Light Open Mappings On Compact n-Manifolds Do Not Raise Dimension And A Proof Of The Hilbert-Smith Conjecture by Louis F. McAuley Suppose that M is a compact n-manifold and OE is a light open mapping of M o* *nto a metric space Y . It is shown that dim Y = n. The symbol æ is used for the metric on both M and Y . Recall that OE is ligh* *t iff for each x 2 M , OE-1OE(x) is totally disconnected. The following lemma is crucial to defining certain coverings of M with disti* *nguished families of open sets. Lemma 1. Suppose that OE is a light open mapping of a compact connected n-manif* *old M onto a metric space Y . For each z 2 Y and ffl > 0, there is a connected ope* *n set U such that (1) diam U < ffl, (2) z 2 U , and [s (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 s, (b) ~Ui\ ~Uj= ; for i 6= j , and (c) OE(Ui) = U for each i. Proof. It follows from Whyburn's theory of light open mappings [5; p. 148] 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) 1 consists of a finite number of components U1, U2, . .,.Us such that ~Ui\ ~Uj= ;* * for i 6= j , and OE(Ui) = U for each i. Standing Hypothesis: In the following, M is a compact metric n-manifold. Sup* *pose 1 also that Y has a countable basis Q = Bi i=1 such that (1) for each i, Bi is * *connected " and uniformly locally connected and (b) if H is any subcollection of Q and * *h 6= ;, " h2H then h is connected and uniformly locally connected (a consequence of a the* *orem due h2H to Bing and Floyd [1]). All of the open sets used in Y below to construct cov* *erings of Y are in Q. The metric æ is chosen such that for each ffl > 0 and x 2 M , Nff* *l(x), the ffl-neighborhood of x, is connected. Similarly, for y 2 Y , Nffl(y) is connecte* *d. One can use the alternative below and assume that n > 1. Alternative Standing Hypothesis: In the following, M is a compact metric n- 1 manifold, n > 1. Suppose also that Y has a countable basis Q = Bi i=1 such th* *at for each i, Bi is connected and has a connected boundary (a consequence of a Theore* *m of Jones [3]). All of the open sets used in Y below to construct coverings of Y * * are in Q. The metric æ is chosen such that for each ffl > 0 and x 2 M , Nffl(x), the ffl-* *neighborhood of x, is connected. Similarly, for y 2 Y , Nffl(y) is connected [5]. If one uses The Alternative Standing Hypothesis, then make note of the follo* *wing: It is known that light open mappings on compact metric 1-manifolds are finit* *e-to-one and do not raise dimension [5]. Furthermore, light open mappings on compact me* *tric n-manifolds do not lower dimension [4]. Consequently, if f is a light open map* *ping on a compact metric n-manifold M onto a metric space Y with n > 1, then Y does n* *ot have any local separating points and, hence, by the theorem of Jones [3] has a * *basis of connected open sets with connected boundaries. Lemma 2. Suppose that OE is a light open mapping of M onto Y and G is an open covering of Y . Then there exists a finite open covering R of Y which refines * *G such that 2 (1) if y 2 Y , then there is r 2 R such that y 2 r , r 2 Q where Q is the b* *asis in The Standing Hypothesis, OE-1(r) = r1[ r2[ . .