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.
References
1.Bing, R.H. and Floyd, E.E., Coverings with connected intersections, TAMS 69 *
*(1950), 387-391.
2.Browder, F., editor, Mathematical development arising from Hilbert Problems,*
* Northern Ill. Univ.,
1974, XXVIII, Parts I and II, Proceedings of Symposia in Pure Mathematics.
3.Jones, F.B., On the Existence of A Small Connected Open Set With A Connected*
* Boundary, BAMS
68 (1962), 117-119.
4.Nagata, J., Modern Dimension Theory, revised and extended edition, Helderman*
*n Verlag, Berlin,
1983.
5.Whyburn, G.T., Analytic Topology, AMS Colloq. Publications 28 (1942).
6.Wilder, R.L., Topology of Manifolds, AMS Colloquium Publications, Vol. 32 (1*
*949).
15
7.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.
8.______, p-adic transformation groups, Mich. Math. J. 7 (1960), 201-218.
9.Whyburn, G.T., Topological Analysis, Princeton University Press, 1958.
10.Nagami, K., Dimension Theory, Academic Press, 1970.
16
*