James Maissen An equivalence to the Hilbert-Smith conjecture University of Florida

An equivalence to the Hilbert-Smith conjecture
James Maissen
University of Florida
47th Spring Topology and Dynamics Conference
Central Connecticut State University
New Britain, CT
March 24th, 2013
This work is in collaboration with:
This work is in collaboration with:
Jed Keesling
This work is in collaboration with:
Jed Keesling
David C. Wilson
History
Hilbert’s Problems
On August 8th, 1900 at the Second International Congress in
Paris, David Hilbert gave his famous speech. He challenged the
world with ten unsolved problems.
David Hilbert
James Maissen
Hilbert-Smith conjecture
History
Hilbert’s Problems
On August 8th, 1900 at the Second International Congress in
Paris, David Hilbert gave his famous speech. He challenged the
world with ten unsolved problems.
When it made it to publication, the number had grown to 23. The
fifth of those is often written as:
David Hilbert
Hilbert in 1885
James Maissen
Hilbert-Smith conjecture
History
Hilbert’s Problems
On August 8th, 1900 at the Second International Congress in
Paris, David Hilbert gave his famous speech. He challenged the
world with ten unsolved problems.
When it made it to publication, the number had grown to 23. The
fifth of those is often written as:
Hilbert’s Fifth Problem (1900)
Is every (finite-dimensional) locally Euclidean topological group
necessarily a Lie group?
David Hilbert
Hilbert in 1885
James Maissen
Picture from 1912 postcard
Hilbert-Smith conjecture
History
Hilbert’s Fifth problem was generalized in 1940 by Paul A. Smith:
Paul Althaus Smith (1900-1980)
James Maissen
Hilbert-Smith conjecture
History
Hilbert’s Fifth problem was generalized in 1940 by Paul A. Smith:
Hilbert-Smith Conjecture (1940s)
If G is a locally compact group which acts effectively on a manifold as a
(topological) transformation group, then G is a Lie group.
Paul Althaus Smith (1900-1980)
James Maissen
Hilbert-Smith conjecture
History
Hilbert’s Fifth problem was generalized in 1940 by Paul A. Smith:
Hilbert-Smith Conjecture (1940s)
If G is a locally compact group which acts effectively on a manifold as a
(topological) transformation group, then G is a Lie group.
Which he proved is equivalent to:
Paul Althaus Smith (1900-1980)
James Maissen
Hilbert-Smith conjecture
History
Hilbert’s Fifth problem was generalized in 1940 by Paul A. Smith:
Hilbert-Smith Conjecture (1940s)
If G is a locally compact group which acts effectively on a manifold as a
(topological) transformation group, then G is a Lie group.
Which he proved is equivalent to:
Hilbert-Smith Conjecture
There does not exist an effective action of a p-adic group on a manifold.
Paul Althaus Smith (1900-1980)
James Maissen
Hilbert-Smith conjecture
History
Hilbert’s Fifth problem was generalized in 1940 by Paul A. Smith:
Hilbert-Smith Conjecture (1940s)
If G is a locally compact group which acts effectively on a manifold as a
(topological) transformation group, then G is a Lie group.
Which he proved is equivalent to:
Hilbert-Smith Conjecture
There does not exist an effective action of a p-adic group on a manifold.
Paul Althaus Smith (1900-1980)
”was tall and thin,
with lots of hair early gone white,
making him seem much older than he was.”
(from Sherman Stein, one of Smith’s students)
James Maissen
Hilbert-Smith conjecture
History
Hilbert’s Fifth problem was generalized in 1940 by Paul A. Smith:
Hilbert-Smith Conjecture (1940s)
If G is a locally compact group which acts effectively on a manifold as a
(topological) transformation group, then G is a Lie group.
Which he proved is equivalent to:
Hilbert-Smith Conjecture
There does not exist an effective action of a p-adic group on a manifold.
Paul Althaus Smith (1900-1980)
”was tall and thin,
with lots of hair early gone white,
making him seem much older than he was.”