[.rq, for some natural number* * q , such that for each i = 1, 2, . .,.q , ri is a component of OE-1(r), ri maps * *onto r under OE, and __ri\ __rj= ; for i 6= j , and (2) R is irreducible. Proof. Since Y is compact, use Lemma 1 to obtain a finite irreducible covering* * R of Y of sets r satisfying the conditions of the lemma. Let V 1= {c | c is a component of OE-1(r) such that r 2 R = R1}. [t1i 1 Definition. For each r1i2 R1, OE-1(r1i) = f1ijand f1ijtij=1= Fi1is called a* * distin- j=1 guished family of open sets in V 1 where f1ijis a component of OE-1(r1i). Construction Of U1 Of Order n + 1 Which Star Refines V 1 List the distinguished families of V 1as F11, F21, . .,.Fn11where the degene* *rate families, if any, are listed last in the ordering. Recall from the Standing Hypothesis that Y has a countable basis Q with cer* *tain properties. Use Q to obtain a finite open star refinement R^ Q of R1 which c* *overs Y such that each r 2 ^Ris connected and uniformly locally connected (ulc) and * *inherits certain other properties from Q. Let U^1= {c | c is a component of OE-1(r) where r 2 ^R}. Clearly, U^1 star r* *efines V 1. Recall also that if c is a component of OE-1(r), then OE(c) = r . 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) = {r | r 2 ^Rand y 2 r}. There are at most a finit* *e number of such sets distinct from each other. Order these sets as Q1, Q2, . .,.Qm1 such t* *hat for i < j , [ ____ Qi6= Qj and card Qi card Qj. Let Oi= \Qi- (\Qj ). For each i, 1 i m1, let j cardDi and Bs 2 Dj for some j , 1 j < i. 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 [6]. The open coverings below of subsets N of M such that dim N n* * - 1 consist of open subsets of M . q1 Let D1i i=1 be the collection of all Dt such that card Dt = c1 = cardD1 (ma* *ximum cardinality). Observe_that (\D1i) \ (\D1j) = ; for i 6= j . q1[ ! Cover OE-1 (\D1i) with a finite irreducible open covering H1 such that i=1 (1) if h 2 H1, then diam OE(h) < e, (2) H1 star refines ^U1 and OE(H1) = {OE(h) | h 2 H1} star refines R^, * * _____ (3) if H1i is the subcollection of H1 which covers OE-1(\D1i) irreducibly, * *then [H1i \ _____ [H1j = ; for i 6= j , and (4) if for each i, 1 i q1, h 2 H1i, w 2 H1i, and u 2 ^U1such that (a) OE(h) \ OE(u) 6= ;, 4 (b) OE(h) \ OE(w) 6= ;, and (c) Bs 62 D1ifor all Bs such that @OE(u) Bs (alternatively, OE(u)\(\* *D1i) 6= ;), then OE(u) OE(h) [ OE(w). Now, (1) and (2) should be clear. To see (3), observe that if i 6= j , then OE-1(\D1i) and OE-1(\D1j) are disj* *oint compact sets. To see (4), assume the hypothesis of (4). For each h 2 H1i there exists y 2* * OE(h) \ (\D1i). It follows from (1) that Ne(y) OE(h). Since Bs 62 D1i, æ(\D1i), Bs) >* * 4e. Thus, OE(u) N4e(y) since N4e(y) is connected. Also, diam OE(w) < e and OE(w) \ OE(* *h) 6= ;. Hence, OE(u) OE(h) [ OE(w) since OE(u) N4e(y) OE(h) [ OE(w). Let c1, c2, . .,.cm4 denote the cardinal numbers (distinct from each other)* * of the car- dinality of the collections Dt, 1 t m3, such that ci> ci+1 for 1 i < m4. qi Let Dij j=1 denote the collection of all Dt such that card Dt = ci. If Hj* *-1 has been defined, let fflj > 0 be such that fflj < (1_4) min{e, fflj-1, ffit, 1 t* * < j , where ffit = æ æ ` t-1[ ' fifi oe æ ` t-1[ (1_4) min æ \ Dti, Y - OE([Hs) fifi1 