The Solution of Hilbert's Fifth Problem for Transitive Groupoids
In the following paper, we investigate the question: when is a transitive topological groupoid continuously isomorphic to a Lie groupoid? We present many results on the matter, which may be considered generalizations of Hilbert's fifth problem to this context. Most notably, we present a "s...
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| Cite this: | The Solution of Hilbert's Fifth Problem for Transitive Groupoids / P. Raźny // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 12 назв. — англ. |
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| description | In the following paper, we investigate the question: when is a transitive topological groupoid continuously isomorphic to a Lie groupoid? We present many results on the matter, which may be considered generalizations of Hilbert's fifth problem to this context. Most notably, we present a "solution" to the problem for proper transitive groupoids and transitive groupoids with compact source fibers.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 070, 11 pages
The Solution of Hilbert’s Fifth Problem
for Transitive Groupoids
Pawe l RAŹNY
Institute of Mathematics, Faculty of Mathematics and Computer Science,
Jagiellonian University in Cracow, Poland
E-mail: pawel.razny@student.uj.edu.pl
Received May 11, 2018, in final form July 10, 2018; Published online July 17, 2018
https://doi.org/10.3842/SIGMA.2018.070
Abstract. In the following paper we investigate the question: when is a transitive topo-
logical groupoid continuously isomorphic to a Lie groupoid? We present many results on
the matter which may be considered generalizations of the Hilbert’s fifth problem to this
context. Most notably we present a “solution” to the problem for proper transitive groupoids
and transitive groupoids with compact source fibers.
Key words: Lie groupoids; topological groupoids
2010 Mathematics Subject Classification: 22A22
1 Introduction
In this short paper we refine the generalization of Hilbert’s fifth problem (which states that
a locally Euclidean topological group is isomorphic as a topological group to a Lie group and
was famously solved in [4] and [7]) to the case of transitive groupoids. The main results of our
paper [10] gave a partial solution to this problem via the following results:
Theorem 1.1. Let G be a transitive topological groupoid with a smooth base G0, Tychonoff
source fibers Gx and first countable space of morphisms G1 for which the topological groups Gxx
are locally Euclidean and the map tx is a quotient map (or equivalently has the sequence covering
property). Then G is continuously isomorphic to a unique Lie groupoid through a base preserving
isomorphism.
Theorem 1.2. Let G be a transitive topological groupoid with a smooth base G0 for which the
spaces Gxx , G1 and Gx are topological manifolds and the map tx is a quotient map (or equivalently
has the sequence covering property). Then G is continuously isomorphic to a unique Lie groupoid
through a base preserving isomorphism.
Using these results we were able to solve the problem for proper transitive groupoids:
Theorem 1.3. Let G be a proper transitive topological groupoid with a smooth base G0, for
which the spaces Gxx , Gx and G1 are topological manifolds. Then G is continuously isomorphic to
a unique Lie groupoid through a base preserving isomorphism.
In this paper we provide the solution of the general case by dropping the assumption “tx is
a quotient map” in Theorem 1.2 (as it turns out it follows from other assumptions in Theo-
rem 1.2) and in doing so obtain Theorem 3.6 which is the main result of this paper. We also
present some versions of the Gleason–Yamabe theorem for transitive groupoids. Throughout
the paper we assume manifolds to be second-countable and Haudorff (in particular they are
metrizable).
mailto:pawel.razny@student.uj.edu.pl
https://doi.org/10.3842/SIGMA.2018.070
2 P. Raźny
2 Preliminaries
2.1 Some topology
In order to make this short paper as self-contained as possible we recall some basic topological
notions and properties which are used in subsequent sections. We begin with some well known
properties of quotient maps (identifications). A continuous surjective map p : X → Y is a quo-
tient map if a subset U of Y is open in Y if and only if p−1(U) is open in X. Equivalently,
Y is the quotient space of X with respect to the relation ∼ given by x ∼ y if and only if there
exists a point z ∈ Y such that x, y ∈ p−1(z). Quotient maps are characterized by the following
universal property:
Proposition 2.1. Given a quotient map p : X → Y and another continuous map f : X → Z
which is constant on the fibers of p there is a unique continuous map f : Y → Z such that the
following diagram commutes:
X Z
Y.
p
f
f
We also recall the following result which will be useful in the proof of the main theorem:
Theorem 2.2 ([9, Corollary 5]). Let f : M → N be a continuous bijection between topological
manifolds. Then dim(M) = dim(N) and f is a homeomorphism.
