * * OE(u)A u2U(y) u2* *U^(y) ~r2y, (3) diam r2y< (1_8) min{æ(y, @OE(v)) | v 2 ^U1and y 62 @OE(v)}, and (4) OE-1(r2y) = r2y1[ r2y2[ . .[.r2yq, q = ptq where tq 1, r2yimaps onto * *r2yunder OE, ~r2yi\ ~r2yj= ; for i 6= j , and r2yiis homeomorphic to r2yjfor each i * *and j with a homeomorphism compatible with the projection OE (indeed, there is an el* *ement of Ap which takes r2yionto r2yj). It follows that for each such y , there is u 2 U10 U1 such that u ~r2y. See * *the proof of Lemma 4 for the existence of r2y. Let R21denote a finite irreducible collection of such sets r2ywhich covers B* * - OE([WF2). 27 If y 2 Y and y 62 [R21, and y 62 [OE(WF2), then choose r2ysatisfying (1)-(4) a* *bove such that ~ry\ B = ; and let R22denote a finite irreducible cover of Y - ([R21[ ([OE* *(WF2)), consisting of such r2y. Let R2 = R21[ R22[ OE(WF2) which is an irreducible cove* *r of Y . Define V 2= {c | c is a component of OE-1(r2yi) for some i, where r2yi2 R21[* * R22} [ OE(WF2), which is an irreducible cover of M that star refines V 1 and U1 . The * *collection t2 of components, f2ijji=1, of OE-1(r2yi), r2yi2 R21[ R22, is a non degenerate di* *stinguished family in V 2, whereas each w 2 WF2, where w = OE-1OE(w), is a singleton distin* *guished family in V 2. Definitions Of ff1, fi1, And ß1 = fi1ff1 * * t2 Case (1): Take any r2i= r2yi2 R21chosen for some yi2 B-[OE(WF2). Let Fi2= f* *2ijji=1 be the (non degenerate) distinguished family in V 2 generated by r2yi. Now, H0* *1is the x subcollection of H which covers0OE-1(\D1)0-1([W 2F)1and H0iis the subcollectio* *n of H i-1[ x [ which covers OE-1(\Di) - @ @ ([H0j)A [ ([W 2F)A. Either (a) yi 2 \Dt - ([* *H0j) j=1[ j 1, then Xpk Xp 2pX X3p (pk-1)pX ci= ci+ ci+ ci+ . .+. ci with pk-1 summations each of len* *gth i=1 i=1 i=p i=2p+1 i=(pk-1-1)p p. Now, ci cj mod p if i j mod p gives that each of the pk-1 summations is Xpk _Xp ! congruent to 0 mod p. Thus, ci = pk-1 ci 0 mod p if k > 1. If k = 1, * *then i=1 i=1 40 p X xq = ciffii, i=1 oe(xq)= c1(ffi1 + ffi2 + . .+.ffip) + c2(ffi2 + ffi3 + . .+.ffip+1) .. . + cp(ffip + ffi1 + ffi2 + . .+.ffip-1), rearrange as (c1 + c2 + . .+.cp)ffi1+ (c1 + c2 + . .+.cp)ffi2+ .. . (c1 + c2 + . .+.cp)ffip, Xp and ci 0 mod p. i=1 X9 It is instructive to consider a simple example. Let p = 3 and xq = ciffi* *i. Thus, i=1 oe(xq) = c1(ffi1 + ffi2 + ffi3) + c2(ffi2 + ffi3 + ffi4) + c3(ffi3 + ffi4 + ffi* *5) + c4(ffi4 + ffi5 + ffi6) + c5(ffi5 + ffi6 + ffi7) + c6(ffi6 + ffi7 + ffi8) + c7(ffi7 + ffi9 + ffi9) + c8(c8 + c9 + c* *1) + c9(c9 + ffi1 + ffi2) = (by rearrangement) = (c1 + c8 + c9)ffi1 + (c1 + c2 + c9)ffi2 + (c1 + c2 + c3)ffi3 +* * (c2 + c3 + c4)ffi4 + (c3 + c4 + c5)ffi5 + (c4 + c5 + c6)ffi6 + (c5 + c6 + c7)ffi7 + (c6 + c7 + c8)ff* *i8 + (c7 + c8 + c9ffi9. For each i, 1 i 9, the coefficient of ffii = 0 mod 3. Observe that from the* * coefficients of ffi1 and ffi2, it follows that c2 c8 mod 9. The coefficients of ffi2 and * *ffi3 yield that c3 c9 mod 3. Continuing, c1 c4, c2 c5, c3 c6, c4 c7, c5 c8, c6 