i qt and min æ \ Dia- O* *E([Hs), s=1 s=1 t-1[ ' fifi` t-1[ ' ` t-1[ ' oeoe \Djb- OE([Hs) fifi\ Dia- OE([Hs) \ \ Djb- OE([Hs) = ; . Let Hj* * be s=1 s=1 ` s=1 [qj ' j-1[ a finite irreducible open covering of OE-1 (\Dji) - ([Ht) such that i=1 _q t=1 ! j-1[ (1) if h 2 Hj, then diam OE(h) < fflj and OE(h) \ (\D(j-1)i) = ;, i=1 (2) Hj star refines ^U1 and OE(Hj) = {OE(h) | h 2 Hj} star refines OE(U^1) * *= ^R, j-1[ (3) if Hji is the subcollection of Hj which covers OE-1(\Dji) - ([Ht) irr* *educibly, _____ _____ t=1 then [Hji \ [Hjs = ; for i 6= s, and (4) if for each i, 1 i qi, h 2 Hji, w1 2 Hji, w2 2 Hji, and u 2 ^U1such* * that (a) OE(h) \ OE(u) 6= ;, (b) OE(h) \ OE(w1) 6= ;, 5 (c) OE(w1) \ OE(w2) 6= ;, and ` * * j-1[ (d) Bs 62 Djifor all Bs such that @OE(u) Bs (alternatively, OE(u)\ * *\Dji- ' * * t=1 OE([Ht) 6= ;); then OE(u) OE(h) [ OE(w1) [ OE(w2). j-1[ * *j-1[ To see (3), observe that if i 6= s, then OE-1(\Dji)- ([Ht) and OE-1(\Djs)-* * ([Ht) t=1 * *t=1 are disjoint compact sets. To see (4),_assume the hypothesis of (4). Since h 2 Hji, it follows that th* *ere is j-1[ ! _ j-1[ ! z 2 OE(h) \ \Dji- OE([Ht) . Now, æ \Dji- OE([Ht), @OE(u) > 4fflj. Eit* *her (a) t=1 ____ t=1 OE(u) N4e(z) or (b) Y -OE(u) N4fflj(z) since N4fflj(z) is connected. Since O* *E(u)\OE(h) 6= ; and diam OE(h) < fflj, it follows that OE(u) N4fflj(z) and OE(u) OE(h). By* * hypothesis, diam OE(wi) < fflj, i 2 {1, 2}. Hence, OE(u) OE(h) [ OE(w1) [ OE(w2). m4[ It follows by mathematical induction that H = Hi exists which covers OE-* *1(B) i=1 irreducibly such that _ qj-1[ ! (1) if h 2 Hj, then diam OE(h) < fflj, OE(h) \ (D(j-1)i) = ;, i=1 0 * * 1 qi[ (2) for each i, 1 < i m4, Hi is an irreducible open covering of OE-1 @ * *(\Dij)A - j=1 i-1[ _ q1[ ! ([Ht) and H1 is an irreducible open covering of OE-1 (\D1i) , t=1 i=1 (3) Hj star refines ^U1 and OE(Hj) star refines OE(U^1) = ^R, j-1[ * * _____ (4) if Hjiis the subcollection of Hj which covers OE-1(\Dji)- ([Ht), then* * [Hji \ _____ t=1 [Hjs = ; for i 6= s, and (5) if for each i, 1 i qj, h 2 Hji, w1 2 Ht, j t m4, w2 2 Hs, j s* * m4, and u 2 ^U1such that (a) OE(h) \ OE(u) 6= ;, (b) OE(h) \ OE(w1) 6= ;, (c) OE(w1) \ OE(w2) 6= ;, and 6 * * ` (d) Bs 62 Dji for all Bs such that @OE(u) Bs (alternatively, OE(u) \* * \ Dji- j-1[ ' OE([Ht) 6= ;; then OE(u) OE(h) [ OE(w1) [ OE(w2). t=1 It should be clear from the arguments above how to obtain properties (1)-(5). The next step is to shrink H to H0 which has order n and retains Properties * *(1)-(5). Order the elements of H as {h11, h12, . .,.h1x1; h21, h22, . .,.h2x2; . .,.h* *m41, hm42, . .,.hm4xm4 } where the elements of Hi are ordered before the elements of Hi+1, 1 i < m4. A* * corol- lary to a Theorem [10; p. 90] states that: A metric space (X, d) has covering d* *imension n if and only if X has a sequence {Gi} such that (1) for each i, Gi is an ope* *n covering of X , (2) for each i, Gi+1 refines Gi, and (3) {mesh Gi} ! 