By
JAMES R. MAISSEN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
3
ACKNOWLEDGMENTS
I would like to thank my advisor, Dr. James Keesling, for his wonderful insight and
support. I am grateful to have had the pleasure of working with him. After each and
every meeting, I leave with the energy and enthusiasm that he projects to everyone
around him.
I am also grateful for all the wonderful support that I have found in the Mathematics
Department at the University of Florida. I would like to thank all the members of my
committee for all of the help that they have given to me over the years. They are part
of a wonderful tradition of nurturing mathematicians, and I can only hope to soon join
the ranks of such a loving family. In particular, I would like to thank Dr. David C. Wilson
for joining Dr. Keesling and me in discussions, as I have found that three is a perfect
number for such talks. But, moreover, I would like to thank him for not only sharing his
mathematical acumen, but also in keeping my writing on track. To Dr. Beverly Brechner,
I owe a great deal of thanks, as without her I would not have found the love for topology
that I have today. I must also thank her for all of the countless meetings we had in years
gone by, and for introducing me to the Hilbert-Smith conjecture in the first place. I would
also like to thank Dr. Alex Dranishnikov. He was such a wonderful teacher when I first
started as a graduate student in the department and solidly reaffirmed my love of the
subject. I wish to especially thank him for directing and maintaining the student topology
seminar. I always find myself looking forward to the seminar and especially the problem
session associated with it. Beyond that, I wish to thank him for being such a wonderful
resource and wealth of mathematical knowledge; he is a treasure I will sorely miss
having elsewhere. I would finally like to thank the two faculty members that have served,
sequentially, as my outside committee member. Serving as an outside member has to
be the most daunting of roles to fill on a committee, and I appreciate both of them for
doing so on my behalf. To Dr. Malay Ghosh, I wish to extend my thanks for his time and
kindness, and I am just sorry that scheduling issues prevented him from seeing me to
4
the very end with this. Thanks also go out to Dr. Paul Gader for his willingness to join
my committee so late in the process, and for being so eager and able to get up to speed
on it. I have found myself calmed in two otherwise very stressful times as a result of the
manner in which they conduct themselves, and I sincerely wish to thank both of them for
it.
I would like to thank my late mother for fostering a love of learning in me. I wish that
she could have seen this day. If she were able to hear me, I would tell her ”I love and
miss you, always.”
Lastly, but most importantly, I wholeheartedly thank my fiancee Marie. She has
turned my life around with her unending love and support. I do not think that I would
have had the strength to go back to finish my PhD without her.
5
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1 Notation and Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.3 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.4 Inverse Limit of Finite Groups and p-adic Notation . . . . . . . . . . 16
2 HISTORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 COMPACTIFICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1 Actions on the Space of Irrational Numbers . . . . . . . . . . . . . . . . . . 30 3.2 Equivariant Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Extending Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5 Rings of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5.1 Group Actions on C∗(X ) and a Metric on C∗(X ) . . . . . . . . . . . 48 3.5.2 Dimension in C ∗(X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5.3 Hilbert-Smith in C ∗(X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 PEANO SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1 Equivariant Partitions of Peano Continua . . . . . . . . . . . . . . . . . . . . 51 4.2 Lifting Arcs and Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 EXAMPLES AND INVARIANT SETS . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1 Simple Effective ANR Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2 Menger Curve Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.1 Pasnynkov Partial Product Description of µn . . . . . . . . . . . . . . 62 5.2.2 Dranishnikov’s Action on µn . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6 HILBERT SPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.1 Space of Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.2 Viewing Simple Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6
6.3 Some Special Simplices inM([0, 1],G) ≅ 2 . . . . . . . . . . . . . . . . . 78 6.4 Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2-3 Alfred Haar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2-4 J. VonNeumann, L.S. Pontrajagin, C. Chevalley, and H. Fredudenthal . . . . . . 20
2-5 A. Gleason and K. Iwasawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2-6 S. Bochner, D. Montgomery, L. Zippin, and H. Yamabe . . . . . . . . . . . . . . . 21
2-7 One of only two photos of P.A. Smith known to exist . . . . . . . . . . . . . . . . 22
2-8 M.H.A. Newman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2-10 S. Eilenberg and N.E. Steenrod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2-11 R.D. Anderson, L.V. Keldys, and D.C. Wilson . . . . . . . . . . . . . . . . . . . . . 24
2-12 C.T. Yang, F. Raymond, and R.F. Williams . . . . . . . . . . . . . . . . . . . . . . . 25
2-13 A.N. Dranishnikov and Z. Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2-14 Pictured: B.A. Pasynkov (sitting), M. Bestvina, and R.D. Anderson . . . . . . . . 27
2-15 Pictured: E.V. Scepin, D. Repovs, J.F. Nash, and J. Pardon . . . . . . . . . . . . 29
3-1 Pre-compact quotient space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3-2 Compactification of N ×Z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3-3 L. Gillman, M. Jerison, and M. Henriksen . . . . . . . . . . . . . . . . . . . . . . . 46
3-4 M.H. Stone, E. Hewitt, A. N. Kolmogoroff, and I.M. Gelfand . . . . . . . . . . . . 47
3-5 M. Katetov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5-1 Cantor Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5-2 Deformation retraction of a triangle to the Cantor Tree . . . . . . . . . . . . . . . 60
5-3 Constructing a simple closed curve with disjoint orbits . . . . . . . . . . . . . . . 68
5-4 Invariant p-adic solenoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5-5 pk invariant p-adic solenoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8
5-6 Comparing quotient spaces for free actions on µ1 . . . . . . . . . . . . . . . . . . 74
9
Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
EXTENSIONS OF GROUP ACTIONS AND THE HILBERT-SMITH CONJECTURE
By
The Hilbert-Smith Conjecture proposes that every effective compact group action
on a compact manifold is a Lie group. The conjecture is the generalization of Hilbert’s
fifth problem, and it is still open for dimensions 4 and higher. It is well known that the
conjecture is equivalent to postulating that there is no effective action of a p−adic group
on a compact manifold.
We explore several well-known examples of free and effective p-adic group actions
on spaces that are not manifolds. We provide constructions for other actions on similar
spaces by the group of p-adic numbers and other pro-finite groups. We prove that
for any Peano continuum admitting an effective p-adic action, the continuum can be
equivariantly partitioned. While the quotient map of the action is not generally a covering
map, we extend many standard results on covering maps to it.
We also present a different approach to the Hilbert-Smith conjecture by looking
internally at the space under the action of the group. We show that free p-adic actions
on the space of irrationals are unique up to conjugation. We also show that should a
counter-example to the Hilbert-Smith conjecture exist, the counter-example would be an
extension of this unique free p-adic group action on the space of irrationals.
This motivates the investigation of compact extensions of compact metric group
actions on separable metric spaces. While we show that there is always an extension
of a p-adic action to some metric compactification, a free action does not necessarily
10
extend to a free action. We give sufficient conditions to guarantee an extension of a
group action to a given compactification. We present examples of group actions failing to
extend without those conditions.
The conjecture is translated into terms of the ring of continuous functions on the
space of irrationals. We give an equivalent version of the conjecture in these terms,
and explore how the new setting facilitates our investigation of extending actions and
answering the conjecture. We give sufficient conditions for ensuring that being able to
extend each homeomorphism in a group action to a compactification will have the group
action extend.
The Hilbert-Smith conjecture has captured the minds of mathematicians for
generations. In 1940, Paul Althaus Smith generalized Hilbert’s Fifth Problem, which
had asked whether every (finite-dimensional) locally Euclidean topological group is
necessarily a Lie group. Smith proposed that every effective compact group action on
a compact manifold is by a Lie group. This is well known to be equivalent to asking
whether or not a p-adic group can act effectively on a compact manifold.
This dissertation is focused by the desire to answer the conjecture. We give a
multi-pronged attack on the problem. We study related, known, p-adic actions on
non-manifold spaces and construct similar, new ones. We explore the quotient spaces
of such actions to draw conclusions as to the nature of any possible counter-example to
the conjecture. And we also approach the problem in a different way by looking internally
at a space under the action of the p-adic group. This last approach leads us to some
equivalent ways of seeing the conjecture.
In studying a p-adic action on a space, the quotient space of the action is quite
useful to understand in whole or in part. Many before us have extensively studied what
properties such quotient spaces must have. While such quotient maps are not, except in
the simplest of cases, covering maps, they do share similar properties to covering maps.
We prove some similar results to classical covering space theory for these quotient
maps. These results help us to prove that every p-adic action on a Peano continuum
admits equivariant partitions; which, in turn, helps to further our understanding of the
nature of p-adic actions on such continua.
It is well known that there are effective, and even free, p-adic actions on non-manifold
spaces. One example was known even prior to Smith making the conjecture itself,
though perhaps not realized as such at the time. In fact, free p-adic actions on Menger
manifolds of all dimensions have been constructed. We discuss some of those classic
12
partial topological product based constructions to present a more geometric view
of them. We recap how to construct free p-adic actions on Hilbert space from easy
to construct effective p-adic actions on the Hilbert Cube. In attempts to solve the
conjecture, more exotic Peano continua of all dimensions have been constructed in
such a way to admit p-adic actions with desired properties. One of our goals is to make
many of these somewhat arcane constructions more accessible. We demonstrate
easier constructions of some, provide new p-adic actions on the Menger curve, and
show interesting properties of such in regards to any potential counter-example to the
Hilbert-Smith conjecture.