Furthermore, we note some facts about the dimension of a separable metric space from [3].
Definition 2.3. Let X be a set and U a family of subsets of X. By the order of the family U
we mean the largest integer n such that the family U contains n + 1 sets with a non-empty
intersection, if no such integer exists, we say that the family U has order ∞.
Definition 2.4. To a separable metric space X one assigns the dimension of X, denoted by
dim(X) defined by the following conditions:
1) dim(X) ≤ n, where n ∈ N if every finite open cover of the space X has a finite open
refinement of order ≤ n,
2) dim(X) = n if dim(X) ≤ n and it is not true that dim(X) ≤ n− 1,
3) dim(X) =∞ if dim(X) ≥ n for any n ∈ N
We also put dimx(X) = inf{dim(U) |U neighbourhood of x}.
Remark 2.5. The invariant defined above is called the covering dimension of X. We recall that,
in the separable metric case all three definitions of dimension (small inductive, large inductive
and covering; see [3]) coincide by [3, Theorem 1.7.7]. Hence, we simply call this invariant the
dimension of X and restate the theorems from [3] in a manner suitable to this convention.
Theorem 2.6 ([3, Theorem 1.1.2]). If X is a separable metric space and A ⊂ X is a subspace
then dim(A) ≤ dim(X).
Theorem 2.7 ([3, Theorem 1.5.3]). If a separable metric space X can be represented as the
union of a sequence {F1, F2, , . . . } of closed subspaces such that dimFi ≤ n for i ∈ N then
dim(X) ≤ n.
The Solution of Hilbert’s Fifth Problem for Transitive Groupoids 3
It is also well known that the dimension of a manifold is equal to its dimension as a separable
metric space and hence no ambiguity can arise (this is actually a consequence of the previous
two theorems and the fact that dim(Rn) = n; see [3, Theorem 1.8.2] for the latter).
In the final section of this paper we will be using the following important theorem:
Theorem 2.8 (Gleason–Yamabe theorem for compact groups). Let G be a compact Hausdorff
topological group. For every neighbourhood U of the identity of G there exists a compact normal
subgroup K contained in U such that G/K is isomorphic as a topological group to a Lie group.
The proof of this theorem can be found, e.g., in [12, Theorem 1.4.14].
2.2 Groupoids
We give a brief recollection of some basic notions concerning groupoids. Let us start by giving
the definition:
Definition 2.9. A groupoid G is a small category in which all the morphisms are isomorphisms.
Let us denote by G0 the set of objects of this category (also called the base of G) and by G1
the set of morphisms of this category. This implies the existence of the following five structure
maps:
1) the source map s : G1 → G0 which associates to each morphism its source;
2) the target map t : G1 → G0 which associates to each morphism its target;
3) the identity map Id: G0 → G1 which associates to each object the identity over that object;
4) the inverse map i : G1 → G1 which associates to each morphism its inverse;
5) the multiplication (composition) map ◦ : G2 → G1 which associates to each composable
pair of morphisms its composition (G2 is the set of composable pairs).
A groupoid endowed with topologies on G1 and G0 which make all the structure maps continuous
is called a topological groupoid. If additionally G0 and G1 are smooth manifolds, the source map
is a surjective submersion and all the structure maps are smooth, then the topological groupoid
is called a Lie groupoid.
Remark 2.10. Note that the identity map and the target map restricted to the image of the
identity map are inverse to each other and so the identity map is an embedding. Hence, we can
identify G0 with the image of the identity structure map.
We denote by Gx the fibers of the source map (source fibers) and by Gx the fibers of the target
map (target fibers). We also denote by Gyx the set of morphisms with source x and target y. For
a Lie groupoid Gx, Gx and Gyx are all closed embedded submanifolds of G1. What is more Gxx are
Lie Groups. We also denote the image of G0 through the identity structure map by IdG .
Definition 2.11. A morphism of groupoids is a pair (F, f) : G → H where F : G1 → H1
and f : G0 → H0 are functions which commute with the structure maps. If in addition G
and H are topological (resp. Lie) groupoids and both F and f are continuous (resp. smooth)
then (F, f) is called a continuous (resp. smooth) morphism. If both F and f are bijections
(resp. homeomorphisms, diffeomorphisms) then (F, f) is an isomorphism (resp. continuous iso-
morphism, smooth isomorphism) of groupoids. Furthermore, if f is the identity on G0 then the
morphism (F, f) is called a base preserving morphism.