0. It follows that * *if (X, d) is a compact metric space with dim X n and Q is any finite irreudcible coveri* *ng of X , then there is a natural number i such that Gi refines Q where {Gi} is the squen* *ce in the statement above from [10]. Now, dim OE-1(B) n - 1. Using [10], it follows that there exists a finite * *irreducible open covering G = {g1, g2, . .,.gm } of OE-1(B) (the elements of G are open in * *M ) with sufficiently small mesh such that 0 1 m4[ (1) the subcollection G1 of G which covers OE-1(B)\@ [H1 - ([Hj)A refi* *nes H1, 0j=1 0 * *1 1 qi[ * * i-1[ (2) if i > 1, then the subcollection G2 of G which covers @ OE-1 @ (\Dij)* *A - ([Gt)A- j=1 * * t=1 m4[ m4[ ([Hj) [where ([Hj) = ; when i = m4] refines Hi, j=i+1 j=i+1 (3) for each i and k , there is some j such that hki gj and hst6 gj for a* *ny ordered pair (s, t) 6= (k, i) (this follows using the irreducibility of H ), and (4) order G n. Let h011be the union of all gi such that h11 gi and let h01ibe the union o* *f all gj such that h1i gj and h1t 6 gj for 1 t < i, that is, h01t6 gj. It follows t* *hat H01= 7 _ q1 ! [ {h011, h012, . .,.h01q1} is an irreducible open cover of OE-1 \ D1i . Cont* *inue. Let h021 i=1 be the union of all gi such that h21 gi and h1t6 gi, 1 t q1. For each i,* * 1 < i q2, let h02ibe the union of all gj such that h2i gj, h1t 6 gj, 1 t q1, and h* *2t 6 gj for 1_ t < i. It follows that H02= {h021, h022, . .,.h02q2} is an irreducible* * open cover of q2[ ! _ q2[ ! OE-1 \ D2i - [H01. To see that H02covers OE-1 \ D2i - [H01suppose * *that i=1 i=1 there is some x in this set0such0that x 621h02tfor1any t, 1 t q2. By Proper* *ty (2) above, q2[ m4[ G2 refines H2 and covers @ OE-1 @ (\D2j)A - [G1A - ([Hj). Now, if x 2 g 2 * *G1, j=1 j=2 0 * * 1 m4[ then x 2 h1s for some s since G1 refines H1 and covers OE-1(B) \ @ [H1 - ([H* *j)A . 0 1 0 1 j=1 m4[____ q2[ Hence, x 62 g for any g 2 G1. Recall that @ [Hj A \ OE-1 @ (\D2j)A = ;. T* *hus, by j=1 j=1 definition of G2, x 2 g for some g 2 G2 and, consequently, x 2 h2t for some t s* *ince G2 refines H2. That is, there is a smallest t such that0h2t1 g . It can be shown i* *n a similar qi[ i-1[ * *m4[ way that for each i, 1 i m4, H0icovers OE-1 @ DijA - ([H0t). Let H0 = * * H0i. j=1 t=1 * *i=1 Clearly, order H0 n since order G n and no member of G is in two different * *elements of H0t, 1 t m4. This is similar to a theorem of Nagata in [4]. It is not difficult to see that Properties (1)-(5) are true where H0ireplace* *s Hi, 1 i m4. For convenience of notation, suppose that Properties (1)-(5) are true for H* *i as stated above. Shrinking the elements of Hi as above does not violate any of the other prop* *erties. [ ____ xi Recall the definition of Qi and Oi = \Qi - (\Qj ). Let Qi = rit t=1. * *For j 0 such that (a) ffi < (1_4) min{æ(x(* *u), @u) | u 2 U1(y)} and (b) if x 2 OE-1(y), then there is some v 2 U1(y) such that x 2 v and æ(x, @v) < 4ffi . By [9; p. 78, Theorem (1.3)], it follows that r2ycan be * *chosen with sufficiently small diameter such that for each component c of OE-1(r2y), diam ~* *c< ffi . Hence, if cx(u)is the component of OE-1(r2y) which contains x(u), then (a) u ~cx(u),* * (b) diam ~cx(u)< (1_4)æ(cx(u), @u), and (c) if k is a component of OE-1(r2y), then there* * is v 2 U1(y) such that v ~kand diam ~k< (1_4)æ(~k, @v). 