c9* *, and c7 c1 all mod 3. Thus, (c1 + c2 + c3) + (c4 + c5 + c6) + (c7 + c8 + c9) = 0 m* *od 3 since (c4 + c5 + c6) (c1 + c2 + c3) mod 3, (c7 + c8 + c9) (c4 + c5 + c6) (c1 + * *c2 + c3) mod X9 3 and ci 3(c1 + c2 + c3) 0 mod 3. i=1 Case (b): oem (xq) = xq. Choose notation as in Case (a). Write oe(xq) as in * *Case (a), but in this case, xq = oe(xq) rather than 0 = oe(x). Consider first the example p =* * 3 and xq = 41 9 X ciffii where k = 2. Now, oe(xq) = (c1+c8+c9)ffi1+(c1+c2+c9)ffi2+(c1+c2+c3)ff* *i3+(c2+ i=1 c3+c4)ffi4+(c3+c4+c5)ffi5+(c4+c5+c6)ffi6+(c5+c6+c7)ffi7+(c6+c7+c8)ffi8+(c7+c8+c* *9)ffi9 = xq = c1ffi1+c2ffi2+c3ffi3+c4+ffi4+c5ffi5+c6ffi6+c7ffi7+c8ffi8+c9ffi9. This is a* *n identity. Thus, the coefficient of ffii on one side is equal mod p to the coefficient of ffii on th* *e other side. Hence, c1 + c8 + c9 c1 mod p, c1 + c2 + c9 c2 mod p, c1 + c2 + c3 c3 mod p, and * *so forth. X9 Thus, ci (c1+c8+c9)+(c1+c2+c9)+(c1+c2+c3)+(c2+c3+c4)+(c3+c4+c5)+(c4+ i=1 c5+c6)+(c5+c6+c7)+(c6+c7+c8)+(c7+c8+c9) = 3(c1+c2+c3+c4+c5+c6+c7+c9) 0 mod p. Consider the general case as in Case (a) but with xq = oe(xq) rather than 0 * *= oe(xq). Hence, xq = oe(xq)= (c1 + cpk + cpk-1 + . .+.cpk-p+2)ffi1+ (c2 + c1 + cpk + . .+.cpk-p+1)ffi2+ .. . (cp + cp-1 + cp-2 + . .+.c1)ffip+ (cp+1 + cp + cp-1 + . .+.c2)ffip+1+ .. . pkX (cpk + cpk-1 + . .+.cpk-p+1)ffipk = ciffii. i=1 It follows from this identity that the coefficient of ffii on one side is equal* * mod p to the Xpk Xpk coefficient of ffii on the other side. Consequently, ci = p ci mod p = 0 * *mod p p _ i=1 ! i=1 _ ! X Xp Xp as claimed where k > 1. For k = 1, xq = ciffii = ci ffi1 + ci ffi* *2 + . .+. _ p ! i=1 i=1 _ i=1 ! X Xp Xp Xp ci ffip, ct ci mod p for each t, 1 t p, and ci p ci mod* * p = 0 i=1 i=1 i=1 i=1 mod p. Lemma 8 is proved. Lemma 9. If W is any domain in M (connected non-empty open set), then H~n(M - 42 W ) = 0 and if 2 ~Hn(M) with 6= 0, then there is a natural number m0 such t* *hat for Xr each m > m0, the carrier of zm ( ) meets W , that is, if zm = gjffinjwhere 1* * gj < p, j=1 then for some j , N[ffinj] \ W 6= ; where N[ffinj] is the nucleus of ffinj. Proof. Here, the fact that H~n(M - W ) = 0 is used. Suppose that the lemma is * *false. There is a subsequence {V mi} cofinal in the collection of all open coverings o* *f M such that zmi( ) is an n-cycle in N(V mi | (M - W )). Thus, zmi( ) is not a bounding chain in N(V mi) and, hence not a bounding chain in N(V mi| (M - W )). Thus, zm* *i( ) determines a non zero element in Hn(V mi | (M - W )) and {zmi( )} determines a * *non zero element of H~n(M - W ). This yields a contradiction. The lemma is proved. Lemma 10. [cf. 27] If K = interior FOE, then K = ; and FOEis nowhere dense. Proof. Suppose that K 6= ;. Let be a non zero n-cycle in H~n(M) ~=Zp and znm(* * ) = fi*mßUm ( ) 2 Hn(Vnm). Let znm( ) = Cm1+ Cm2 where Cm1 is an n-chain such that * *each n-simplex in Cm1 has a nucleus in M -K~ and Cm2 is an n-chain such that each n-* *simplex in Cm2 has a nucleus