We prove that any potential counter-example to the Hilbert-Smith conjecture would
have a dense subset homeomorphic to the space of irrational numbers upon which
the given p-adic action would act freely as a sub-action. In fact for a large class of
examples of effective p-adic actions on non-manifolds (including those on Menger
continua) this is indeed the case. We prove that there is, up to conjugation, only
one such free p-adic action on the space of irrational numbers. Thus, any potential
counter-example to Hilbert-Smith, as well as these numerous other effective p-adic
actions on non-manifolds, can be seen as an extension of the unique, free, p-adic action
on the space of irrationals.
In exploring the nature of extending group actions to compactifications, potential
obstructions were found. We give a clear example of these obstructions by means
of a simple space with some fairly simple 0-dimensional groups acting upon it. We
then give sufficient conditions for a group action to extend, as a group action, to a
given compactification. Likewise, given a 0-dimensional compact group acting on a
non-compact space we demonstrate the existence of a compactification to which the
action extends (though not necessarily freely). While our focus is on Hilbert-Smith, and
thus on p-adic groups, we treat this in more generality as it can have other applications.
13
When studying compactifications, it is natural to attempt to translate the problem
into the setting of rings of continuous functions. We reformulate the Hilbert-Smith
conjecture here as well as translate other related properties. It is our hope that this new
setting where compactifications can be so easily described will prove more tractable.
1.1 Notation and Terms
For our terms and notations and unless otherwise specified, we will assume that we
are dealing with a metric space (X ,dx)
1.1.1 Sets
We let R denote the real numbers, Q the rational numbers, and I = R Q the
irrational numbers. Let the unit interval be given by I = [0, 1].
We will use Zk to mean the abelian group of integers with addition modulo k , that is
Zk = Z/kZ.
1.1.2 Topology
For Y ⊂ X , we will denote its interior by Y , it’s boundary (or frontier) by ∂Y ,
and it’s closure in X by Y . Denote the open ball about a set A ⊂ X to be the open
set Br(A) = {x ∈ X dx(x ,A) < r} ⊂ X and denote the diameter of a subset A by
diam(A) = sup{dx(a,b)a,b ∈ A}.
Definition 1.1. A map f X → Y is said to be
1. An open (or interior) map if, for every open subset U ⊂ X , the image f (U) ⊂ Y is open.
2. A closed map if, for every closed subset V ⊂ X , the image f (V ) ⊂ Y is closed.
3. A perfect mapping, if it is a continuous, closed, surjective map such that for each y ∈ Y the set f −1(y) is compact.
4. A light map if, for every y ∈ f (X ), the pre-image f −1(y) is totally disconnected.
1.1.3 Partitions
Definition 1.2. [8] A subset K ⊂ X has property S if for each number > 0, there is an
N > 0 such that K = Nn=1Kn, where each Kn is connected and diam(Kn) < .
14
Definition 1.3. [8] A subset K ⊂ X can be partitioned if, for each number > 0, there is
an N > 0 such that there is a collection of disjoint subsets {Un}Nn=1 of K , where Un ⊂ K is
connected and open for each n ≤ N, diam(Un) < for each n ≤ N, Uj ∩Uk = ∅ for all j ≠ k ,
and Nn=1Un is dense in K . We shall say that U = {Un}Nn=1 is an -partitioning of K .
Definition 1.4. [9] A partitioning U of K ⊂ X is a brick partitioning if:
(a) Each domain containing a point of K which is a limit point of each of two elements of U also contains a point of K which is a limit point of each of these same two elements of U but of no other element of U .
(b) Each element of U is uniformly locally connected under the connected distance metric within K .
(c) Each boundary point in K of an element of U is a boundary point of another element of U (i.e. regular)
For a partition U of a set K ⊂ X we define the mesh of the partition to be mesh(U) =
sup{diam(U)U ∈ U}. We also define the star of a subset A ⊂ X (or a point a ∈ X ) with
respect to a partition U to be St(A,U) = {x ∈ X x ∈ U,U ∈ U , U ∩A ≠ ∅}.
We say that a partition V refines a partition U if, for each V ∈ V, there is an element
UV ∈ U with V ⊂ UV . If mesh(V) ≤ , then V is an -refinement of U . If both partitions
U and V are brick partitions, then we say that V brick refines U . We denote V refining
U by V U . We will use the same notation when dealing with open covers. And if a
refinement V of an open cover U is such that for each V ∈ V there is an element UV ∈ U
with St(V ,V) ⊂ UV , then we say that V star-refines U and denote that by V ∗ U .
Definition 1.5. [10] The brick partitioning V of a set K ⊂ X is a core refinement of the
brick partitioning U of K ⊂ X if:
(a) V is a refinement of U (i.e. V U).
(b) for each pair of adjacent elements U ′,U ′′ of U there is a pair of adjacent elements V ′,V ′′ of V in U ′ and U ′′ respectively such that V ′ ∪ V ′′ is a subset of the interior of U ′ ∪ U ′′, and
15
(c) for each element U of U , the elements of V in U may be ordered V0,V1, ... ,Vn such that V0 intersects each Vi , while Vi intersects the boundary of U if and only if i > 0.
We call V0 a core element and Vi , i > 0 border elements.
1.1.4 Inverse Limit of Finite Groups and p-adic Notation
When we write G = lim←Ð{Gi , i+1 i } we will have the following understandings:
1. Each Gi is a finite group
2. G0 = {e}
3. Gi ≤ Gi+1 with equality if and only if Gi = G
4. For each i ≥ 0, i+1i Gi+1 → Gi is an onto homomorphism
In the case where the group G is finite, for simplicity, we will take Gi = G and i+1i G → G
to be the identity map for all i ≥ 1.
By the p-adic group, p, we mean lim←Ð{Zpn ,ψ i+1 i } where the bonding maps ψi+1i
Zpi+1 → Zpi are the surjections obtained by taking the modulus by pi . In the literature this
zero-dimensional compact group is sometimes also denoted by Ap or Zp.
While topologically one element of p generates the group (that is that one element
algebraically generates a dense set of p), this element is by no means unique. When
we have need to pick one of these generators, we will denote it by τ . Proper, nontrivial
subgroups of p are then of the form τ p k p for each choice k ∈ N. We will use the
notation k = τ pkp to conserve space and eyestrain.
When p acts freely on a space X then there naturally arises a system of maps on
the quotients:
X p5ÐÐ→ X/4 p4ÐÐ→ X/3 p3ÐÐ→ X/2 p2ÐÐ→ X/1 p1ÐÐ→ X/p (1–1)
Where the p-to-1 covering maps pn X/n → X/n−1 were induced by the multiplicative
map µn n−1 → n (where µn is simply composition by τ ). When p acts effectively
16
rather than freely, then the same system can be obtained; however, the maps pn are
branched coverings rather than full covering maps.
For each n ∈ N, let the pn-to-1 (branched) covering map Pn X/n → X/p be given by
Pn = p1 p2 ⋅ ⋅ ⋅ pn−1 pn. We will take π0 X → X/p to be the quotient map of the p
action and for each n ∈ N we will let the map πn X → X/n be the quotient map induced
by the subgroup action, meaning the composition Pn πn = π0.
17
CHAPTER 2 HISTORY
At the start of the prior century in August of 1900, David Hilbert spoke at the Paris
conference of the Internation Congress of Mathematics where he challenged the world
with a set of, at the time, ten unsolved problems in various disciplines of mathematics
saying “We hear within us the perpetual call: There is the problem. Seek its solution”
[27]. His list of problems expanded to number twenty-three by the time he published
his speech later that year in a German journal [26]. In 1902, Dr. Mary Winston Newson
translated this article into English for the Bulletin of the American Mathematics Society
[27].
Figure 2-1. David Hilbert as he appeared in 1912 for postcards of the faculty members at the University of Gottingen, which were sold to students at the time.
David Hilbert’s challenge that “just as every human undertaking pursues certain
objects, so also mathematical research requires its problems. It is by the solution of
problems that the investigator tests the temper of his steel” [27] inspired mathematicians
for generations to come. Not all of the listed problems were clearly defined however,
so there is ambiguity on whether a given solution to a problem completely solves that
problem. Thus when one considers a Hilbert Problem, one must first work to understand
the problem and the contexts in which it can be viewed.
The fifth problem of Hilbert’s 23 was translated into English as “How far Lie’s
concept of continuous groups of transformations is approachable in our investigations
without the assumption of the differentiability of the functions” [27]. Over the years since
then many partial solutions and interpretations of this problem have been given by a
cornucopia of the world’s brightest mathematicians. The names and pictures that appear
below should find resonance with any mathematician for the sheer breadth of their works
18
is awe inspiring. That each touched and were touched in return by this problem is what
makes it such a treasure.
A Brouwer B Kerekjarto
Figure 2-2. LEJ Brouwer and Bela Kerekjarto independently solved the case for dimension 2 by exhaustion
Early advances on the problem were made in 1919 by Luitzen Egbertus Jan
Brouwer in the special case when dealing with a manifold of dimension 2. After the
First World War, when he turned back to Topology, he showed that a group acting
effectively on a 2-manifold is a Lie group [13]. Similar results to Brouwer’s were obtained
independently by Bela Kerekjarto [33], whom interestingly enough Brouwer had once
described derogatorily as “rash” [51]. It is, also, perhaps interesting to note that both
papers appeared one after the other in the same issue of the Math Annalen journal.