(from Sherman Stein, one of Smith’s students)
photo courtesy of University Archives, Columbia University in the City of New York (for $20)
James Maissen
Hilbert-Smith conjecture
Definitions and Notation
Definition: p-adic group
For a given prime number p, the p-adic group is an abelian group
of the form Ap := lim{Zpk , φk+1
} where Zpk := Z/pk Z, and
k
←
−
φk+1
: Zpk+1 → Zpk are p-fold group homomorphisms. In other
k
words, this is just the p-odometer group, and is topologically a
Cantor Set.
James Maissen
Hilbert-Smith conjecture
Definitions and Notation
Definition: p-adic group
For a given prime number p, the p-adic group is an abelian group
of the form Ap := lim{Zpk , φk+1
} where Zpk := Z/pk Z, and
k
←
−
φk+1
: Zpk+1 → Zpk are p-fold group homomorphisms. In other
k
words, this is just the p-odometer group, and is topologically a
Cantor Set.
If a group, G with identity element e, acts on a space X then
Definition: Effective Action (a.k.a. Faithful Action)
We say G acts effectively on the space X if and only if for every
g ∈ G \ {e}, there exists a point x ∈ X such that g (x) 6= x.
James Maissen
Hilbert-Smith conjecture
Definitions and Notation
Definition: p-adic group
For a given prime number p, the p-adic group is an abelian group
of the form Ap := lim{Zpk , φk+1
} where Zpk := Z/pk Z, and
k
←
−
φk+1
: Zpk+1 → Zpk are p-fold group homomorphisms. In other
k
words, this is just the p-odometer group, and is topologically a
Cantor Set.
If a group, G with identity element e, acts on a space X then
Definition: Effective Action (a.k.a. Faithful Action)
We say G acts effectively on the space X if and only if for every
g ∈ G \ {e}, there exists a point x ∈ X such that g (x) 6= x.
Definition: Free Action (a.k.a. Strongly Effective Action)
We say G acts freely on the space X if and only if for every
g ∈ G \ {e} and for every point x ∈ X , we have g (x) 6= x.
James Maissen
Hilbert-Smith conjecture
Partial Solutions
There are affirmative solutions to the Hilbert-Smith conjecture
proving that Ap cannot act effectively on a manifold M, when
James Maissen
Hilbert-Smith conjecture
Partial Solutions
There are affirmative solutions to the Hilbert-Smith conjecture
proving that Ap cannot act effectively on a manifold M, when
dim M = 2 (L.E.J. Brouwer 1919, separately B. Kerékjártó)
LEJ Brouwer
(voutsadakis.com)
Bela Kerékjártó
(history.mcs.st-and.ac.uk)
James Maissen
Hilbert-Smith conjecture
Partial Solutions
There are affirmative solutions to the Hilbert-Smith conjecture
proving that Ap cannot act effectively on a manifold M, when
dim M = 2 (L.E.J. Brouwer 1919, separately B. Kerékjártó)
dim M = 3 (J. Pardon 2011)
John Pardon
(paw.princeton.edu)
James Maissen
Hilbert-Smith conjecture
Partial Solutions
There are affirmative solutions to the Hilbert-Smith conjecture
proving that Ap cannot act effectively on a manifold M, when
dim M = 2 (L.E.J. Brouwer 1919, separately B. Kerékjártó)
dim M = 3 (J. Pardon 2011)
The group Ap acts by diffeomorphisms (Bochner-Montgomery
1946)
The group Ap acts by Lipschitz homeomorphisms
(S̆c̆epin-Repovs̆ 1997), et al.
Salomon Bochner Deane Montgomery
(pictures from history.mcs.st-and.ac.uk)
James Maissen
Evengy S̆c̆epin and Dus̆an Repovs̆
(from Jed Keesling and www.pef.uni-lj.si)
Hilbert-Smith conjecture
What if?
Suppose the Hilbert-Smith conjecture were false
Then there would be an n-dimensional manifold, M, admitting an
effective action by the p-adic group, Ap .