9 Let R21denote a finite irreducible collection of such sets r2ywhich covers B* * . If y 2 Y and y 62 [R21, then choose r2ysatisfying (1)-(5) above such that ~ry\ B = ; and* * let R22 denote a finite irreducible cover of Y - ([R21), consisting of such r2y. Let R2* * = R21[ R22 which is an irreducible cover of Y . Define V 2 = {c | c is a component of OE-1(r2yi) for some i, where r2yi2 R21* *[ R22}, which is an irreducible cover of M . Observe that Property (5) implies that V 2* *star refines U1 . Now U1 is constructed so that U1 refines U^1 which star refines V 1. Hen* *ce, U1 t2 star refines V 1. The collection of components, f2ijji=1, of OE-1(r2yi), r2yi2* * R21[ R22, is a distinguished family in V 2. Definitions Of ff1, fi1, And ß1 = fi1ff1 Case (1): yi2 B . t2 Take any r2i= r2yi2 R21chosen for yi 2 B . Let Fi2= f2ijji=1be the distingu* *ished family in V 2 generated by r2yi. Definition of eyi for yi2 B Let eyi= min{t | yi2 OE([Htq) for some unique q}. eyi-1[ Now, H(eyi)q Heyi and H(eyi)qcovers OE-1(\D(eyi)q) - Hs irreducibly. Ei* *ther s=1 (a) OE-1(yi) is covered by H(eyi)qor (b) OE-1(yi) is not covered by H(eyi)q. Case (a): By the choice of r2yiwhich generates Fi2, it follows that for each* * j , 1 j t2i, there is Uij2 H(eyi)qsuch that Uij ~f2ij. This follows from Property (5) above* *. In this case, there is no u 2 O such that u ~f2ij. Case (b): For some j , 1 j t2i, choose Uij2 H(eyi)qif possible such that* * Uij ~f2ij; if not, then choose Uij 2 Ht for the smallest t (where, of course, t > eyi) suc* *h that Uij f~2ij. By definition of eyi, there is some u 2 H(eyi)qsuch that u \ OE-1* *(yi) 6= ;. By Property (5), there is some t, 1 t t2i, such that u = Uit f~2it. If th* *ere is no Uij2 H(eyi)qsuch that Uij ~f2ij, then clearly there is u = Uij2 U1(yi) H suc* *h that 10 Uij ~f2ijby Property (5) above. Now, Uij2 Ht H for the smallest subscript t.* * By the definition of eyi, it follows that t > eyi. Definition of dyi for yi2 B Let dyi= min{j | Djt D(eyi)qfor the smallest t}. If yi62 B , then neither eyi nor dyi is defined. Definition of ^V(yi) If yi 62 B , then V^(yi) = ^U(yi). For yi 2 B , let V^(yi) = {u | u 2 ^U1 a* *nd OE(u) \D(dyi)t}. Property P If yi2 B , then ^V(yi) 6= ; and if u 2 ^V(yi), then (1) u 2 ^U(yi), that is, ^U(yi) ^V(yi), [t2i (2) OE(u) OE(Uis), s=1 (3) OE(u) \D(dyi)t, ____ (4) OE(u) [D(dyi)t, ____ (5) OE(u) [D(eyi)q, and eyi-1[ (6) OE(u) \D(eyi)q- OE([Ht). t=1 Proof. Recall that eyi= min{t | yi 2 OE([Htq) for some q}, dyi= min{j | Djt D(* *eyi)q for the smallest t, and ^V(yi) = {u | u 2 ^U1and OE(u) \D(dyi)t}. Take u 2 ^U1 such that OE(u) \ (\D(dyi)t) 6= ;. It follows that Ba 62 D(dyi* *)tfor each Ba such that @OE(u) Ba. To see this, suppose that @OE(u) \ (\D(dyi)t) 6= ;. T* *hen there exists Bx such that @OE(u) Bx and Bx \ (\D(dyi)t) 6= ;. There exists c < dyi * *such that Dcs D(dyi)tfor some unique s where Bx 2 Dcs. This contradicts the choice of dy* *i since Dcs D(dyi)t D(eyi)q. Hence, Ba 62 D(dyi)tas claimed. For any Bx 2 D(dyi)t, Bx = Bjr = (@Oj) \ @r where r 2 Qj for some j , OE(u) * * Oj, ____ and OE(u) 2 Qj. Thus, OE(u) [D(dyi)tand OE(u) \D(dyi)tsince @OE(u)\(\D(dyi)t* *) = ;. 