Figure 2-3. Alfred Haar was a student of David Hilbert at the University of Gottingen
In 1933, Alfred Haar introduced the world to Haar Measure [23]. This allows us to
define quite a natural metric on a space upon which a compact group acts. Suppose
(X ,d) is a metric space and that a compact group G acts on it, then we may define a
metric using the Haar measure:
ρG(x , y) = ∫ G d((g(x),g(y))dg (2–1)
19
This produces a G -invariant metric on the space that is topologically equivalent to
the original metric. This makes the metric induced by Haar measure the perhaps most
natural metric on which we should investigate a compact group action. Not surprisingly,
some nice uses of this metric have occurred in the history of the problem [40, 46, 52].
A VonNeumann B Prontrjagin C Chevalley D Freudenthal
Figure 2-4. J. VonNeumann, L.S. Pontrajagin, C. Chevalley, and H. Fredudenthal each gave solutions for various special cases of Hilbert’s Fifth Problem
In fact, in the same journal in the article immediately following that of Haar, John
Von Neumann used Haar measure to prove Hilbert’s fifth problem in the case of compact
groups [52]. In the following year, Lev Semenovich Pontrjagin extended the result for the
case of abelian groups [43]. In 1941, Claude Chevalley published a solution for solvable
groups [14]. And a few years prior to that, Hans Freudenthal introduced the idea of a
locally compact group being approximated by Lie groups [18], namely:
Definition 2.1. [30] A locally compact group G can be approximated by Lie groups, if G
contains a system of normal subgroups {Nα} such that G/Nα are Lie groups and that the
intersection of all Nα coincides with the identity e.
A Gleason B Iwasawa
Figure 2-5. A. Gleason and K. Iwasawa independently paved the way to the solution of Hilbert’s Fifth Problem
This idea would inspire Andrew M. Gleason with his concept of ‘’Groups without
Small Subgroups”. And, after the Second World War, progress on the problem
20
expanded under the separate works of Kenkichi Iwasawa and Andrew Gleason, who
both conjectured that the following form of Hilbert’s fifth problem held true:
Conjecture 1 (Gleason-Iwasawa [21, 30]). Every locally compact group is a generalized
Lie group
As an aside, it is interesting to see the ability of the world wars to hamper research
in this problem through the restriction of travel, in restricting communications between
mathematicians, as well as simply drawing them to other things. In the case of the First
World War, it was restricting the travel of the Dutch Brouwer into Germany, and in the
case of the Second World War, it was both by limiting the availability of journal papers
to the Japanesse Iwasawa, and by taking the American Gleason from his research
into breaking Japanese codes for the Allies. Germany’s politics of the 1930s brought
Salomon Bochner to Princeton and, after the war, he and Deane Montgomery proved
Hilbert’s fifth problem for groups of diffeomorphisms [11].
A Bochner B Montgomery C Zippin D Yamabe
Figure 2-6. S. Bochner, D. Montgomery, L. Zippin, and H. Yamabe gave solutions for different versions of Hilbert’s Fifth Problem
These successes culminated in 1952, when Gleason proved that a locally compact
finite dimensional group without small subgroups was a Lie group [22]. Using this Deane
Montgomery and Leo Zippin receive credit for solving Hilbert’s fifth problem in the
following form:
compact group is a Lie group
In the following year, Hidehiko Yamabe extended their result for infinite dimensional
groups, generalizing the results of Montgomery and Zippin. Depending on how you view
21
Hilbert’s fifth problem determines to whom you will credit for the final answer, although it
is most typically viewed as having been solved by Montgomery and Zippin.
However, this is not the only way that one may view Hilbert’s fifth problem. In
the period of the late 1930s a more generalized version of Hilbert’s fifth problem was
considered and put forth by Paul Althaus Smith.
Figure 2-7. One of only two photos of P.A. Smith known to exist. Photo courtesy of University Archives, Columbia University in the City of New York (for $20)
Conjecture 2 (Smith [48]). If G is a compact group which acts effectively on a manifold,
then G is a Lie group.
He showed that this conjecture is equivalent to the following, which is known as the
Hilbert-Smith Conjecture that inspires research towards an answer to this day:
Conjecture 3 (Hilbert-Smith). A p-adic group cannot act effectively on a manifold.
It is towards this problem that P. A. Smith developed his Smith Theory in studying
transformations by groups of finite periods. He also provided his own proof of Newman’s
theorem [48] different from MHA Newman’s original. The theorem as originally stated by
Maxwell Herman Alexander Newman is:
Theorem 2.2 (Newman [39]). Let M be a connected manifold with metric d , then there
exist an > 0 such that, for every action of a finite group G on M, there exists an orbit of
diameter larger than .
Figure 2-8. M.H.A. Newman, the mathematical father of Colossus (the world’s first digital programable computer)
22
One key element in attempting to determine whether or not a p-adic group can act
effectively (or freely) on a manifold is to consider the quotient space that arises from
such a potential p-adic action. Being the image of an open mapping of a manifold, the
quotient space of the action is a compact, connected, locally connected, metric space.
However, this hypothetical space does not have many more of the nice qualities that one
associates with manifolds.
Figure 2-9. A. N. Kolmorgoroff ”failed” to prove that open maps could not raise dimension
In 1941, Smith proved that if a p-adic group Ap acted freely on a manifold X , then
the dimension of the quotient space dim X/Ap ≠ dimX [47]. One natural question is
whether or not a quotient map, or in fact any open map can raise dimension. Just a
few years prior, in 1937, Andrey Nikolaevich Kolmogoroff showed this to be possible
by constructing an open map from a 1-dimensional space to a 2-dimensional space
[34]. This example, though motivated simply to show that an open map could raise
dimension, has a direct impact on the study of the Hilbert-Smith conjecture, even though
neither of the spaces involved is a manifold.
A Eilenberg B Steenrod
Figure 2-10. S. Eilenberg and N.E. Steenrod were involved in carrying on the spirit of Hilbert’s Problems
In 1949 and in the spirit of Hilbert’s problems, Samuel Eilenberg published a list of
open problems in Topology [16] arising from a mathematics conference held as part of
23
the Bicentennial Celebration of Princeton University. The list included the Hilbert-Smith
conjecture and some problems motivated by it. Those related questions posed by
Norman Earl Steenrod were:
Conjecture 4 (Steenrod [16]). Does there exist an interior map of a manifold on a space
of higher dimension?
Conjecture 5 (Steenrod [16]). Does there exist a light interior map of a manifold such
that the inverse image of some point is a Cantor set?
In both of these the reader should be aware that by ‘interior’ the meaning is ‘open ’as the
former term has fallen out of practice in favor of the later. It should, also, be noted that
a negative answer to either would also suffice to answer the Hilbert-Smith conjecture in
the negative.
A Anderson B Keldys C WIlson
Figure 2-11. R.D. Anderson, L.V. Keldys, and D.C. Wilson answered Steenrod’s conjectures
The first conjecture was shown to hold true, in 1956 by Richard Davis Anderson,
who announced far more than what was asked by Steenrod [2].
Theorem 2.3 (Anderson [2]). If for any n ≥ 3 and m ≥ 2, M is an n-cell or n-sphere and Y
is an m-cell or m-sphere, then there exists a monotone open mapping of M onto Y .
(While I say ‘shown ’I cannot find a proof of it published by R. D. Anderson, though
Lyudmilla Vsevoldovna Keldys published a such poof in 1957 in Russian [32]).
And the second conjecture was solved (also in the affirmative) in 1973 by David C.
Wilson [57] and his solution also exceeded the demands of the conjecture motivating it:
24
Theorem 2.4 (Wilson [57]). If M3 is a compact triangulated 3-manifold and m ≥ 3, then
there exists a light open mapping of M3 onto Im such that each point-inverse set is
homeomorphic to the standard Cantor set
In 1960, Chung-Tao Yang extended the work of P.A. Smith, and in particular made
use of ‘a modified special homology theory of Smith in which reals modulo 1 are used
as cofficients’ [58]. He proved if a p-adic group acts effectively on an n-dimensional
manifold X , then the orbit space X/G is of homology dimension n + 2. Moreover, if X is
merely a locally compact Hausdorff space, then the homology dimension of the orbit
space is no more than n + 3. This result was also proven independently, and in a different
manner, by Glen Eugene Bredon, Frank Raymond, and Robert Fones Williams in a joint
paper using spectral sequences though they give primacy to CT Yang for the result [12].
A Yang B Raymond C Williams
Figure 2-12. C.T. Yang, F. Raymond, and R.F. Williams determined the dimension of the quotient space for any possible counter-example to the conjecture
If a counter-example to the Hilbert-Smith conjecture were to exist, then the
quotient by such an action would perforce be a dimension raising open map. Towards
understanding what such actions might be like, Raymond and Williams constructed
examples of p-adic actions on spaces where the quotient map raised dimension [45].
One example, done by Williams [55], was a modification of the first dimension raising
open map: the example Kolmogoroff gave in 1933 [34]. Kolomogoroff’s example is, in
reality, the quotient map of an effective dyadic action on the Menger curve where the
quotient (or orbit) space is the Pontrjagin 2-surface [42].
Williams modified the construction in order that the action would be free (as well as
for an arbitrary prime p) and formalized a functor [55] to more readily describe not only
25
the dimensionally deficient Pontrjagin surfaces, but the example of Pontrjagin’s student
Vladimir Grigorevich Boltyanskii.