And there would be a quotient space (or orbit space) of this action.
James Maissen
Hilbert-Smith conjecture
What if?
Suppose the Hilbert-Smith conjecture were false
Then there would be an n-dimensional manifold, M, admitting an
effective action by the p-adic group, Ap .
And there would be a quotient space (or orbit space) of this action.
What could be said about such a quotient space?
(Image taken from http://www.lorisreflections.com)
James Maissen
Hilbert-Smith conjecture
Features of the quotient space
If the p-adic group, Ap , acted effectively on an n-dimensional
manifold M, then:
dimZ M/Ap 6= dim(M) (P.A. Smith 1940)
Paul Althaus Smith
photo of P.A. Smith courtesy of Columbia University Archives
Columbia University in the City of New York
James Maissen
(hey for $20 I’m going to use it more than once)
Hilbert-Smith conjecture
Features of the quotient space
If the p-adic group, Ap , acted effectively on an n-dimensional
manifold M, then:
dimZ M/Ap 6= dim(M) (P.A. Smith 1940)
dimZ M/Ap = 2 + dim(M) (C.T. Yang 1960)
(math.upenn.edu)
James Maissen
Hilbert-Smith conjecture
Features of the quotient space
If the p-adic group, Ap , acted effectively on an n-dimensional
manifold M, then:
dimZ M/Ap =
6 dim(M) (P.A. Smith 1940)
dimZ M/Ap = 2 + dim(M) (C.T. Yang 1960)
Frank Raymond
(math.lsa.umich.edu)
Robert Fones Williams
(ma.utexas.edu)
James Maissen
Hilbert-Smith conjecture
Features of the quotient space
If the p-adic group, Ap , acted effectively on an n-dimensional
manifold M, then:
dimZ M/Ap 6= dim(M) (P.A. Smith 1940)
dimZ M/Ap = 2 + dim(M) (C.T. Yang 1960)
The quotient space M/Ap cannot be dimensionally full-valued
(Raymond-Williams 1963)
Frank Raymond
(math.lsa.umich.edu)
Robert Fones Williams
(ma.utexas.edu)
James Maissen
Hilbert-Smith conjecture
Not Dimensionally Full-Valued
Definition: Dimensionally Full-Valued
(a.k.a. not Dimensionally Deficient)
A topological space X is called dimensionally full-valued, if for all
topological spaces Y , the following equality holds:
dim(X × Y ) = dim X + dim Y
James Maissen
Hilbert-Smith conjecture
Not Dimensionally Full-Valued
Definition: Dimensionally Full-Valued
(a.k.a. not Dimensionally Deficient)
A topological space X is called dimensionally full-valued, if for all
topological spaces Y , the following equality holds:
dim(X × Y ) = dim X + dim Y
In 1930, L.S. Pontryagin constructed 2-dimensional spaces Πp (for
each prime p) such that products between them are 3-dimensional.
And his student V.G. Boltyanskii constructed a 2-dimensional
space where its product with itself is only 3-dimensional.
Lev S. Pontryagin
Valdimir G. Boltyanskii
James Maissen
Hilbert-Smith conjecture
So if a counter-example were to exist. . .
If a p-adic group, Ap , acted effectively on an n-dimensional
manifold M then:
James Maissen
Hilbert-Smith conjecture
So if a counter-example were to exist. . .
If a p-adic group, Ap , acted effectively on an n-dimensional
manifold M then:
n≥4
James Maissen
Hilbert-Smith conjecture
So if a counter-example were to exist. . .
If a p-adic group, Ap , acted effectively on an n-dimensional
manifold M then:
n≥4
dimZ M/Ap = n + 2
James Maissen
Hilbert-Smith conjecture
So if a counter-example were to exist. . .
If a p-adic group, Ap , acted effectively on an n-dimensional
manifold M then:
n≥4
dimZ M/Ap = n + 2
M/Ap
is dimensionally deficient
James Maissen
Hilbert-Smith conjecture
So if a counter-example were to exist. . .