11 This establishes (3), (4), and (5) since D(dyi)t D(eyi)q. ____ Take any such u as above. It follows that OE(u) [D(eyi)t. Observe that if* * Bs \ (\D(eyi)q) 6= ;, Bs 62 D(eyi)q, and @OE(u) Bs, then Bs 2 Dam for some a < eyi* * and the eyi-1[ smallest m. Thus, OE([Ht) \Dam . Note also that Bs 2 D(dyi)t. t=1 Now, Ba 62 D(eyi)qfor each Ba such that @OE(u) Ba. Consequently, OE(u) \* *D(eyi)q- eyi-1[ OE([Ht). Hence, (6) is established. t=1 There is0x, 1 x t2i, such that1Uix 2 H(eyi)q. Hence, OE(u) \ OE(Uix) 6=* * ; since eyi-1[ OE(Uix) \ @ \D(eyi)q- OE([Ht)A 6= ;. Take any Uiz, 1 z t2i. Let h = U* *ix, t=1 w1 = w2 = Uiz2 Hs, eyi s m4. Also, yi 2 OE(Uix) \ OE(Uiz). By Property (5) o* *f H , [t2i OE(u) OE(Uix) [ OE(Uiz) and OE(u) OE(Uit). It follows that yi 2 OE(u) an* *d u 2 ^U(yi). t=1 Thus, V^(yi) exists and U^(yi) ^V(yi). Hence, (1) and (2) are true and Prope* *rty P is established. Definition of si for yi2 B ____ t1 Let si = min{s | r1s OE(u)for all u 2 ^V(yi)}. Take Fs1i= f1sijjsi=1for t* *he given Fi2. Each f2ij2 Fi2is in one and only one member of Fs1i, say f1sizij. To see t* *hat there is a unique zij, 1 zij t1si, such that f1sizij Uij ~f2ij, recall that U1 star* * refines U^1, that U^1 star refines V 1, the definition of U(yi), and Property (2) in the con* *struction of [ ____ " " V 2. Since OE(u) r1si, r1si2 R1, OE(u) ~r2yi, OE(u) ~r2y* *i, r2yi2 R2, u2V^(yi) u2U(yi) u2U^(yi) and Uij ~f2ij, we have r1si OE(Uij) and f1sizij Uij for some unique zij. t2 Observe that the chosen collection Uij ji=1has the property that Uij 2 Ht * *where t eyi. Let ff1(f2ij) = Uij, fi1(Uij) = f1sizij, and ß1(f2ij) = fi1ff1(f2ij) = f1siz* *ij. Case (2): yi62 B . t2 Take r2yi2 R22; here yi62 B and r2yigenerates Fi2= f2ijji=1in V 2. Let si= * *min{s | 12 [ ____ 1 * * [ ____ r1s OE(u)}. Take Fs1i= f1sijtsij=1for the given Fi2, then OE(f1sij) * * OE(u)}. u2U^(yi) u* *2U^(yi) For each j , 1 j t2i, choose Uij 2 O U1 such that Uij f2ij(there is su* *ch a Uij by Property (2) of r2yiin the construction of V 2 above). Then Uij 2 U(yi)* * and there is a unique zij, 1 zij t1si, such that f1sizij Uij f2ij. Let ff1(* *f2ij) = Uij, fi1(Uij) = f1sizij, and ß1(f2ij) = fi1ff1(f2ij) = f1sizij. It will be shown now that the mappings ff1 and fi1 are well defined. We fix the choice of open sets in U1 , which are images of elements of V 2 u* *nder the mapping ff1, and the question is: whether the definition of fi1 is correct (wel* *l defined)? Suppose that fi1 is not well defined and there exist Fi2and Fk2, two differe* *nt distin- guished families in V 2 such that (a) si 6= sk (if si = sk, then fi1 is well de* *fined), Fs1iis * * t2 chosen for Fi2, Fs1kis chosen for Fk2, and (b) Uij= Ukt f2ij[ f2ktwhere Fi2= * *f2ijji=1, t2 * * ___ Fk2= f2kjjk=1, where for some j , 1 j t2i, Uij2 U1 is chosen such that Uij* * f2ij, ___ and for some t, 1 t t2k, Ukt = Uij2 U1 is chosen such that Ukt f2ktas des* *cribed above. Case A: Uij = Ukt 2 O U1 . Then yi 62 B and yk 62 B . Indeed, yi 2 Om a* *nd yk 2 Om for some m, 1 m n1. In this case, it follows from the definition * *of O that for each u 2 U^(yi), OE(u) Om and for each v 2 U^(yk), OE(v) Om . * *Thus, U^(yi) = ^U(yk) = ^V(yi) = ^V(yk), and si= sk contrary to the assumption above. Case B: Uij= Ukt2 H U1 , yi2 B , and yk 2 B . There is x, 1 x t2i, such that Uix 2 H(eyi)qand there is z , 1 z t2k* *, such that Ukz 2 H(eyk)s. Since Uij= Ukt, yi 2 OE(Uij) \ OE(Uix) and yk 2 OE(Ukt) \ * *OE(Ukz). Now, æ(\D(eyi)q, \D(eyk)s) < e since diam OE(Uix) < _1_4e, diam OE(Uij) < _1_4e* *, and diam OE(Ukz) < _1_4e. Consequently, (\D(eyi)q) \ (\D(eyk)s) 6= ; by the definition o* *f e. Observe that D(dyi)tis the collection of maximal cardinality dyi that contains De(yi)qw* *ith the smallest subscript t as all those collections of cardinality dyi are ordered) a* *nd D(dyk)ris the collection of maximal cardinality dyk that contains D(eyk)swith smallest su* *bscript r . 