In a later paper [56], Williams details out a construction to create for any n ≥ 1, an
n-dimensional space X , acted upon freely by a p-adic group p, in such a way that the
orbit space X/p has dimension n + 1. Moreover, these examples have a second property,
namely that Hn+1c (U) = Zp∞ for any open U ⊂ X/p. This is something that must occur
whenever X is a Peano continuum, and something which I will directly evidence later in
this paper, but which follows from the spectral sequence.
Constructing a space X that has the p-adic group, p act upon it freely such that
dim(X/p) = 2 + dim(X ) is far more difficult. Frank Raymond was able to construct such
an example [45]. Now, in Raymond’s example the second property that Williams was
able to obtain in his examples that just raised the dimension of the orbit space by 1 did
not hold. In fact, neither was able to construct an example with both features, and went
so far as to conjecture that it was not possible when dim X/p < ∞. They, indeed, proved
that their methods of construction could certainly not produce such an example in any
case, and this serves to highlight the stringent requirements that such a p-adic action
would have to have in order to act effectively on a manifold.
Williams and Raymond obtained their dimensional bounds by studying the
classifying space for the p-adic group, and it has been a source of interest for many
others in attacking the problem. In 1992, Sergei Mikhailovich Ageev proved that µn is
n-universal for free actions of p [1]. He showed that for given free p actions on µn and
an ANR X , such that dim X/+p ≤ n and for any p-invariant closed subset A ⊂ X that any
p-equivariant mapping f A → µn admits an equivariant extension f X → µn. And in the
same article conjectured
Conjecture 6 (Ageev). For any 0-dimensional compact group G , if µn+m and µn are free
G -spaces, then there is no equivariant map µm+n → µn implying that the orbit space R/p
26
has infinite dimension, where R is any compact ANR-space with free action of the group
p of p-adic integers.
This would handle one of the two possible cases for the dimension of the orbit space of
a manifold should it admit a free p-adic action.
A Dranishnikov B Yang
Figure 2-13. A.N. Dranishnikov and Z. Yang constructed and explained non-dimension raising free p-adic actions on Menger continuua
The study of p-adic actions on Menger universal spaces µn is then very reasonable
and connected to the problem. In addition to Kolmogoroff’s example and Williams
modification of it to a free action, Alexander (Sasha) Dranishnikov produced a free
action on general Menger compacta [15] in the late 1980s. Now Dranishnikov’s action on
the menger curve was different from that of Kolmogoroff’s in that the quotient space was
not a higher dimensional space.
A Pasynkov B Bestvina C Anderson
Figure 2-14. Pictured: B.A. Pasynkov (sitting), M. Bestvina, and R.D. Anderson. They found descriptions and characterizations for µn
Dranishnikov’s action made use of Boris Pasynkov’s concept of partial topological
products, specifically their use in describing the universal compacta µn. Zhiquing Yang’s
paper [59] recreates this action in a more readily accessible version. The Pasynkov
partial product description of µn can be seen from first principles in the later paper,
27
without any knowledge of partial topological products. Further, Yang makes use Mladen
Bestivina’s characterization for µn [7] rather than R.D. Anderson’s [4] which was all
that Pasynkov had with which to work at the time. From the Pasynkov partial product
description of Menger continua, it is plain why such continua are a natural setting for
p-adic actions.
In this dissertation, I will demonstrate a similar construction of µ1 to obtain a free p
action on the menger curve5.4. The construction, that I will use, will form the menger
curve out of p-adic solenoids. It will be different from the action given by Dranishnikov
[15], but the construction can easily be modified in order to produce that action if it is
desired.
Recent times have shown interesting partial solutions to the Hilbert-Smith
conjecture. As the history of the problem is marked by such groupings of advancements,
it is reasonable to expect more, and similar partial solutions to occur in the near future.
In 1997, one of Dranishnikov’s advisors, Eugine Vitalievich Scepin together with
Dusan Repovs were able to put together a most elegant proof that the Hilbert-Smith
conjecture holds true for actions by Lipchitz maps [46]. Their proof is a dimension
theoretic argument that resolves down to a string of half a dozen inequalities that can
be represented in a single chain. The proof utilizes the metric formed using the Haar
measure mentioned earlier (2–1) from the 1930s, John Forbes Nash Jr’s embedding
theorem from the 1950s, some classical dimension theory results [28], and C.T. Yang’s
bound on the Integral Cohomological dimension of the quotient space, as well as even
a Baire Category argument. Following in the wake of this proof are derivative results for
Holder actions and quasi-conformal maps [36, 37].
Just this past year, a young mathematician named John Pardon announced
via Arχiv, a solution to Hilbert-Smith in the case of manifolds of dimension 3 [40].
Interestingly enough, it does not approach the problem by attacking through the quotient
28
A Scepin B Repovs C Nash D Pardon
Figure 2-15. Pictured: E.V. Scepin, D. Repovs, J.F. Nash, and J. Pardon. Recent advances on the Hilbert-Smith Conjecture
space of a potential p action, but rather takes the spirit of Gleason’s concept of groups
without small sub-groups and combines it with known facts of 3-manifolds.
29
CHAPTER 3 COMPACTIFICATIONS
We will show that for a given zero-dimensional compact group G , there is only
one free G -action, up to conjugation by a homeomorphism, on the space of irrationals.
Moreover, this G -action will be a sub-action of any effective G -action on a complete
metric space where the periodic points contain no open set and the action has no
isolated orbits. These results are of interest since any potential counter-example to the
Hilbert-Smith conjecture would contain this unique free action on the space of irrationals
as a sub-action. We turn to investigate how group actions on non-compact spaces can
be extended, as group actions, to compactifications of those spaces. First, we show
that every zero-dimensional compact group action on a separable metric space has
at least one compactification to which the action extends. Second, we give conditions
to ensure that such a group action will extend for a given compactification. Requiring
that each group element extends to a homeomorphism on the compactification is
generally not sufficient to ensure that the group action extends to the compactification.
We have examples and theorems that illustrate exactly what can happen. We provide a
straightforward example of a fixed-point free effective action on a separable metric space
that cannot be extended non-trivially to an action on any compactification. We then
conclude the chapter giving an equivalent formulation of the Hilbert-Smith conjecture in
light of the results found herein.
3.1 Actions on the Space of Irrational Numbers
The main goal of this section is to prove Theorem 3.3. We begin by showing
that the quotient space of any compact zero- dimensional group action on I is
homeomorphic to I. This result will follow from Theorem 3.1, where we show that
the image of any perfect open map on I is homeomorphic to I.
Lemma 3.1. If (X ,d) is a complete metric space and the mapping p X → Y is a perfect
open surjection, then the space Y is topologically complete.
30
Proof. Since p is a perfect mapping, let ρ be the metric for Y induced by the Hausdorff
distance between pre-images of points under p using the complete metric d . In other
words for α,β ∈ Y , let A = p−1(α), B = p−1(β) and define
ρ(α,β) =max{sup a∈A inf b∈B d(a,b), sup
b∈B inf a∈A d(a,b)}.
We shall show that (Y ,ρ) is complete. Let {αn}∞n=1 ⊂ Y be a Cauchy sequence with
respect to the metric ρ. For = 1/21, there is an N1 > 0 such that n,m ≥ N1 implies that
ρ(αn,αm) < 1/21. Pick b1 ∈ p−1(αN1) and observe that the open ball B1 = B 1
21 (b1) intersects
p−1(αm) for every m > N1. Inductively pick bn+1 such that Nn+1 > Nn corresponds
to a choice of = 1/2n+1 and bn+1 ∈ p−1(αNn+1) ∩ Bn then observe that the open ball
Bn+1 = B 1
2n+1 (bn+1) intersects p−1(αm) for every m > Nn+1. Let N0 = 0 and B0 = I.
Choose a sequence {an}∞n=1 by picking ai = bk if i = Nk for some k ≥ 1 and by choosing
ai ∈ p−1(αi) ∩Bk when Nk < i < Nk+1 for some k ≥ 0. By construction the sequence {an}∞n=1
is Cauchy with respect to d . Since d is a complete metric this later sequence has a limit
a = liman. Let α = p(a) and observe that α = limαn and thus the metric ρ is a complete
metric for Y . Since there exists a complete metric on it, the space Y is topologically
complete.
Theorem 3.1. If the mapping p I → Y is a perfect open surjection, then the space
Y ≅ I, the space of irrationals.
Proof. Hausdorff characterized the irrationals as the 0-dimensional, nowhere locally
compact, separable metric space that is topologically complete [24].
Since p is both an open and closed mapping, any subset O ⊂ I that is both open
and closed will have the property that p(O) is both open and closed. Since p is a
surjection the open-closed basis of I is mapped to an open-closed basis of Y , hence Y
is a 0-dimensional separable metric space.
31
Further, since p is a perfect and light surjection and the space I is nowhere locally
compact, the image space p(I) = Y is as well.
Since the space I is topologically complete (say viewed as NN) there is a metric d
for which I is complete. By Lemma 3.1 the space Y is topologically complete.
Hence by Hausdorff’s characterization of the irrationals the space Y ≅ I = R Q as
stated.
Now we apply this result to 0−dimensional compact metric group actions on the
space of irrationals. Pontrjagin showed that every 0-dimensional compact group is
the inverse limit of finite groups [44]. It is easy to see that for a given 0-dimensional
compact metric group G , either G is finite or G has the topology of a Cantor set. If G is
not finite, we call G a Cantor Group. In either case, we can write G = lim←Ð{Gi , i+1 i } with
the conventions established in section 1.1.4. The zero-dimensional compact group of
p-adic numbers is a Cantor group of the form lim←Ð{Zpi , i+1 i }, hence the results below will
hold for them as well.