Throughout this paper we are going to use several special classes of groupoids:
4 P. Raźny
Definition 2.12. A groupoid G is said to be transitive if for each pair of points x, y ∈ G0
there exists a morphism with source x and target y. Conversely, if all the morphisms of G are
automorphisms (i.e., s(g) = t(g) for all g ∈ G) we say that G is totally intransitive. A topological
groupoid is said to be proper if the map (s, t) : G1 → G0 × G0 is proper.
Definition 2.13. A topological groupoid is principal if:
1) the restriction of the target map to any source fiber is a quotient map (we write tx for the
restriction of the target map to the source fiber over x ∈ G0);
2) for each x ∈ G0 the division map δx : Gx×Gx → G1 defined by the formula δx(g, h) = g◦h−1
is a quotient map.
Remark 2.14. This notion is more commonly used in a different sense (see, e.g., [11]). The
above definition (used, e.g., in [5]) is convenient to the study of transitive topological groupoids
whilst in the Lie case it is equivalent to local triviality of a transitive groupoid. However, the
terminology of locally trivial Lie groupoids is itself rarely used due to a result of Pradines stating
that every transitive Lie groupoid over a connected base is locally trivial.
Example 2.15. Let G be a topological (resp. Lie) group and let M be a topological space (resp.
smooth manifold). M × G ×M is a topological (resp. Lie) groupoid over M with source and
target maps given by projections onto the first and third factor and composition law given by
the formula
(x, g, y) ◦ (y, h, z) = (x, gh, z).
Groupoids of this form are called trivial groupoids. If G is the trivial group then the above
groupoid is called the pair groupoid of M . Such groupoids are principal topological groupoids.
More examples of principal groupoids can be easily provided by the gauge groupoid construc-
tion presented in the subsequent section.
Proposition 2.16 ([5, Proposition 3.2]). A transitive groupoid G is isomorphic (not in a con-
tinuous manner) to the trivial groupoid G0 × Gxx × G0.
Remark 2.17. It is worth noting that for a transitive topological groupoid G and a given source
fiber Gx composition with h : x→ y gives a homeomorphism h̃ : Gy → Gx since it has an inverse
(composition with h−1) and conjugation by h (hgh−1 for g ∈ Gxx) gives a continuous isomor-
phism of topological groups h∗ : Gxx → G
y
y since it has an inverse (conjugation by h−1). Hence,
we write “Gx (resp. Gxx) has property P” as shorthand for “Gx (resp. Gxx) has property P for
some x ∈ G0 and equivalently for any x ∈ G0” whenever this remark can be applied. More-
over, since ty(h̃) = tx and δy(h̃, h̃) = δx we write “tx (resp. δx) has property P” as shorthand
for “tx (resp. δx) has property P for some x ∈ G0 and equivalently for any x ∈ G0” whenever
this remark can be applied.
We are going to use the following theorem concerning groupoid morphisms with principal
source:
Theorem 2.18 ([5, Proposition 1.21]). Let G be a principal topological groupoid and G′ be
any topological groupoid. Additionally, let (φ, f) : G → G′ be a morphism of groupoids (not
necessarily continuous). If there is an x ∈ G0 such that φx : Gx → G′f(x) is continuous then (φ, f)
is continuous.
The final notions we need to introduce here are subgroupoids and quotient groupoids:
The Solution of Hilbert’s Fifth Problem for Transitive Groupoids 5
Definition 2.19. A subgroupoid of G is a pair of subset G′1 ⊂ G1 and G′0 ⊂ G0, such that
s(G′1) ⊂ G′0, t(G′1) ⊂ G′0, Id(G′0) ⊂ G′1 and G′1 is closed under the composition and inversion
structure maps. A subgroupoid G′ of G is called wide if G′0 = G0. A subgroupoid N of G is said
to be normal if it is wide and for any g ∈ N1 and any h ∈ G1 satisfying s(h) = s(g) = t(g) we
have hgh−1 ∈ N1.
To introduce the notion of a quotient topological groupoid we first need do specify the un-
derlying groupoid structure on the appropriate quotient space. Since in this paper we are only
interested in quotients by totally intransitive groupoids, we provide an appropriately restricted
definition (see [5] for a more general discussion of this topic).
Definition 2.20. Let N be a totally intransitive normal subgroupoid of the groupoid G. Define
an equivalence relation ∼ on G1 by g ∼ h if and only if there exists n1, n2 ∈ N such that
n1gn2 = h. We then define the quotient groupoid as G/N := (G1/ ∼,G0) with the structure
maps induced from G.