in K . It will be shown that both Cm1 and Cm2 are n-cycle* *s. Let z = znm( ). Xk Let z = ciffini. If int FOE N[ffini] for some i, then gs(ffini) = ffinif* *or each s. Suppose i=1 that ffinjshares an (n - 1)-face ffin-1 with ffiniwhere int FOE6 N[ffinj]. Now* *, ffinjbelongs to a nondegenerate distinguished family Smj of n-simplices in N(V m). Furthermore* *, each member of Smj shares the same (n - 1)-face ffin-1 . Let Cj = {ffinj1, ffinj2, * *. .,.ffinjq} denote the collection of all n-simplices in Smj such that ffinjt, 1 t q , is in z * *(with a non zero Xq coefficient). By Lemma 8, cjt= 0 mod p. Thus, the coefficient of ffin-1 in @* *Cm1 is 0 t=1 mod p. To show that @Cm1 is 0, it suffices to show that each such (n - 1) simplex f* *fin-1 in @Cm1 which is a face of some ffiniwhere int F N[ffini] is 0. As shown above, this* * is the case and Cm1 is an n-cycle. Thus, @Cm1 = 0 and Cm1 is an n-cycle. It follows that * *Cm2 is 43 an n-cycle. From the definition of the special projections, ß*m(Cm+1i) = Cmi fo* *r i = 1, 2. Thus, we can write = 1 + 2 where znm( i) = Cmi for i = 1, 2. Now, the nucle* *us of each simplex in znm( 2) misses the nonempty open set M - ~Kfor each m, so by Le* *mma 7, 2 = 0. Similarly, the nucleus of each simplex in znm( 1) misses the non-emp* *ty open set K for each m. Hence, by Lemma 7, 1 = 0. Consequently, we have = 0, a contradiction. The lemma is proved. 5. A proof that a p-adic group Ap can not act effectively on a compact connect* *ed n- manifold where OE : M ! M=Ap is the orbit mapping. Remarks. If the compact connected n-manifold M has a non empty boundary, then two copies of M can be sewed together by identifying the boundaries in such a w* *ay that the result is a compact connected n-manifold M0 without boundary. If Ap acts ef* *fectively on M , then Ap acts effectively on M0. If M is not an orientable n-manifold, th* *en we can take the double cover of M on which Ap acts effectively if it acts effectively * *on M . There is no loss of generality in assuming that M is a compact connected orientable n* *-manifold without boundary. Definition. An n-manifold (M, d) is said to have Newman's Property w.r.t. the * *class L(M, p) (as stated above) iff there is ffl > 0 such that for any OE 2 L(M; p), * *there is some x 2 M such that diam OE-1OE(x) ffl. Generalizations can be made to metric spaces (X, d) which are locally compac* *t, con- nected, and lcn [4] which have domains D such that D~ is compact, lcn , and Hn(* *X, X - D), Zp) ~=Zp. Theorem. If L(M, p) is the class of all orbit mappings OE : M ! M=Ap where Ap a* *cts effectively on a compact, connected, and orientable n-manifold M , then M has N* *ewman's Property w.r.t. L(M, p). 