Corollary 3.1.1. Let G = lim←Ð{Gi , i+1 i } be a zero-dimensional compact group. If the group
G acts freely on the space of irrationals I, then the quotient space I/G ≅ I.
Proof. Let the map p I → I/G denote the quotient map induced by the free action. The
mapping p is a perfect open surjection, so by Theorem 3.1 the quotient space I/G ≅ I.
We will use this result to show that for any given zero-dimensional compact group
G , there is only one free action by G on the space of irrationals. In the case where G is
finite, this is done in one step and the map s1 initially obtained will be the map s that is
eventually desired to define the homeomorphism in the statement of the theorem below.
Otherwise when G is infinite (and hence a Cantor group), the map s is obtained by a
sequence of maps {si}∞i=1.
32
Theorem 3.2. Let G = lim←Ð{Gi , i+1 i } be a zero-dimensional compact group. If the
mapping A G × I → I is a free topological group action, then there is a homeomorphism
h I → I ×G such that
A(g,w) = h−1(π1h(w),gπ2h(w)),
where π1 I ×G → I and π2 I ×G → G are the projection maps.
Proof. Let G = lim←Ð{Gi , i+1 i }. For i > 0 let Ki = ker ii−1 Gi and denote by ei the identity
element for the group Ki . Thus we can write G = {(gi)∞i=1gi ∈ Ki}. For every n > 0,
let G n = {(gi)∞i=1 ∈ G gi = ei ∈ Ki for all i < n} G . Let B = {Bn}n∈N be a countable
open-closed basis of I.
Claim 1. For a given i ≥ 1 and w ∈ I there is a set Bnw ,i ∈ B such that w ∈ Bnw ,i and
G i(Bnw ,i ) ∩ (G G i)(Bnw ,i ) = ∅
Proof. Since G acts freely on I, the set G i(w) ∩ (G G 1)(w) = ∅. Since G and G i
are both compact and B is a basis of I there is an open-closed set B ∈ B such that
w ∈ B but (G G i)(w) ∩ B = ∅. Further there is a Bnw ,i ∈ B such that w ∈ Bnw ,i ⊆ B and
(G G i)(B) ∩ Bnw ,i = ∅ thus as G is a group we have G i(Bnw ,i ) ∩ (G G i)(Bnw ,i ) = ∅ as
claimed.
Since the point w ∈ I was arbitrary, for a given i ≥ 1 there is a map ti I → N by
ti(w) = nw ,i ∈ N from Claim 1 above. For i = 1, let N1 = (nj)ωj=1 be the strictly increasing
(possibly infinite) sequence of natural numbers such that we have m ∈ (nj)∞j=1 if and only
if m ∈ t1(I).
Now using the order on this subset of the naturals and the open-closed nature
of the sets Bnk , we will define a continuous map s1 I → K1. For k ≥ 1 and for all
w ∈ G(Bnk G(m<k Bnm)), let s1(w) = η ∈ K1 if and only if w ∈ gη(Bnk) for some
gη = (gη,i)∞i=0 ∈ G 1 = G such that gη,1 = η. Since all of the sets Bnm are both open and
closed, so are sets of the form G(Bnk G(m<k Bnm)) and the mapping s1 is continuous.
33
Proceed with the space s−11 (e1), form a sequence N2, and use it to define a mapping
s2 I → K2 such that it is consistent with our mapping s1 and then proceed inductively to
define sn I → Kn.
Define the map s I → G by s(g) = s((gi)∞i=0) = (si(gi))∞i=0 ∈ G , let the map f I/G → I
be the homeomorphism guaranteed by corollary 3.1.1, and let the map p I → I/G be the
perfect light open quotient map induced by the G action.
Define the map h I → I ×G by h(w) = (f p(w), s(w)).
We now show that for a given zero-dimensional compact group G , there is only one
free G -action on the space of irrationals, as all others are conjugate to the action by
homeomorphisms.
Corollary 3.2.1. If A G × I → I is a free zero-dimensional compact group action on I,
then it is conjugate by homeomorphism with the free zero-dimensional compact group
action B G × (I ×G)→ I ×G , where B is defined by B(j , (w ,g)) = (w , j g).
Proof. Let G = lim←Ð{Gi , i+1 i } be a zero-dimensional compact group then I ≅ I ×G .
Define B G × (I ×G) → I ×G by B(j , (w ,g)) = (w , j g). If A G × I → I is a free
zero-dimensional compact group action on I, then it is conjugate by homeomorphism
with B.
Let G be a zero-dimensional compact group. Not every effective G -action on a
space X will contain, as a sub-action, the free G -action on the space of irrationals. A
trivial example would be when X is finite. Slightly less trivial would be letting G be a
Cantor group and the space X = G with G acting on X by the group operation. In this
second example I ⊂ X , but as X/G is a singleton, obviously I X/G. Another easy
example will be discussed later when it appears in Figure 5.1. To avoid this type of
34
problem, we restrict our attention to Cantor groups, and impose restrictions on the action
and the space.
Theorem 3.3. Let G = lim←Ð{Gi , i+1 i } be a Cantor group. If X is a complete metric space
upon which G acts effectively such that the set of periodic points contains no open set
and the action has no isolated orbits, then there is a subspace Y ⊂ X , with Y ≅ I, such
that G acts freely on Y .
Proof. Let π X → X/G be the quotient map induced by the effective action on X by G .
Since X is a complete metric space, the quotient space X/G is as well. Denote by
Pk ⊂ X the collection of periodic points of X of period k under the G action. Let P = Pk .
Since X has no isolated orbits, the space X/G π(P) has no isolated points.
There is a set Z ⊂ X/G such that Z is a dense Gδ set and Z ≅ I. Since, for each k ,
the set Pk is closed and contains no open set, the subset π(P) ⊂ X/G is an Fσ set and
X/G π(P) is a dense Gδ set. Thus the set Z π(P) = Z ∩ (X/G π(P)) is a dense Gδ set
and Z π(P) ≅ I.
Since π is an open mapping, the set Y = π−1(Z π(P))) is a dense Gδ set of the
complete metric space X . Since Y has no isolated points, Y ≅ I. Finally the group G
acts freely upon Y , since Y ⊂ X P and Y = π−1(Z π(P))) was the full pre-image of a
set in X/G.
If there is a counter-example to the Hilbert-Smith conjecture, then it will contain this
unique free p-adic action on the irrationals. Moreover this counter-example will be the
extension of that p-adic group action to a manifold compactification of the irrationals.
This observation is why we turn to examine equivariantly extending group actions and
understand the potential obstacles that might be found therein.
3.2 Equivariant Compactifications
In this section, we will use the fact that a Cantor group is an inverse limit of finite
groups to provide conditions when a group action can be extended from a pre-compact
metric space to a compact metric space. In the case of a p-adic group p, if τ ∈ p is a
35
generator, then the sequence of sub-groups {τ pip}∞i=0 is an infinite descending normal
sequence and the order of (τp i+1 p)/(τpip) is p. In the case of the Cantor group Π∞i=1Z2,
which we will use as an example in a later section, consider the sequence {Hi}∞i=1 where
Hi ≅ Π∞j=iZ2 Π∞i=1Z2 is the subgroup where the first i − 1 coordinates are 0 ∈ Z2. This
sequence is an infinite descending normal sequence where the order of Hi+1/Hi is 2. The
theorem below demonstrates a construction for an extension of a group action on a
separable metric space to an effective group action on its compactification. All that is
required is the choice of a pre-compact metric and choice of a sequence of finite open
covers of the space.
Theorem 3.4. Let G = lim←Ð{Gi , i+1 i } be a Cantor group and X be a separable space with
a pre-compact metric distX . If G acts effectively on X , then there is a compactification C
of X such that G extends to an effective action on C .
Proof. Let ρG be the metric defined in equation 2–1 induced from the totally bounded
metric distX . Let {Ui}∞i=1 be a sequence of finite open covers of (X ,ρG) such that
mesh(Un)→ 0 and Un ∗ Un−1.
Since G is a Cantor group, G has an infinite descending normal sequence [5]. To
whit, there is a sequence {Gk G}∞k=1 such that Gk = {e} and for each k ∈ N we
have that {e} ≠ Gk Gk−1 with index Gk/Gk−1 = nk < ∞. Since G is compact, take
Gk = {gGk g ∈ G} to be the finite cover of G by open-closed cosets.
For each m ∈ N, define a finite cover Vm = {HU H ∈ Gm,U ∈ Um} and note that,
obviously, we have Vm ∗ Vm−1. Please also note that while both mesh(Un) → 0 and
mesh(Gn) → 0, we do not know that mesh(Vn) → 0 and, indeed, it might not be the case
for the given pre-compact metric.
Let µ denote the collection {Vm}∞m=1.
Claim 1. The collection µ strongly separates points in X (for any pair of distinct points
x , y ∈ X only finitely many elements of µ contain both).
36
Pick x , y ∈ X distinct.
Case 1: If x ∈ G(y) but x ≠ y , then there is a K > 0 such that x ∉ GK(y). Let
d = min{ρG(GK(y),gGK(y)) g ∈ G GK} > 0. Since mesh(Um) → 0, there is an n ∈ N
such that n > K and mesh(Un) < d/4.