If a p-adic group, Ap , acted effectively on an n-dimensional
manifold M then:
n≥4
dimZ M/Ap = n + 2
M/Ap
is dimensionally deficient
the group Ap cannot act by diffeomorphisms, Lipschitz
homeomorphisms, etc
James Maissen
Hilbert-Smith conjecture
So if a counter-example were to exist. . .
If a p-adic group, Ap , acted effectively on an n-dimensional
manifold M then:
n≥4
dimZ M/Ap = n + 2
M/Ap
is dimensionally deficient
the group Ap cannot act by diffeomorphisms, Lipschitz
homeomorphisms, etc
What else can we say about such a theoretical counter-example?
James Maissen
Hilbert-Smith conjecture
The Space of Irrationals
Hausdorff Characterization of the Irrationals
The space of irrational numbers I is characterized as the
0-dimensional, nowhere locally compact, separable metric space
that is topologically complete.
Felix Hausdorff
James Maissen
Hilbert-Smith conjecture
The Space of Irrationals
Hausdorff Characterization of the Irrationals
The space of irrational numbers I is characterized as the
0-dimensional, nowhere locally compact, separable metric space
that is topologically complete.
Theorem (Exercise/ Easy Homework)
If the mapping p : I → Y is a perfect open surjection, then the
space Y ∼
= I, the space of irrationals.
James Maissen
Hilbert-Smith conjecture
The Space of Irrationals
Hausdorff Characterization of the Irrationals
The space of irrational numbers I is characterized as the
0-dimensional, nowhere locally compact, separable metric space
that is topologically complete.
Theorem (Exercise/ Easy Homework)
If the mapping p : I → Y is a perfect open surjection, then the
space Y ∼
= I, the space of irrationals.
Corollary
If a zero-dimensional compact group, G , acts freely on the space of
irrationals I, then the quotient space I/G ∼
= I.
James Maissen
Hilbert-Smith conjecture
Unique Free Ap Action on I
Theorem (Maissen)
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group.
i
←
−
James Maissen
Hilbert-Smith conjecture
Unique Free Ap Action on I
Theorem (Maissen)
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group.
i
←
−
If the mapping A : G × I → I is a free topological group action,
then there is a homeomorphism h : I → I × G such that
James Maissen
Hilbert-Smith conjecture
Unique Free Ap Action on I
Theorem (Maissen)
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group.
i
←
−
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 (π1 h(w ), g π2 h(w )),
with π1 : I × G → I and π2 : I × G → G the projection maps.
James Maissen
Hilbert-Smith conjecture
Unique Free Ap Action on I
Theorem (Maissen)
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group.
i
←
−
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 (π1 h(w ), g π2 h(w )),
with π1 : I × G → I and π2 : I × G → G the projection maps.
In other words, every free G -action on I is conjugate to the action
B : G × (I × G ) → I × G (merely acting on the second factor).
James Maissen
Hilbert-Smith conjecture
Finding the free Irrational sub-action in spaces
To find a space that will contain our sub-action on the space of
Irrational numbers, we need to impose some conditions:
James Maissen
Hilbert-Smith conjecture
Finding the free Irrational sub-action in spaces
Theorem (Keesling, Maissen, Wilson)
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group.
i
←
−
James Maissen
Hilbert-Smith conjecture
Finding the free Irrational sub-action in spaces
Theorem (Keesling, Maissen, Wilson)
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group.
i
←
−
If X is a complete metric space upon which G acts effectively such
that
James Maissen
Hilbert-Smith conjecture
Finding the free Irrational sub-action in spaces
Theorem (Keesling, Maissen, Wilson)
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group.
i
←
−
If X is a complete metric space upon which G acts effectively such
that the set of periodic points contains no open set and
James Maissen
Hilbert-Smith conjecture
Finding the free Irrational sub-action in spaces
Theorem (Keesling, Maissen, Wilson)
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group.
i
←
−
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
James Maissen
Hilbert-Smith conjecture
Finding the free Irrational sub-action in spaces
Theorem (Keesling, Maissen, Wilson)
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group.
i
←
−
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 dense subspace Y ⊂ X , with
Y ∼
= I, such that G acts freely on Y .