44 Proof. There is no loss of generality in assuming that M is orientable and has* * empty boundary. By hypothesis, H~n(M) ~=Zp. Consider a finite open covering U = W1 where W1 * *and W2 satisfy Lemma 2. If z( ) is the V -coordinate of a non-zero n-cycle 2 H~n* *(M) where V refines W2, then ßV Uz( ) 6= 0. Let ffl be the Lebesque number of W2. * *Choose OE 2 L(M, p) such that diam OE-1OE(x) < ffl for each x 2 M . Construct the spec* *ial coverings {V m} and the special refinements {Um } as in Lemma 5 such that the star of eac* *h distin- guished family in V 1lies in some element of W2. Furthermore, the special proje* *ctions ßm tm are such that if ffinsjjs=1is a distinguished family of n-simplices in N(Vnm),* * then ßV mU takes ffinsj, 1 j ts, to the same simplex ffis in N(U). Now, let zm = znm( * *). Xk Let zm = ciffini. By Lemma 10, F \ N[ffini] = ; for each i since int F = * *;. Hence, for i=1 each j , 1 j k , ffinjis in a non degenerate distinguished family Smj of n-* *simplices in N(V m). Let Cj = {ffinj1, ffinj2, . .,.ffinjq} denote the collection of all n-s* *implices in Smj such * * Xq that ffinjiappears in zm for 1 i q with non zero coefficients. By Lemma 8,* * cji= 0 * * i=1 mod p. Since the n-simplices in Cj are sent by ßV mU to a single simplex ffij i* *n N(U), it follows that the coefficient of ffij is 0 mod p and, therefore, z is sent by ßV* * mU to the zero n-cycle in N(U). Thus, the projection of zm by ßV mU : Hn(Vnm) ! Hn(U) takes * *the nontrivial n-cycle zm ( ) to the 0 n-cycle mod p. This violates the conclusion * *of Lemma 2. Thus, M has Newman's Property w.r.t. the class L(M, p). Hence, ffl is a Ne* *wman's number and the Theorem is proved. It is well known that if Ap acts effectively on a compact connected n-manifo* *ld M , then given any ffl > 0, there is an effective action of Ap on M such that diam OE-1O* *E(x) < ffl for each x 2 M . That is, M fails to have Newman's property w.r.t. L(M, p). It foll* *ows that Ap can not act effectively on a compact connected n-manifold M . 45 6. How to obtain a proof that a p-adic group can not act effectively on a conn* *ected n- manifold. As indicated above, there is no loss of generality in assuming that M is a c* *onnected orientable n-manifold without boundary. If Ap acts effectively on M (which is* * locally compact), then the orbit map OE : M ! M=Ap is open and closed with OE-1OE(x) co* *mpact for each x 2 M . Hence, OE is a proper map (if M=Ap A and A is compact, then * *OE-1(A) is compact). Construct sequences {V m} and {Um } of locally finite open coverings of M b* *y con- structing locally finite open coverings Rm of M=Ap in the same manner as in Le* *mma 3, 4, and 5 where each r 2 Rm has a compact closure. Since OE is proper, the dist* *inguished families of open sets in V m generated by members of Rm have the same properti* *es as in Lemmas 1, 3, 4, and 5. The proof follows as in the compact case. Consequently, the Hilbert Smith Conjecture is true. 46 References 1.Bochner, S. and Montgomery, D., Locally compact groups of differentiable tra* *nsformations, Annals of Math 47 (1946), 639-653. 2.Bredon, Glen E., Orientation In Generalized Manifolds And Applications To Th* *e Theory of Trans- formation Groups, Mich. Math. Jour. 7 (1960), 35-64. 