If x ∈ HU ∈ Vn, then there is a h ∈ G such that H = hGn. Now the orbit G(y) is
partitioned by GK(y) that separates the points x and y . Since U ∈ Un and diam(U) < d ,
then the set {J ∈ GK J(y) ∩ U ≠ ∅} is a singleton consisting of the sole representative
member of GK for the partition of the orbit GK(y) intersecting with U. Since H = hGn ⊂
hGK ∈ GK merely permutes the cosets of GK we have only one member of the partition of
the orbit GK(y) intersecting the set HU = hGnU ⊂ hGKU thus x ∈ HU implies y ∉ HU as
they occur in different members of the partition of their orbit GK(y).
Case 2: If x ∉ G(y), then let d = ρG(G(x),G(y)) > 0. Since mesh(Um) → 0, there is
an n ∈ N such that mesh(Un) < d/4. If x ∈ HU ∈ Vn, then G(x) ∩ U ≠ ∅. Since U ∈ Un, we
have diam(U) < d/4. Thus G(y) ∩U = ∅ and likewise y ∉ HU.
Therefore the collection µ strongly separates points in X as claimed.
Claim 2. The collection µ is a basis for uniformity on X .
For each n ∈ N, we have, for every U ∈ Un, an open set V ∈ Vn ∈ µ such that U ⊂ V .
Since each Un is a uniform covering, we have each Vn is a uniform covering.
Since Vm ∗ Vm−1 for all m ∈ N, we have µ is a normal family (in fact, forms a normal
sequence) and, thus is a sub-basis for a pre-uniformity [29]. Since pairwise intersections
of members of any Vm can be represented as unions of later members of µ, we have that
µ is, in fact, a basis rather than just a sub-basis.
We have µ is a basis for uniformity rather than just a pre-uniformity, since given any
two points x , y ∈ X , there is a covering Vm, such that for every V ∈ Vm, if x ∈ V , then
y ∉ V .
Claim 3. The group G is equi-uniformly continuous with respect to µ.
37
Let τ be one of the generators of G .
Since for each k ∈ N the cover of G by Gk consists of the subgroup Gk of finite
index and its cosets, the collection {τ−nGk}∞n=0 is finite. Thus it is trivially true that for
each m ∈ N the number H(∨n−1i=0 τ−i(Vm)) is bounded for all n ∈ N. Since µ is a basis for
uniformity on X , the family {τ n} is equiuniformly continuous with respect to µ.
Let C be the compactification determined by the basis of uniformity µ. Since each
Vm is a finite cover and the normal sequence of those covers separates points, the
compactification C is metric.
Claim 4. The group G acts effectively on the compactification C determined by the basis
of uniformity µ.
Again, let τ be one of the generators for G . Since cover elements permute under
τ , the homeomorphism τ extends to a homeomorphism τ C → C . Since τ forms a
equiuniformly continuous family and τ was arbitrary, the entire group G extends to C and
furthermore extends as a group action on C .
Even if the group action given in Theorem 3.4 happened to be a free action, the
extended group action on the compactification might not be free. This depends, in part,
on the pre-compact metric given for the space.
We will demonstrate this possibility with the following example. We start by
constructing a non-compact quotient space of a Cantor group action.
Let the space Y be the space consisting of the half-open interval (0, 1] together with an
infinite sequence of tangent circles. Impose upon Y the pre-compact metric in order that
the compactification of Y is the 1 point compactification. Thus the sequence of tangent
circles will limit down to the compactification point 0.
Let G = lim←Ð{Gi , i+1 i } be a Cantor group acting freely on a non-compact metric
space X such that Y = X/G. There are infinitely many possible spaces X that will satisfy
38
( a c g
Figure 3-1. Pre-compact quotient space Y with a given pre-compact metric
that requirement. However, if C is a compactification of X constructed by Theorem 3.4
where C/G is the 1-point compactification of Y , and if the extended G action on C is a
free action, then X ≅ Y ×G .
To see what happens with some of the other possibilities for the space X , fix a
nontrivial group element g ∈ G {e}. To avoid confusion, denote this new possible X by
X ′. Let X ′ ≅ (((0, 1]×G)Z)/, where Z ≅ ([0, 1]×N)×G . All that remains is to describe
how each copy of [0, 1] is attached to (0, 1] ×G . Picture that traversing once around any
circle downstairs in Y lifts to a path in X ′ from one copy of (0, 1] to it’s image under the
fixed group element g.
Applying our construction from Theorem 3.4 , we will obtain the 1-point compactification
of X ′, and as such the extended group action on it will be, perforce, only merely effective
rather than free. This is not to say that X ′ does not have compactifications where G
extends freely, but rather to say that with this pre-compact metric it fails to extend freely.
For example if Z = ([0, 1] (S1×(N∪{∞})))/ is a compactification of Y , then there is a
compactification, C ′ of X ′, and a free extension of G so that C ′/G ≅ Z . This leaves open,
the following question, for which there likely is a positive answer:
Question 3.1 (Maissen). Let G = lim←Ð{Gi , i+1 i } be a Cantor group and X be a separable
space. If G acts freely on X , then is there a compactification C of X such that G can be
extended to a free action on C?
39
For sake of completeness, we mention that the remaining possible choices for our
space X in this example can be obtained by individual choices of group elements for
each circle in the quotient space Y . Hopefully this can start to give some insight in the
way in which one can construct Cantor group actions, and the choices involved. There
are some minor obstacles in doing this for group actions in general, which we address in
the following sections.
3.3 Extending Group Actions
The theorem below will show sufficient conditions to guarantee the extension of the
group action as a group action. Later, we will discuss possible consequences from not
imposing these conditions.
Theorem 3.5. Let X be a metric space, G be a compact metric group, and A G ×X → X
be a topological group action. If (C ,dc) is a metric compactification of X such that for all
g ∈ G the map A(g, ⋅) = g(x) X → X extends continuously to A(g, ⋅) = g(x) C → C , then
A G ×C → C is a continuous group action.
Proof. For ease of notation, define for each g ∈ G the map g C → C by g(x) = A(g, x).
By way of a contradiction, suppose that A is not continuous. With this assumption
we will show that there is an element h ∈ G such that the function h = A(h, ⋅) C → C is
not continuous.
Since A is not continuous, C is metric, and X is dense in C there is a sequence
{(gi , xi)}∞i=1 ⊂ G ×X such that (gi , xi)→ (g, x) ∈ G ×C but gi(xi) g(x) ∈ C .
Claim 1. Without loss of generality we may assume g = e ∈ G .
If the sequence g−1gi(xi) → x ∈ C , then gg−1gi(xi) → g(x) since g ∈ G implies g is
continuous. But then gi(xi) → g(x) since gg−1gi(xi) = gi(xi) as xi ∈ X and A is a group
action on X . Thus the sequence g−1gi(xi) x ∈ C and we may assume that g = e ∈ G .
Claim 2. Without loss of generality we may assume that gi(xi) converges to y ≠ x .
40
Since gi(xi) x ∈ C and C is a compact metric space the sequence {gi(xi)}∞i=1 has
a convergent subsequence such that gik(xik) → y ∈ C {x}. So we can assume that
gi(xi)→ y ≠ x .
Claim 3. There is a sequence of group elements {hj ∈ G}∞j=0, sets {Wj ⊆ X}∞j=1, and
points {uj ∈ X}∞j=1, {vj ∈ X}∞j=1 such that
1. diam(Wj)→ 0, 2. uj ∈Wj ∩X and vj ∈Wj ∩X , and 3. hj(ui) ∈ U and hj(vi) ∈ V for all i ≤ j .
Let 0 < r < dc(x , y) ≠ 0, let n0 = 0, let U = B r 2 (x) and let V = B r
2 (y), so U ∩ V = ∅. For
sake of induction, we define h0 = e ∈ G andW1 = B r 2 (h−10 (x)) = U.
Since xi → x and gi(xi) → y in C , there exists a number M0 > 0 such that whenever
n >M0 both xn ∈ U and gn(xn) ∈ V . Let m1 =M0 + 1 and let u1 = h−10 (xm1) = xm1 ∈ U =W1.
Since gi → e in G , for the compact (finite) set {xm1}, there is a number N1 > m1 such
that whenever n > N1 the point gn(xm1) ∈ U. Let n1 = N1 + 1 and let v1 = h−10 (xn1) = xn1 ∈
U =W1.
Let h1 = gn1 h0 = gn1 ∈ G , then h1(u1) = gn1(xm1) ∈ U and h1(v1) = gn1(xn1) ∈ V .
Suppose for a number k ≥ 1 and all 1 ≤ j ≤ k that
1. hj ∈ G 2. Wj = B r
j (h−1j−1(x))
3. uj ∈Wj ∩X and vj ∈Wj ∩X 4. nj >mj > nj−1 5. hj−1(uj) = xmj and hj−1(vj) = xnj 6. hj(ui) ∈ U for all 1 ≤ i ≤ j and hj(vi) ∈ V for all 1 ≤ i ≤ j
LetWk+1 = B r k+1 (h−1k (x)).
Since gi → e in G , for the compact (finite) set {hk(ui),hk(vi) 1 ≤ i ≤ k} there
is a number Mk > nk such that whenever n > Mk the point h−1k (xn) ∈ Wk+1, the points
gn(hk(ui)) ∈ U for all 1 ≤ i ≤ k and the points gn(hk(vi)) ∈ V for all 1 ≤ i ≤ k . Let
mk+1 =Mk + 1 and let uk+1 = h−1k (xmk+1) ∈Wk+1.