Example 2.21. Simplest examples of normal subgroupoids and quotient groupoids are provided
by the trivial groupoids. A normal subgroupoid N of a trivial groupoid G = M × G ×M is
of the form
⋃
x∈M
{x} × N × {x} for some normal subgroup N of G whilst the quotient G/N is
simply M × (G/N)×M .
It was proven in [2] that if G is a topological groupoid then G/N can be endowed with a unique
topology such that it makes G/N into a topological groupoid, the projection π : G → G/N is
continuous and for every continuous morphism of topological groupoids (φ, f) : G → G′ such
that φ(N ) ⊂ IdG′ there is a unique continuous morphism (φ, f) : G/N → G′ satisfying φ ◦ π = φ
(the last condition is called the universal property of topological quotient groupoids). From
now on quotient groupoids of a topological groupoid will be considered with this topology. It
is important to note that this topology need not coincide with the quotient topology of G1/ ∼
(cf. [5]). This will give rise to minor difficulties when proving the Gleason–Yamabe theorem for
transitive groupoids.
We conclude this section by recalling an important technical result from [10] which also finds
a use in the current paper:
Proposition 2.22. Let G be a transitive topological groupoid with G1 first countable. Then the
following conditions are equivalent:
1) tx is a quotient map;
2) tx is open;
3) δx is a quotient map;
4) δx is open.
Furthermore, if (s, t) : G1 → G0 × G0 is a quotient map, then the above properties hold.
2.3 Cartan principal bundles
In this section we give a brief recollection of principal bundles, Cartan principal bundles, how
they relate to each other as well as some results from [8] which are of key importance to the
present paper. A more detailed exposition of this subject can be found in [5] and [8].
Definition 2.23. A Cartan principal bundle is a quadruple (P,B,G, π), where P and B are
topological spaces, G is a topological group acting freely on P and π : P → B is a surjective
continuous map, with the following properties:
6 P. Raźny
1) π is a quotient map with fibers coinciding with the orbits of the action of G on P ,
2) the division map δ : Pπ → G with domain Pπ := {(u, v) ∈ P × P |π(u) = π(v)} defined by
the property δ(ug, u) = g is continuous.
We are also going to need a notion of morphism between such bundles:
Definition 2.24. A morphism of Cartan principal bundles is a triple
(F, f, φ) : (P,B,G, π)→ (P ′, B′, G′, π′),
where F : P → P ′ and f : B → B′ are continuous functions and φ : G → G′ is a continuous
morphism of topological groups such that
π′ ◦ F = f ◦ π, F (pg) = F (p)φ(g)
for p ∈ P and g ∈ G. A morphism of Cartan principal bundles is said to be base preserving if f
is the identity on B.
Definition 2.25. A Cartan principal bundle (P,B,G, π) is called proper if the action of G on P
is proper.
A better known and stronger notion is the following:
Definition 2.26. A principal bundle is a quadruple (P,B,G, π), where P and B are topological
spaces, G is a topological group acting freely on P and π : P → B is a surjective continuous
map, with the following properties:
1) the fibers of π coincide with the orbits of the action of G;
2) (local triviality) there is an open covering Ui of B and continuous maps σi : Ui → P such
that π ◦ σi = IdUi .
A principal bundle is said to be smooth if P and B are smooth manifolds, G is a Lie group, and
the action, projection and σi are smooth maps.
We are now going to present important constructions from [5] and [6] which relate the notion
of Cartan principal bundles to principal groupoids. Given a principal groupoid G the quadruple
(Gx,G0,Gxx , tx) constitutes a Cartan principal bundle for any point x ∈ G0 (this is called the
vertex bundle of G at x). It is easy to see that given a morphism of groupoids (F, f) : G → G′
the restriction of the map F to Gx gives a morphism of bundles F |Gx : Gx → Gf(x). It is also
worth noting that even though this construction is dependent on the choice of x all the vertex
bundles are continuously isomorphic by use of translations (cf. [5]). In the other direction given
a Cartan principal bundle (P,B,G, π) there exists a structure of a topological groupoid over B
on (P × P )/G (this is called the gauge groupoid of (P,B,G, π)). Furthermore, a morphism of
Cartan principal bundles
(F, f, φ) : (P,B,G, π)→ (P ′, B′, G′, π′)
induces a morphism of gauge groupoids F ∗ defined by F ∗([(u, v)]) = [F (u), F (v)]. It is apparent
from the form of the induced morphisms that a base preserving morphism of Cartan principal
bundles induces a base preserving morphism of the corresponding gauge groupoids and that
a base preserving morphism of principal groupoids induces a base preserving morphism of vertex
bundles. We give the following theorem which was proven in [5]:
Theorem 2.27. The constructions above are mutually inverse (up to a continuous base pre-
serving isomorphism) and give a one to one correspondence between continuous isomorphism
classes of Cartan principal bundles and continuous isomorphism classes of principal groupoids.