James Maissen
Hilbert-Smith conjecture
Finding the free Irrational sub-action in spaces
Theorem (Keesling, Maissen, Wilson)
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group.
i
←
−
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 dense subspace Y ⊂ X , with
Y ∼
= I, such that G acts freely on Y .
If there is a counter-example to the Hilbert-Smith conjecture, then
it will contain the unique free p-adic action on the space of
irrationals.
James Maissen
Hilbert-Smith conjecture
Finding the free Irrational sub-action in spaces
Theorem (Keesling, Maissen, Wilson)
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group.
i
←
−
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 dense subspace Y ⊂ X , with
Y ∼
= I, such that G acts freely on Y .
If there is a counter-example to the Hilbert-Smith conjecture, then
it will contain the unique free p-adic action on the space of
irrationals. Thus the Hilbert-Smith conjecture becomes:
James Maissen
Hilbert-Smith conjecture
Finding the free Irrational sub-action in spaces
Theorem (Keesling, Maissen, Wilson)
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group.
i
←
−
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 dense subspace Y ⊂ X , with
Y ∼
= I, such that G acts freely on Y .
If there is a counter-example to the Hilbert-Smith conjecture, then
it will contain the unique free p-adic action on the space of
irrationals. Thus the Hilbert-Smith conjecture becomes:
Hilbert-Smith
Can we extend the unique p-adic group action on the space of
irrationals to a manifold compactification of that space?
James Maissen
Hilbert-Smith conjecture
Extending Group Actions to Compactifications
Theorem (Maissen)
There is an extension to some compactification
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group and
i
←
−
X be a separable space.
James Maissen
Hilbert-Smith conjecture
Extending Group Actions to Compactifications
Theorem (Maissen)
There is an extension to some compactification
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group and
i
←
−
X be a separable space.
If G acts effectively on X , then there is a metric compactification
C of X such that G extends to an effective action on C .
James Maissen
Hilbert-Smith conjecture
Extending Group Actions to Compactifications
Theorem (Maissen)
There is an extension to some compactification
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group and
i
←
−
X be a separable space.
If G acts effectively on X , then there is a metric compactification
C of X such that G extends to an effective action on C .
Theorem (Maissen)
Sufficient conditions to extend to a given compactification
James Maissen
Hilbert-Smith conjecture
Extending Group Actions to Compactifications
Theorem (Maissen)
There is an extension to some compactification
Let G = lim{Gi , φi+1
} be a zero-dimensional compact group and
i
←
−
X be a separable space.
If G acts effectively on X , then there is a metric compactification
C of X such that G extends to an effective action on C .
Theorem (Maissen)
Sufficient conditions to extend to a given compactification
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
Â(g , ·) = ĝ (x) : C → C , then  : G × C → C is a continuous
group action.
James Maissen
Hilbert-Smith conjecture
Compactifications of the space of Irrationals
We seek to translate the problem into a setting where
compactifications are less cumbersome:
James Maissen
Hilbert-Smith conjecture
Rings of Continuous Functions
We seek to translate the problem into a setting where
compactifications are less cumbersome:
The ring of bounded continuous functions C ∗ (X )
Let X be a completely regular topological space.
James Maissen
Hilbert-Smith conjecture
Rings of Continuous Functions
We seek to translate the problem into a setting where
compactifications are less cumbersome:
The ring of bounded continuous functions C ∗ (X )
Let X be a completely regular topological space.
Define C ∗ (X ) := {f : X → R|f is bounded and continuous}.
The ring structure of R defines a ring structure for C ∗ (X ).
James Maissen
Hilbert-Smith conjecture
Rings of Continuous Functions
We seek to translate the problem into a setting where
compactifications are less cumbersome:
The ring of bounded continuous functions C ∗ (X )
Let X be a completely regular topological space.
Define C ∗ (X ) := {f : X → R|f is bounded and continuous}.