41
Since gi → e in G , for the compact (finite) set {xmk+1} there is a number Nk+1 > mk+1
such that whenever n > Nk+1 the point gn(xmk+1) = gn(hk(uk+1)) ∈ U. Let nk+1 = Nk+1 + 1
and let vk+1 = h−1k (xnk+1).
Let hk+1 = gnk+1 hk ∈ G then hk+1(ui) ∈ U and hk+1(vi) ∈ V for 1 ≤ i ≤ k + 1. Then as
claimed there is the following for all n ∈ N and all k ≤ n
1. hn ∈ G 2. Wn = B r
n−1 (h−1n−1(x)) ⊆ X , thus diam(Wn)→ 0
3. un ∈Wn ∩X and vn ∈Wn ∩X 4. hn(uk) ∈ U and hn(vk) ∈ V
Claim 4. There is an h ∈ G such that h(x) = A(h, ⋅) C → C is not continuous which
contradicts our assumption that A is continuous.
Since C is a compact metric space, there is a subsequence of {uk}∞k=1 which
converges to a single point in C . Without loss of generality ignore subindices. Let z ∈ C
such that uk → z in C . Since diam(Wn)→ 0, we also have vk → z in C as well.
Since G is a compact metric group, the sequence {hk}∞k=1 has a convergent
subsequence to an element in G . Without loss of generality ignore subindices. Let
h ∈ G such that hk → h in G .
For a fixed i ∈ N and the corresponding compact (finite, two point) set {ui , vi} since
hk → h in G it follows that hk(ui) → h(ui) and hk(vi) → h(vi). For k > i the points hk(ui) ∈ U
and hk(vi) ∈ V thus h(ui) ∈ U and h(vi) ∈ V .
Since h ∈ G we should have h C → C being continuous, so h(ui) → h(z) ∈ U
and h(vi) → h(z) ∈ V . Thus we have h(z) ∈ U ∩ V = ∅ which means that h C → C is
discontinuous as claimed.
This is a contradiction and thus it must be the case that A is continuous as desired.
While the prior section demonstrated a construction for choosing a compactification
to which the group action would perforce extend, with Theorem 3.5 we have criteria to
42
guarantee an extension to a compactification of our choice. In the next section of the
chapter below, we will explore and justify the criteria we require in the theorem.
3.4 Examples
One might think that simply requiring every element of a group action upon a space
to extend would be sufficient to force the group action to extend. However, we impose
by Theorem 3.5 two additional criteria, namely that the group be compact and that the
compactification be metric. We now give an example of a simple space, provide effective
group actions on that space, and demonstrate that they will not extend when one of the
criteria from the theorem is not met.
Consider the space X = N × Z2, where N denotes the natural numbers and Z2
the group of two elements. Let C be a compactification of N and let Y = C × Z2 be the
corresponding compactification of X .
0 ... ∴ C N × {0} 1 2 3 4 . . .
N C N
Figure 3-2. Compactification of N ×Z2, which will be referred to as the space Y
We will first show that the condition that the group be compact is not a spurious
choice. We will define an effective group action on X by a non-compact group, and show
that this group action does not extend to any compactification, Y , defined above. Define
for each i ∈ N, the homeomorphism fi X → X given by:
fi(n, z) =
(n, z) n ≠ i
43
Let e X → X be the identity, and let Gw be the group generated by {fi}∞i=1 (in other words
the countable weak product of Z2 actions). For any compact set K , let NK = max{n ∈
N(n, z) ∈ K}, then for i > NK we have fi K = e. We have that fi → e with the topology
generated by convergence on compact sets.
For each i ∈ N the homeomorphism fi extends to a fi Y → Y by:
fi(α) =
α α ∈ Y X
Let e Y → Y be the identity on Y . The extension, fi , is also a homeomorphism since
the function fi is a homeomorphism, the composition fi fi = e, and for any neighborhood
Ny of Y X where Ny ⊂ Y {(n, z)n ≤ i , z ∈ Z2} we have fi Ny = e and, thus fi is
continuous. Since every g ∈ Gw is the finite composition of elements from {fi}∞i=1, the map
g X → X extends to a homeomorphism g Y → Y as well.
Yet, in the sequence {fi}∞i=1, we do not have fi → e. To see this, consider α ∈ Y X
and any neighborhood, Nα ⊂ Y , of α. For every N ∈ N, there is an nN > N such that
(nN , z) ∈ Nα for some z ∈ Z2. Now fnN(nN , z) ≠ e(nN , z) = (nN , z), so lim fi ≠ e.
Even though every element of the group Gw extends to a homeomorphism from Y
to itself and the group structure on Gw is maintained, the topology of the group action
is not. It is possible to define extensions of Gw to some compactifications of X , but not
those compactifications in the form that we defined for Y . Given a Y defined above,
define Z = Y/∼ by (c , z) ∼ (c ′, z ′) if and only if c = c ′ ∈ C N. The group action Gw extends
as a group action to Z , where for every f ∈ Gw we have fZX ≡ 1Z , the identity on the
space Z .
Next, we will justify our decision to require also that the compactification be a metric
compactification. Let us consider a larger group, a compact group that was mentioned
earlier, namely the Cantor group Gs ≅ Π∞i=1Z2. For every γ = (γi) ∈ Π∞i=1Z2, define
fγ X → X by fγ(n, z) = (n, z ⊕Z2 γn). These functions yield a Gs group action on X . If, for
44
every γ ∈ Gs , the homeomorphism fγ extends to a homeomorphism fγ Y → Y , then we
will show that C = βN, the Stone-Cech compactification of N.
For each γ = (γi) ∈ Π∞i=1Z2, define a function Fγ N → {0, 1} by Fγ(n) = γn. Since
C × {0} is open in Y and since fγ extends to a homeomorphism fγ, there is a continuous
extension Fγ C → {0, 1} given by
Fγ(α) =
0 α ∈ C N, fγ(α, 0) ∈ C × {0}
1 α ∈ C N, fγ(α, 0) ∈ C × {1}
Since every function F N → {0, 1} extends continuously to a function F C → {0, 1}
by definition of the Stone-Cech compactification, we have that the compactification
C = βN. Since Y = C × Z2, we have Y ≅ βN. Since the only compact sets of βN that are
metrizable are finite sets, the orbits of a compact group action on it would be perforce
trivial. Consider the extension fγ Y → Y of fγ where γ = (1)∞i=1 which would perforce
have to swap (C N) × {0} with (C N) × {1}. Thus Gs does not act on Y as a group
action.
Again, as was the case with Gw acting on X , there are extensions of the group
action Gs to compactifications of X , but we cannot mandate that those compactifications
be of the form given to Y . Since Gs is a Cantor group, we can use Theorem 3.4 to
construct a compactification, Z , of the space X where there is an extension of the given
Gs group action to a Gs group action on Z . This compactification will be the Z = Y/∼ that
was defined by (c , z) ∼ (c ′, z ′) if and only if c = c ′ ∈ C N that we described earlier in
the section. There will be an extension of the given free Gs action on X to a Gs-action on
Z . For every γ ∈ Gs , we extend the homeomorphism fγ X → X to the homeomorphism
fγ Z → Z by fγ(α) = α for every α ∈ Z X .
Moreover the use of Z2 as a factor of X in our example could have been replaced
by a more interesting compact group such as the p-adic numbers or some other Cantor
45
group without difficulty. Likewise, given a zero-dimensional compact group action on a
separable metric space, it is not hard to see an example of this kind occurring within it as
a sub-action. Thus the obstacles that this simple example highlights are, in some sense,
fundamental.
This example demonstrates the importance of understanding how group actions
on spaces can be extended to group actions on compactifications of those spaces.
Every effective Cantor group action on a complete metric space that does not have
an open set of periodic points and that the action does not have isolated orbits, is an
extension of the unique free action, by that Cantor group, on the space of irrationals.
Since the group of p-adic numbers is just a Cantor group, if a counter-example to the
Hilbert-Smith conjecture were to exist, we have shown that this counter-example would
have to be an extension of the unique free p-adic action on the space of irrationals. Thus
the Hilbert-Smith conjecture can be viewed as a question on the possible extensions of
this one free p-adic action of the space of irrationals to manifold compactifications.
3.5 Rings of Continuous Functions
A Gillman B Jerison C Henriksen
Figure 3-3. L. Gillman, M. Jerison, and M. Henriksen ran the Purdue seminar that lead to the famous book on rings of continuous functions
Since, in the earlier sections of this chapter, we have established the Hilbert-Smith
Conjecture as a question of extensions of a group action from a metric space to a
manifold compactification of it, it is natural to seek a setting in which compactifications
and extending maps to them are reasonably well-suited. For the reader unfamiliar with
Rings of Continuous Functions, I recommend the well-known book similarly entitled
by Leonard Gillman and Meyer Jerison [20]. It is a very useful book that sprang from
46
a seminar at Purdue in 1954-1955 that was organized by Melvin Henriksen. The roots
of the subject lie in the works of Marshall Harvey Stone[50], Edwin Hewitt[25], and
Kolmogorov’s paper with his student Izrail Moisevich Gelfand[19].
A Stone B Hewitt C Kolmogoroff D Gelfand
Figure 3-4. M.H. Stone, E. Hewitt, A. N. Kolmogoroff, and I.M. Gelfand laid the foundations for studying rings of continuous functions
Definition 3.1. Let X be a completely regular topological space. Define C(X ) = {f
X → Rf is continuous} and C ∗(X ) ⊂ C(X ) be the subset of bounded functions. From
the ring structure on R, the collections C(X ) and C∗(X ) both inherit a ring structure.
