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
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Istanbul Bilgi University
49
*