The Solution of Hilbert’s Fifth Problem for Transitive Groupoids 7
We also need the following important results from [8]:
Theorem 2.28 ([8, Proposition 1.2.5]). A Cartan principal bundle (P,B,G, π) with P and B
Tychonoff is proper.
Theorem 2.29 ([8, Section 4.1]). A Cartan principal bundle (P,B,G, π) with G a Lie group
and P Tychonoff is locally trivial.
Theorem 2.30 ([8, Proposition 1.3.2]). Given a proper Cartan principal bundle (P,B,G, π)
with P Tychonoff and a normal closed subgroup N of G, (P/N,B,G/N, π) is also a proper
Cartan principal bundle with P/N Tychonoff.
Theorem 2.31 ([8, Theorem 4.3.4]). A proper Cartan principal bundle (P,B,G, π) with G a Lie
group and P separable and metrizable admits an invariant metric. Hence, P/G is metrizable.
Moreover, dim[x](P/G) = dimx(P )− dim(G).
3 Hilbert’s fifth problem for transitive groupoids
3.1 Some preliminary theorems
This part is devoted to establishing two theorems which are crucial to the solution to Hilberts
fifth problem for groupoids given in the next subsection. We feel that the results of this section
should be known, however we were unable to find a suitable reference and so we provide their
proofs for the readers convenience.
Theorem 3.1. Let f : M1 → M2 be a continuous bijection between separable metric spaces.
Additionally let M1 be locally compact of dimension m. Then dim(M2) = m.
Proof. We first prove dim(M2) ≤ m. Let us take around each point x ∈ M1 its relatively
compact open neighbourhood Ux. Since a separable metric space is Lindelöf we can choose
a countable subcover {Ui}i∈I from {Ux}x∈M1 . Now f |Ui
: Ui → f(Ui) is a homeomorphism
since for i ∈ I the set Ui is compact. Moreover, this implies that f(Ui) are closed in M2 of
dimension at most m (since homeomorphisms preserve dimension and dim(Ui) ≤ m due to the
fact that they are subspaces of an m dimensional space). Hence the family f(Ui) satisfy the
conditions of Theorem 2.7 which implies that M2 has dimension no greater than m. On the
other hand at least one of Ui has to have dimension m since otherwise by Theorem 2.7 we have
m = dim(M1) ≤ m − 1. Hence, at least one of f(Ui) is of dimension m which implies that
dim(M2) ≥ m. �
Theorem 3.2. Let S ⊂ Rm be such that for some open V ⊂ Rn the set V ×S is homeomorphic
to an (n+m)-dimensional manifold. Then S is open in Rm.
Proof. Let us assume that S is not open in Rm. Then there exists a point x ∈ S such that
no neighbourhood of x is contained in S. Let us take a neighbourhood U(x,y) of (x, y) for some
y ∈ V which is homeomorphic to an open ball and let us denote by i : S × V → Rm × Rn the
inclusion given by the inclusions of S into Rm and V into Rn. Then by the invariance of domain
theorem i(U(x,y)) ⊂ S × V ⊂ Rm × Rn is open in Rm × Rn. But this cannot be the case as it
contains the point (x, y) and it cannot contain any neighbourhood of (x, y) from the product
basis of Rm × Rn (since then S would contain the projection of that set onto Rm which would
be an open neighbourhood of x). �
Remark 3.3. We would like to note that even though this result seems natural and is somewhat
expected it is not trivial since there exists more than one topological space X such that X ×R
is homeomorphic to R4 (cf. [1], where it is shown that X doesn’t even have to be a manifold).
8 P. Raźny
Corollary 3.4. There is no S ⊂ Rk such that for some open V ⊂ Rn the set V × S is homeo-
morphic to an (n+m)-dimensional manifold for some m > k.
Proof. We apply the previous theorem to the set {0}m−k × S ⊂ Rm and note that this set
cannot be open in Rm. �
3.2 The solution to Hilbert’s fifth problem for transitive groupoids
Let us state our main theorem:
Theorem 3.5. Let G be a transitive topological groupoid such that G0 is a smooth manifold,
Gx and Gxx are topological manifolds. Then the map tx is a quotient map.