3.Browder, F., editor, Mathematical development arising from Hilbert Problems,* * Northern Ill. Univ., 1974, XXVIII, Parts I and II, Proceedings of Symposia in Pure Mathematics. 4.Floyd, E.E., Closed coverings in ~Cech homology theory, TAMS 84 (1957), 319-* *337. 5.Gleason, A., Groups without small subgroups, Annals of Math. 56 (1952), 193-* *212. 6.Greenberg, M., Lectures On Algebraic Topology, New York, 1967. 7.Hall, D.W. and Spencer II, G.L., Elementary Topology, John Wiley & Sons, 195* *5. 8.Hilbert, David, Mathematical Problems, BAMS 8 (1901-02), 437-479. 9.Lickorish, W.B.R., The piecewise linear unknotting of cones, Topology 4 (196* *5), 67-91. 10.McAuley, L.F. and Robinson, E.E., On Newman's Theorem For Finite-to-one Open* * Mappings on Manifolds, PAMS 87 (1983), 561-566. 11.______, Discrete open and closed mappings on generalized continua and Newman* *'s Property, Can. Jour. Math. XXXVI (1984), 1081-1112. 12.Michael, E.A., Continuous Selections II, Annals of Math. 64 (1956), 562-580. 13.Montgomery, D. and Zippin, L., Small groups of finite-dimensional groups, An* *nals of Math. 56 (1952), 213-241. 14.______, Topological Transformation Groups, Wiley (Interscience), N.Y., 1955. 15.Newman, M.H.A., A theorem on periodic transformations of spaces, Q. Jour. Ma* *th. 2 (1931), 1-9. 16.Robinson, Eric E., A characterization of certain branched coverings as group* * actions, Fund. Math. CIII (1979), 43-45. 17.Smith, P.A., Transformations of finite period. III Newman's Theorem, Annals * *of Math. 42 (1941), 446-457. 18.Walsh, John J., Light open and open mappings on manifolds II, TAMS 217 (1976* *), 271-284. 19.Whyburn, G.T., Analytic Topology, AMS Colloq. Publications 28 (1942). 20.Wilson, David C., Monotone open and light open dimension raising mappings, P* *h.D. Thesis, Rutgers University, 1969. 47 21.______, Open mappings on manifolds and a counterexample to the Whyburn Conje* *cture, Duke Math. Jour. 40 (1973), 705-716. 22.Yang, C.T., Hilbert's Fifth Problem and related problems on transformation g* *roups, Proceedings of Symposia in Pure Mathematics; Mathematical development arising from Hilbert * *Problems, Northern Illinois University, 1974, edited by F. Browder, XXVIII, Part I, 1976, pp. 1* *42-164. 23.______, p-adic transformation groups, Mich. Math. J. 7 (1960), 201-218. Some Additional References To Work Associated With Efforts Made Towards Solving Hilbert's Fifth Problem (General Version) 24.Anderson, R.D., Zero-dimensional compact groups of homeomorphisms, Pac. Jour* *. Math. 7 (1957), 797-810. 25.Bredon, G.E., Raymond, F., and Williams, R.F., p-adic groups of Transformati* *ons, TAMS 99 (1961), 488-498. 26.Brouwer, L.E.J., Die Theorie der endlichen kontinuierlichen Gruppen unabhagi* *n von Axiom von Lie, Math. Ann. 67 (1909), 246-267 and 69 (1910), 181-203. 27.Chernavsky, A.V., Finite-to-one open mappings of manifolds, Trans. of Math. * *Sk. 65 (107) (1964). 28.Coppola, Alan J., On p-adic transformation groups, Ph.D. Thesis, SUNY-Bingha* *mton, 1980. 29.Kolmogoroff, A., Über offene Abbildungen, Ann. of Math. 2(38) (1937), 36-38. 