The ring structure of R defines a ring structure for C ∗ (X ).
Theorem (Gelfand-Kolmorgoroff)
Let X be a completely regular topological space.
I.M. Gelfand
A.N. Kolmogoroff
James Maissen
Hilbert-Smith conjecture
Rings of Continuous Functions
We seek to translate the problem into a setting where
compactifications are less cumbersome:
The ring of bounded continuous functions C ∗ (X )
Let X be a completely regular topological space.
Define C ∗ (X ) := {f : X → R|f is bounded and continuous}.
The ring structure of R defines a ring structure for C ∗ (X ).
Theorem (Gelfand-Kolmorgoroff)
Let X be a completely regular topological space.
The maximal ideals in C ∗ (X ) are in one to one correspondence
with the points p ∈ βX .
I.M. Gelfand
A.N. Kolmogoroff
James Maissen
Hilbert-Smith conjecture
Compactifications in Rings of Continuous Functions
Theorem (Gillman-Jerison)
Let X be a completely regular topological space.
L. Gillman
M. Jerison
James Maissen
Hilbert-Smith conjecture
Compactifications in Rings of Continuous Functions
Theorem (Gillman-Jerison)
Let X be a completely regular topological space.
The collection of closed subrings F ⊂ C ∗ (X ) such that F contains
the constant functions and generates the topology of X is in one
to one correspondence with the compactifications CF of X , and
C ∗ (CF ) ∼
= F.
L. Gillman
M. Jerison
James Maissen
Hilbert-Smith conjecture
Compactifications in Rings of Continuous Functions
Theorem (Gillman-Jerison)
Let X be a completely regular topological space.
The collection of closed subrings F ⊂ C ∗ (X ) such that F contains
the constant functions and generates the topology of X is in one
to one correspondence with the compactifications CF of X , and
C ∗ (CF ) ∼
= F.
Moreover, a closed subring F that is separable corresponds to a
compactification CF being metric.
L. Gillman
M. Jerison
James Maissen
Hilbert-Smith conjecture
Compactifications in Rings of Continuous Functions
Theorem (Gillman-Jerison)
Let X be a completely regular topological space.
The collection of closed subrings F ⊂ C ∗ (X ) such that F contains
the constant functions and generates the topology of X is in one
to one correspondence with the compactifications CF of X , and
C ∗ (CF ) ∼
= F.
Moreover, a closed subring F that is separable corresponds to a
compactification CF being metric.
We begin to realize that the ring C ∗ (X ) contains all the
topological information of the space X .
James Maissen
Hilbert-Smith conjecture
Compactifications in Rings of Continuous Functions
Theorem (Gillman-Jerison)
Let X be a completely regular topological space.
The collection of closed subrings F ⊂ C ∗ (X ) such that F contains
the constant functions and generates the topology of X is in one
to one correspondence with the compactifications CF of X , and
C ∗ (CF ) ∼
= F.
Moreover, a closed subring F that is separable corresponds to a
compactification CF being metric.
We begin to realize that the ring C ∗ (X ) contains all the
topological information of the space X .
The question simply becomes proper translations of the features in
which we are interested over into the ring.
James Maissen
Hilbert-Smith conjecture
Dimension inside Rings of Continuous Functions
Definition: Analytic Subring, Base, and Dimension
A subring A ⊂ C ∗ (X ) is an analytic subring if all constant
functions belong to A; and f 2 ∈ A implies f ∈ A.
Miroslav Katětov
James Maissen
Hilbert-Smith conjecture
Dimension inside Rings of Continuous Functions
Definition: Analytic Subring, Base, and Dimension
A subring A ⊂ C ∗ (X ) is an analytic subring if all constant
functions belong to A; and f 2 ∈ A implies f ∈ A.
A family B of C ∗ (X ) is the analytic base for A, if A is the
intersection of all analytic subrings of C ∗ (X ) containing B.