Proof. To prove this let us consider the following diagram:
Gx G0
Gx/Gxx .
π
tx
tx
Note that since the fibres of π and tx coincide, tx is a bijection. We shall prove that Gx/Gxx is
a topological manifold and this by Theorem 2.2 will imply that tx is a homeomorphism which
in turn implies that tx is a quotient map. First of all we observe that Gx/Gxx is Hausdorff (given
two points y, z ∈ Gx/Gxx they can be separated by the inverse images through tx of the open sets
separating tx(y) and tx(z)). Furthermore, Gx/Gxx is locally compact since given a point y ∈ Gx/Gxx
the image of a compact neighbourhood of some point in π−1(y) is a compact neighbourhood
of y (we use here the fact that π as a projection onto the orbit space of a group action is
open). Local compactness and being Hausdorff imply that Gx/Gxx is Tychonoff. We also note
that Gx/Gxx is second countable (the countable basis is given by the images of some countable
basis of Gx through π). The group Gxx under our assumptions is a Lie group due to the solution
to the classical Hilberts fifth problem. It is also apparent that (Gx,Gx/Gxx ,Gxx , π) is a Cartan
principal bundle (since π is a quotient map and δ(g, h) = h−1g must be continuous since G1 is
a topological groupoid). Hence, by Theorems 2.29 and 2.28 this bundle is proper and locally
trivial. Theorem 2.31 implies that Gx/Gxx is metrizable with dim(Gx/Gxx) = dim(Gx) − dim(Gxx)
which in turn implies that dim(G0) = dim(Gx) − dim(Gxx) (by Theorem 3.1 and the fact that
a second countable metric space is separable).
We will now show that Gx/Gxx is locally Euclidean. Let us fix a point y ∈ Gx/Gxx along
with its open neighbourhood S such that the bundle (Gx,Gx/Gxx ,Gxx , π) is trivial when restricted
to S. Using local compactness we can assume without loss of generality that S is relatively
compact. This allows us to treat S as a subset of G0 since then tx|S : S → tx(S) is a homeomor-
phism (since tx|S is a continuous bijection with compact domain). If tx(S) is a neighbourhood
of tx(y) in G0 then we are done (by taking the preimage through tx of a neighbourhood of tx(y)
homeomorphic to an open ball contained in tx(S)). Let us consider the set tx(S) ∩ U for some
neighbourhood U of tx(y) homeomorphic to an open ball (this set is now homeomorphic to
a subset of Rdim(G0)). Using the commutativity of the diagram above it is apparent that
tx|t−1
x (tx(S)∩U) : t−1x (tx(S) ∩ U)→ tx(S) ∩ U
is locally trivial (i.e., (t−1x (tx(S)∩U), tx(S)∩U,Gxx , tx|t−1
x (tx(S)∩U)) is a principle bundle). Hence,
for any neighbourhood V of the identity in Gxx we have that (tx(S) ∩ U) × V is homeomorphic
to an open subset in Gx and hence a manifold of the same dimension as Gx. Now by using
Theorem 3.2 we get tx(S) ∩ U is open in U and conclude that tx(S) is indeed a neighbourhood
of tx(y). �
The Solution of Hilbert’s Fifth Problem for Transitive Groupoids 9
From this and Theorem 1.1 it is visible that the assumption that tx is a quotient map is
superfluous in Theorem 1.2 and we get the desired result:
Theorem 3.6. Let G be a transitive topological groupoid with a smooth base G0 for which the
space G1 is first-countable and the spaces Gxx , Gx are topological manifolds. Then G is continuously
isomorphic to a unique Lie groupoid through a base preserving isomorphism.
Remark 3.7. In particular the previous theorem holds if G1 is a topological manifold.
Remark 3.8. We wish to note that if the Hilbert–Smith conjecture (cf. [12]) proves valid then
the assumption on Gxx is unnecessary. However, using this method we see no way of weakening
any more assumptions in order to get a result similar to Theorem 1.1.
4 Gleason–Yamabe theorem for proper transitive groupoids
Our goal in this section is to find an appropriate generalization of Theorem 2.8 to transitive
groupoids. Let us first put our result in the most general way:
Theorem 4.1 (Gleason–Yamabe theorem for transitive groupoids). Let G be a transitive topo-
logical groupoid with G1 first-countable, G0 a smooth manifold, Gx Hausdorff and locally compact,
Gxx compact and tx a quotient map. For every neighbourhood U of the identity subspace IdG and
every point x ∈ G0 there exists a closed totally intransitive normal subgroupoid K such that:
1) Kxx is compact and contained in U ∩ Gxx ,
2) G/K is a Lie groupoid.