30.Ku, H-T, Ku, M-C, and Mann, L.N., Newman's Theorem And The Hilbert-Smith Con* *jecture, Cont. Math. 36 (1985), 489-497. 31.Lee, C.N., Compact 0-dimensional transformation group, (cited in [23] prior * *to publication, perhaps, unpublished). 32.McAuley, L.F., Dyadic Coverings and Dyadic Actions, Hous. Jour. Math. 3 (197* *7), 239-246. 33.______, p-adic Polyhedra and p-adic Actions, Topology Proc. Auburn Univ. 1 (* *1976), 11-16. 34.Montgomery, D., Topological groups of differentiable transformations, Annals* * of Math. 46 (1945), 382-387. 35.Pontryagin, L., Topological Groups, (trans. by E. Lechner), Princeton Math. * *Series Vol. 2, Princeton University Press, Princeton, N.J., 1939. 36.Raymond, F., The orbit spaces of totally disconnected groups of transformati* *ons on manifolds, PAMS 12 (1961), 1-7. 37.______, Cohomological and dimension theoretical properties of orbit spaces o* *f p-adic actions, Proc. Conf. Transformation Groups, New Orleans, 1967, pp. 354-365. 38.______, Two problems in the theory of generalized manifolds, Mich. Jour Math* *. 14 (1967), 353-356. 48 39.Raymond, F. and Williams, R.F., Examples of p-adic transformation groups, An* *nals of Math. 78 (1963), 92-106. 40.von Neumann, J., Die Einfuhrung analytischer Parameter in topologischen Grup* *pen, Annals of Math. 34 (1933), 170-190. 41.Williams, R.F., An useful functor and three famous examples in Topology, TAM* *S 106 (1963), 319-329. 42.______, The construction of certain 0-dimensional transformation groups, TAM* *S 129 (1967), 140- 156. 43.Yamabe, H., On a conjecture of Iwasawa and Gleason, Annals of Math. 58 (1953* *), 48-54. 44.Yang, C.T., Transformation groups on a homological manifold, TAMS 87 (1958),* * 261-283. 45.______, p-adic transformation groups, Mich. Math. Jour. 7 (1960), 201-218. 46.Zippin, Leo, Transformation groups, "Lectures in Topology", The University o* *f Michigan Conference of 1940, 191-221. 47.Nagata, J., Modern Dimension Theory, revised and extended edition, Helderman* *n Verlag, Berlin, 1983. 48.Wilder, R.L., Topology of Manifolds, AMS Colloquium Publications, Vol. 32 (1* *949). 49.Nagami, K., Dimension Theory, Academic Press, N.Y., 1970. 50.Bing, R.H. and Floyd, E.E., Coverings with connected intersections, TAMS 69 * *(1950), 387-391. 51.Repov~s, D. and ~S~cepin, A proof of the Hilbert-Smith conjecture for action* *s by Lipschitz maps, Math. Ann. 308 (1997), 361-364. 52.Shchepin, E.V., Hausdorff dimension and the dynamics of diffeomorphisms, Mat* *h Notes 65 (1999), 381-385. 53.Maleshick, Iozke, The Hilbert-Smith conjecture for Hölder actions, Usp. Math* *. Nauk 52 (1997), 173- 174. 54.Martin, Gaven J., The Hilbert-Smith conjecture for quasiconformal actions, E* *lectronic Res. Announce- ments of the AMS 5 (1999), 66-70. 55.Keldys, L.D., Example of a one-dimensional continuum with a zero-dimensional* * and interior mapping onto the square, Dokl. Akad. Nauk SSSR (N.S.) 97 ((1954)), 201-204. Istanbul Bilgi University 49