We call C(X ) the ring of continuous functions on the space X , and C∗(X ) the ring of
bounded continuous functions on X .
The ring C∗(X ) of bounded continuous functions encodes all of the topological
information of the completely regular topological space X . The following theorems help
to illustrate this:
Theorem 3.6 (Gelfand-Kolmogorov[19]). The maximal ideals in C ∗(X ) are in one to one
correspondence with the points p ∈ βX , the Stone-Cech compactification of X .
Theorem 3.7 ([20]). If a closed subring F ⊂ C∗(X ) separates points in X , then there is a
compactification CF of X such that C∗(CF) ≅ F . Moreover, if the subring F is separable,
then the compactification CF is metric. Conversely, if there is a compactification C of X
then there is a closed subring F ⊂ C ∗(X ) that generates the topology where C∗(C) ≅ F .
We will use the notation CF to denote the compactification associated with the closed
subring F (and note that CC∗(X) ≅ βX ). From the two theorems, it becomes apparent that
the ring of bounded continuous functions on the space of irrationals C ∗(I) is a natural
setting for our problem.
47
3.5.1 Group Actions on C∗(X ) and a Metric on C∗(X )
Every continuous function h X → X naturally induces a ring homomorphism
h∗ C∗(X ) ← C ∗(X ). If G × X → X is a topological group action, then for each g ∈ G
there is a corresponding g∗ C ∗(X ) ← C ∗(X ). One might think that it would be natural
to impose the sup-norm metric on C ∗(X ), however this winds up being problematic.
For one thing, the G action on X does not induce a topological group action on C ∗(X )
equipped with the sup-norm topology even if it perforce retains its algebraic structure.
We can instead define another metric on C ∗(X ), based on a norm taken over maximal
ideals:
Definition 3.2. For every f ∈ C ∗(X ), define d(f , 0) = f = supM(f ), where the
supremum is taken over all maximal ideals M in C ∗ (X ).
Under this metric, the group action G induces a group action G ∗ on C ∗(X ). Moreover,
if the G action on X is effective (free), then the induced G ∗ action is effective (free) on
C ∗(X ).
Given a separable, closed subring F ⊂ C ∗(I) that generates the topology it suffices,
in light of Theorem 3.5 , merely that F is p invariant. In this case, each of the functions
g I → I in the compact group p would extend to the metric compactification CF .
Thus, by Theorem 3.5, the group action p × I → I would extend as a group action to
p × CF → CF . We need not consider a metric in which ∗p acts upon F , but rather just
need to check for each g ∈ p that Fg∗ ⊂ F .
3.5.2 Dimension in C ∗(X )
The ring of bounded continuous functions C∗(X ) encodes the topology of the space
X , and recognizing the dimension of X from it’s bounded ring of continuous functions
is something that is of interest to us. In order to do this, we must first introduce a few
definitions that will be used in defining what is known as the analytic dimension of a
subring of C∗(X ).
Definition 3.3 (Analytic subring[20]). A subring A ⊂ C ∗(X ) is an analytic subring if
48
2. f 2 ∈ A implies f ∈ A.
Definition 3.4. Let B be any subfamily of C∗(X ). Let A be the intersection of all analytic
subrings of C ∗(X ) containing B. Since the intersection of analytic subrings are analytic,
A is an analytic subring. The family B is called the analytic base for A.
In this setting, the notion of analytic dimension was defined by Miroslav Katetov.
Figure 3-5. M. Katetov was not only a famous mathematician, but was also an international chessmaster and the Prague champion in 1942 and 1946.
Definition 3.5 (Analytic Dimension [31]). Let X be a completely regular space. The
analytic dimension (or Katetov dimension), ad C ∗(X ), is the least cardinal m such that
every countable family in C ∗(X ) is contained in an analytic subring having a base of
power ≤m.
From the following theorem, we see that analytic dimension of C∗(X ) will coincide
with the Lebesgue covering dimension of X .
Theorem 3.8 (Katetov [20]). The following are equivalent for any completely regular
space X .
2. ad C ∗(X ) ≤ n,
3. Every finite subfamily of C∗(X ) is contained in an analytic subring having a base of cardinal ≤ n.
3.5.3 Hilbert-Smith in C ∗(X )
Within this framework, we can restate the Hilbert-Smith Conjecture as follows:
Conjecture 7 (Hilbert-Smith). Let p × I → I be the free p-adic group action on the
space of irrationals. If F ⊂ C∗(I) is a closed, separable subring that generates the
49
topology, has analytic dimension n ∈ N, and is generated by n functions {fi}ni=1, then F is
not ∗p invariant.
Should such a closed subring F exist, then the corresponding compactification
CF would imbed in I n using the set {fi}ni=1. Since ad F = n implies dimCF = n, the set
CF ⊂ I n ≠ ∅. Restricting the extended group action to p × CF → CF and then choosing
a sufficiently small non-trivial subgroup of p would produce the counter-example to
Hilbert-Smith. Conversely, should there be a counter-example to Hilbert-Smith for some
dimension n, then there would be a closed, separable subring F ⊂ C∗(I) with the above
properties.
I believe that in this setting the problem will prove to be more tractable and allow
for a complete, rather than merely partial, solution. Translating problems into Rings of
Continuous Functions has proved in the past to be a nice medium for easy solutions
to harder problems[20], and it is our hope that it will happen again. After all, one
can translate the quotient spaces X/k to subrings of C ∗(X ) that do not separate k
orbits, which seems an easier package than the original which would be a collection of
dimensionally deficient Peano continua. Further, it would be wonderful to highlight the
usefulness to topology for which these rings clearly hold the potential to deliver.
50
4.1 Equivariant Partitions of Peano Continua
A Peano continuum, X , is defined as a compact, connected, locally connected
metric space. Since satisfying these criteria is equivalent to saying that the space X
is the continuous image of an arc, Peano continua are also referred to as continuous
curves. In 1949, R.H. Bing proved that every continuous curve is partitionable [8].
If there is an effective p-adic group action on a Peano continuum X , then a natural
question is to ask, whether or not there are partitions of X that respect the group action.
This section demonstrates that, for every > 0, the space X can be partitioned by
partitionable sets of diameter less than such that the group action merely finitely
permutes these sets.
We begin with the proof of an elementary theorem that is by no means anything
new, and can be found in elementary text books (e.g. Theorem 3.7.2 in Engelking[17]).
Theorem 4.1. If f X → Y is a perfect mapping, the set A ⊂ X is closed, the set B ⊂ Y is
compact, and f (A) = B, then the set A is compact.
Proof. Let C = {xi}N ⊂ A be an infinite subset, and let D = f (C).
Case 1: The set D is finite. There is a y ∈ B such that f (xi) = y for infinitely many
values of i . Since f (x) is a perfect mapping, the set f −1(y) is compact. By assumption,
the set A is closed, so the intersection A ∩ f −1(y) is compact, which implies that the set
C admits a limit point in A.
Case 2: The set D is infinite. Since B is compact, the set D admits a limit point,
y ∈ B. Suppose, by way of contradiction, C admits no limit points in A. Since A is a
closed set, the set C is also closed in A, and thus in X . Let E = f −1(y) ∩ A, which is
non-empty and compact since p(x) is a perfect mapping, f (A) = B, and A is closed. The
set C ∩ E must be finite, so C E is infinite and closed in X . Since p(x) is a perfect map,
51
p(x) is a closed map; and the set p(C E) = D {y} is closed. This is a contradiction
with the point y being a limit point of D.
In either case the set C admits a limit point in A, so the set A is compact.
Corollary 4.1.1. If f X → Y is a perfect mapping and B ⊂ Y is compact, then f −1(B) is
compact.
Proof. Let A = f −1(B) ⊂ X . Since f (x) is continuous, the set A ⊂ X is closed, and thus
the set A is compact by Theorem 4.1,.
Lemma 4.1. Let X and Y be connected, locally connected metric spaces. If f X → Y
is a perfect, light open map and U ⊂ Y is open with the property that U is compact and
locally connected, then V = f −1(U) has finitely many components.
Proof. Pick a point y ∈ U. Since f (x) is continuous, the set V ⊂ X is open. Since X is
locally connected, for each point x ∈ W = f −1(y) ⊂ V , there is a connected open set
Ox ⊂ V . Since f (x) is perfect, the setW is compact. The collection {Ox} is an open
cover ofW , so there are a finite sub-collection of open sets Oxi which coverW . Since
each Oxi is connected, there are finitely many components of V containing points ofW ,
which implies that each of those components is both open and closed in V . Since f (x)
is an open map, the image of each component is open. Likewise, since f (x) is a perfect
mapping, it is a closed map, which implies the image of each component is also closed.
Since U is connected, the image of each component is all of U.
Thus, the pre-image V has only finitely many components.
Theorem 4.2. Let X and Y be connected, locally connected metric spaces. If f X → Y
is a perfect, light open map and U ⊂ Y is open, with the property that U is compact and
locally connected, then f −1(U) is also compact and locally connected.
52
Proof. From Theorem 4.1, the pre-image V = f −1(U) is compact. Thus, it suffices
to show that V has property S. By Bing’s partitioning theorems [8], this condition is
equivalent to V being partitionable.
Since U is compact and loca