Proof. First let us note that for a given neighbourhood U of the identity space IdG and a fixed
point x ∈ G0 by Theorem 2.8 there is a compact normal subgroup K ⊂ Gxx such that K is
contained in U and Gxx/K is a Lie group. We use this group to create the desired groupoid K.
We put Kxx = K and Kyy = gKg−1 for some morphism g : x → y. The above construction does
not depend on the choice of the morphism g since given another morphism g̃ : x→ y there exists
an element h ∈ Gxx such that g = g̃h (namely, h = g̃−1g). Then
gKg−1 = g̃hKh−1g̃−1 = g̃Kg̃−1,
K is a topological groupoid with the induced topology. It is obvious from the construction that K
is also a normal subgroupoid.
Note that Gx is Tychonoff since it is locally compact and Hausdorff. By Proposition 2.22
we conclude that G1 is a principal groupoid and so (Gx,G0,Gxx , tx) is a Cartan principal bundle.
Moreover, it is a proper Cartan principal bundle due to Theorem 2.28 and so by Theorem 2.30
the quadruple (Gx/Kxx,G0,Gxx/Kxx, tx) is also a proper Cartan principal bundle and so we can
consider its gauge groupoid H. We will show that H is continuously isomorphic to G/K.
SinceH and G/K are isomorphic (not necessarily in a continuous manner) by a base preserving
isomorphism to G0× (Gxx/Kxx)×G0 and G is isomorphic (not necessarily in a continuous manner)
by a base preserving isomorphism to G0 × Gxx × G0 we have a morphism of groupoids φ : G → H
(the quotient morphism). We note that φx : Gx → Gx/Kxx is then equal to the quotient map
and hence continuous. This combined with Theorem 2.18 implies that φ is in fact a continuous
groupoid morphism. By the universal property of quotient groupoids we have that φ : G/K → H
is continuous.
On the other hand, let us observe that by the universal property of the quotient map the
map φx
−1
: Gx/Kxx → (G/K)x is also continuous since it is equal to the map πx induced by the
10 P. Raźny
groupoid projection map π : G1 → (G/K)1 restricted to Gx:
G H
G/K,
π
φ
φ
Gx Gx/Kxx
(G/K)x.
πx
φx
φx
This again by Theorem 2.18 implies that φ
−1
is continuous and hence H and G/K are conti-
nuously isomorphic.
We now show that H satisfies all the assumptions of Theorem 1.1 and so it is a Lie groupoid
and consequently G/K is also a Lie groupoid. The fact that H is principal (as a gauge groupoid
of a Cartan principal bundle) implies that t′x is a quotient map (where t′ denotes the target map
in H). By Theorem 2.30 we have Hx = Gx/Kxx is Tychonoff. It is also locally compact as an
image of a locally compact space through a continuous open map. As a quotient of a locally
compact space by a group action it is locally compact and first countable (since the projection
onto the orbit space is open and Gx is locally compact and first countable). This also implies
that H1 := (Hx×Hx)/Hxx is first countable. Finally, we note that H0 = G0 is a smooth manifold.
We end the proof by noting that K is closed in G as it is equal to π−1(IdH). �
We also note that under such general assumptions we cannot demand that K is either compact
or contained in U as is shown by the following simple example:
Example 4.2. Let us consider the trivial groupoid G := R × Zp × R, where Zp denotes the
additive group of p-adic integers. Let us now take the open set:
U :=
⋃
n∈N
(−n, n)×Bp
(
0,
1
pn
)
× (−n, n),
where Bp(0, r) denotes the p-adic ball of radius r centered at zero. It is apparent that normal
totally intransitive subgroupoids of G are of the form
⋃
x∈R
{x} ×N × {x} for some subgroup N
of Zp and that such a subgroupoid can be contained in U only if N is trivial. The above example
also highlights the fact that the groupoid K depends on the choice of x.
It is also noteworthy that the G0 has to be smooth for the theorem to work. The following
counterexample shows that assuming that G0 is a topological manifold is insufficient:
Example 4.3. Let G0 be a topological manifold which does not admit any smooth structure
(e.g., the celebrated E8 4-manifold). We then take G to be the pair groupoid over G0. It is
apparent that despite satisfying the assumptions of Theorem 4.1 (except for the smoothness
of G0) the thesis does not hold for this groupoid (since the identity subspace is the only totally
intransitive groupoid).