Miroslav Katětov
James Maissen
Hilbert-Smith conjecture
Dimension inside Rings of Continuous Functions
Definition: Analytic Subring, Base, and Dimension
A subring A ⊂ C ∗ (X ) is an analytic subring if all constant
functions belong to A; and f 2 ∈ A implies f ∈ A.
A family B of C ∗ (X ) is the analytic base for A, if A is the
intersection of all analytic subrings of C ∗ (X ) containing B.
The analytic dimension (or Katětov 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.
Miroslav Katětov
James Maissen
Hilbert-Smith conjecture
Dimension inside Rings of Continuous Functions
Definition: Analytic Subring, Base, and Dimension
A subring A ⊂ C ∗ (X ) is an analytic subring if all constant
functions belong to A; and f 2 ∈ A implies f ∈ A.
A family B of C ∗ (X ) is the analytic base for A, if A is the
intersection of all analytic subrings of C ∗ (X ) containing B.
The analytic dimension (or Katětov 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.
Theorem (Katětov)
The following are equivalent for any completely regular space X .
1
dim X ≤ n,
2
ad C ∗ (X ) ≤ n,
3
Every finite subfamily of C ∗ (X ) is contained in an analytic
subring having a base of cardinal ≤ n.
James Maissen
Hilbert-Smith conjecture
The Hilbert-Smith Conjecture
in the setting of Rings of Continuous Functions
Conjecture (Hilbert-Smith)
Let Ap × I → I be the unique free p-adic group action on the
space of irrational numbers.
James Maissen
Hilbert-Smith conjecture
The Hilbert-Smith Conjecture
in the setting of Rings of Continuous Functions
Conjecture (Hilbert-Smith)
Let Ap × I → I be the unique free p-adic group action on the
space of irrational numbers.
Each F ⊂ C ∗ (I) cannot satisfy all of the following:
James Maissen
Hilbert-Smith conjecture
The Hilbert-Smith Conjecture
in the setting of Rings of Continuous Functions
Conjecture (Hilbert-Smith)
Let Ap × I → I be the unique free p-adic group action on the
space of irrational numbers.
Each F ⊂ C ∗ (I) cannot satisfy all of the following:
F is a closed separable subring,
James Maissen
Hilbert-Smith conjecture
The Hilbert-Smith Conjecture
in the setting of Rings of Continuous Functions
Conjecture (Hilbert-Smith)
Let Ap × I → I be the unique free p-adic group action on the
space of irrational numbers.
Each F ⊂ C ∗ (I) cannot satisfy all of the following:
F is a closed separable subring,
F contains the constants and generates the topology of I,
James Maissen
Hilbert-Smith conjecture
The Hilbert-Smith Conjecture
in the setting of Rings of Continuous Functions
Conjecture (Hilbert-Smith)
Let Ap × I → I be the unique free p-adic group action on the
space of irrational numbers.
Each F ⊂ C ∗ (I) cannot satisfy all of the following:
F is a closed separable subring,
F contains the constants and generates the topology of I,
F is A∗p -invariant,
James Maissen
Hilbert-Smith conjecture
The Hilbert-Smith Conjecture
in the setting of Rings of Continuous Functions
Conjecture (Hilbert-Smith)
Let Ap × I → I be the unique free p-adic group action on the
space of irrational numbers.
Each F ⊂ C ∗ (I) cannot satisfy all of the following:
F is a closed separable subring,
F contains the constants and generates the topology of I,
F is A∗p -invariant,
F has analytic dimension n for some n ∈ N, and
James Maissen
Hilbert-Smith conjecture
The Hilbert-Smith Conjecture
in the setting of Rings of Continuous Functions
Conjecture (Hilbert-Smith)
Let Ap × I → I be the unique free p-adic group action on the
space of irrational numbers.
Each F ⊂ C ∗ (I) cannot satisfy all of the following:
F is a closed separable subring,
F contains the constants and generates the topology of I,
F is A∗p -invariant,
F has analytic dimension n for some n ∈ N, and
F is generated by n functions {fi }ni=1 ⊂ C ∗ (I).
James Maissen
Hilbert-Smith conjecture