13 Since (\D(eyi)q) \ (D(eyk)s) 6= ;, D(dyi)t D(eyk)s, and D(dyk)r D(eyi)q, it f* *ollows that dyi= dyk and t = r . Clearly, ^V(yi) = ^V(yk) by the definitions of ^V(yi) and ^V(yk). Consequent* *ly, si = sk contrary to the assumption above. Case C: yi62 B and yk 2 B . Note that if yi62 B , then Uij62 H and if yk 2 B , then Ukt2 H . Thus, Uij6=* * Ukt. It should be clear that ff1 and fi1 are well defined. Clearly, fi1 is defined on U1 since U1 is irreducible and V 2 refines U1 . O* *bserve that ß1 maps distinguished families onto distinguished families. The Dimension of Y is n For each i, let Zsi denote the union of all Fi2, distinguished families in V* * 2, such that Fs1iis chosen for Fi2 as described above. That is, ß1(Fi2) = Fs1i. Obser* *ve that OE-1OE(Zsi) = Zsi. cardR2 "m Claim: The order Zsi i=1 n + 1. Let x 2 Zsijwhere sij6= sik, k 6= j * *. Now, j=1 for each ij, x 2 f2ijtj Uijtj= ff1(f2ijtj). By the well definedness of fi1, U* *ijtj6= Uiktk for j 6= k . Since Uijtj2 U1 and order U1 n + 1, it follows that m n + 1 an* *d order cardR2 cardR2 Zsi i=1 n + 1. Thus, Z = OE(Zsi) i=1 is a finite open covering of Y o* *f order n + 1. Clearly, if G is any finite open covering of Y , then there is R = R1 * *as defined above which covers Y and refines G. Hence, Z refines G and order Z n + 1. Th* *us, dim Y n. It is not difficult to show that if a p-adic group Ap acts effectively on an* * n-manifold M , then the orbit mapping OE : M ! M=Ap is light open and closed. Use the fact* * that there is a sequence Ap = H0 H1 H2 . . .of open (and closed) subgroups of * *Ap which closes down on the identity e of Ap such that when j > i, Hi=Hj is a cycl* *ic group of order pj-i and Ap=Hi is a cyclic group of order pi. The cyclic group Ap=Hi * *acts effectively on M=Hi with orbit space M=Ap. 14 The following theorem is a consequence of the argument above. Theorem. If a p-adic group Ap acts effectively on a compact connected n-manifol* *d, then the orbit mapping OE : M ! M=Ap = Y is a light open mapping and dim Y = n. The Hilbert-Smith Conjecture 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. 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 [6], then there is a subgroup H of G isomor* *phic to a p-adic group Ap which acts effectively on M . Thus, the Hilbert-Smith Conj* *ecture can be established by proving that there is no effective action by a p-adic gro* *up Ap on a connected n-manifold M . C.T. Yang [7] has shown that if a p-adic group Ap acts effectively on a comp* *act n- manifold M , then the dimension of the orbit space M=Ap = Y is n + 2 or infini* *ty. This contradicts the work in this paper. Hence, there is no such action and the Hilb* *ert-Smith Conjecture is true. 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