Theorem 4.1 leads us to the following two corollaries:
Theorem 4.4 (Gleason–Yamabe theorem for proper transitive groupoids). Let G be a proper
transitive topological groupoid with G1 first-countable, G0 a smooth manifold, Gx Hausdorff and
locally compact. For every neighbourhood U of the identity subspace IdG and every point x ∈ G0
there exists a closed totally intransitive normal subgroupoid K such that:
1) Kxx is compact and contained in U ∩ Gxx ,
2) G/K is a Lie groupoid.
The Solution of Hilbert’s Fifth Problem for Transitive Groupoids 11
Proof. Since Gxx = (s, t)−1(x, x) it is compact. Moreover, since (s, t) is proper it is also closed
and hence it is a quotient map. This by Proposition 2.22 gives us that tx is a quotient map as
well, which in turn together with Theorem 4.1 proves the desired result. �
Theorem 4.5 (Gleason–Yamabe theorem for transitive groupoids with compact source fibres).
Let G be a transitive topological groupoid with G1 first-countable, G0 a smooth manifold, Gx
Hausdorff and compact. For every neighbourhood U of the identity subspace IdG and every point
x ∈ G0 there exists a closed totally intransitive normal subgroupoid K such that:
1) Kxx is compact and contained in U ∩ Gxx ,
2) G/K is a Lie groupoid.
Proof. Gxx is compact as a closed subspace of the compact space Gx. Moreover tx is a quo-
tient map since it is closed as a map with compact domain. We finish the proof by applying
Theorem 4.1. �
References
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[4] Gleason A.M., Groups without small subgroups, Ann. of Math. 56 (1952), 193–212.
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https://doi.org/10.1090/S0002-9904-1958-10160-3
https://doi.org/10.1002/mana.19760710123
https://doi.org/10.2307/1969795
https://doi.org/10.1017/CBO9780511661839
https://doi.org/10.1017/CBO9780511661839
https://doi.org/10.1017/CBO9781107325883
https://doi.org/10.1017/CBO9781107325883
https://doi.org/10.2307/1969796
https://doi.org/10.2307/1970335
https://doi.org/10.1007/BF01098343
https://doi.org/10.3842/SIGMA.2017.098
https://arxiv.org/abs/1710.11440
https://doi.org/10.1007/BFb0091072
https://doi.org/10.1090/gsm/153
1 Introduction
2 Preliminaries
2.1 Some topology
2.2 Groupoids
2.3 Cartan principal bundles
3 Hilbert's fifth problem for transitive groupoids
3.1 Some preliminary theorems
3.2 The solution to Hilbert's fifth problem for transitive groupoids
4 Gleason–Yamabe theorem for proper transitive groupoids
References
|
| id | nasplib_isofts_kiev_ua-123456789-209780 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-03T16:22:59Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Raźny, P. 2025-11-26T12:19:12Z 2018 The Solution of Hilbert's Fifth Problem for Transitive Groupoids / P. Raźny // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 12 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22A22 arXiv: 1805.03066 https://nasplib.isofts.kiev.ua/handle/123456789/209780 https://doi.org/10.3842/SIGMA.2018.070 In the following paper, we investigate the question: when is a transitive topological groupoid continuously isomorphic to a Lie groupoid? We present many results on the matter, which may be considered generalizations of Hilbert's fifth problem to this context. Most notably, we present a "solution" to the problem for proper transitive groupoids and transitive groupoids with compact source fibers. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Solution of Hilbert's Fifth Problem for Transitive Groupoids Article published earlier |
| spellingShingle | The Solution of Hilbert's Fifth Problem for Transitive Groupoids Raźny, P. |
| title | The Solution of Hilbert's Fifth Problem for Transitive Groupoids |
| title_full | The Solution of Hilbert's Fifth Problem for Transitive Groupoids |
| title_fullStr | The Solution of Hilbert's Fifth Problem for Transitive Groupoids |
| title_full_unstemmed | The Solution of Hilbert's Fifth Problem for Transitive Groupoids |
| title_short | The Solution of Hilbert's Fifth Problem for Transitive Groupoids |
| title_sort | solution of hilbert's fifth problem for transitive groupoids |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209780 |
| work_keys_str_mv | AT raznyp thesolutionofhilbertsfifthproblemfortransitivegroupoids AT raznyp solutionofhilbertsfifthproblemfortransitivegroupoids |