On the Generalization of Hilbert's Fifth Problem to 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 the Hilbert's fifth problem to this context. Most notably we present a '...
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Raźny, P. 2019-02-19T19:31:54Z 2019-02-19T19:31:54Z 2017 On the Generalization of Hilbert's Fifth Problem to Transitive Groupoids / P. Raźny // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 12 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22A22 DOI:10.3842/SIGMA.2017.098 https://nasplib.isofts.kiev.ua/handle/123456789/149265 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 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. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Generalization of Hilbert's Fifth Problem to Transitive Groupoids Article published earlier |
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On the Generalization of Hilbert's Fifth Problem to Transitive Groupoids |
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on the generalization of hilbert's fifth problem to transitive groupoids |
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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 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.
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On the Generalization of Hilbert's Fifth Problem to Transitive Groupoids / P. Raźny // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 12 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 098, 10 pages
On the Generalization of Hilbert’s Fifth Problem
to 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 December 02, 2017, in final form December 21, 2017; Published online December 31, 2017
https://doi.org/10.3842/SIGMA.2017.098
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 generalize Hilbert’s fifth problem (which states that a locally Euclidean
topological group is isomorphic as topological groups to a Lie group and was solved in [2] and [7])
to the case of transitive groupoids. We prove a couple of versions of Hilbert’s fifth problem for
transitive groupoids (Theorems 3.6, 3.8 and 3.11). Many of our main results are written in
two versions: one with the weakest assumptions such that the proofs are still valid, and one
in which most assumptions are replaced by demanding appropriate spaces to be topological
manifolds (which is more in the spirit of Hilbert’s fifth problem). We restrict our attention to
groupoids with smooth base as otherwise simple counterexamples to the Hilbert’s fifth problem
for transitive groupoids can be easily found (as described in the final section of this paper). In
light of the examples and results in this paper we feel that topological groupoids with smooth
base are natural objects to investigate when it comes to such theorems. Due to the fact that
transitive groupoids are an extreme case of groupoids (the opposite extreme case being the
totally intransitive groupoids) we feel that the results of this paper combined with a study of
totally intransitive groupoids with smooth base can help in solving Hilbert’s fifth problem for
groupoids.
The paper is split into two parts. The first part is designed to give all the necessary facts
about groupoids and make the paper more self-contained. In the second part we present our
main results and give a brief discussion about the assumptions made throughout the paper.
An interested reader can find a thorough exposition of Hilbert’s fifth problem in [11]. A more
detailed exposition of groupoid theory and it’s vast application (to, e.g., Poisson geometry,
symplectic geometry, foliations) can be found in [4, 5, 6]. Throughout this paper manifolds
(both topological and smooth) are assumed to be Hausdorff and second countable (unless stated
otherwise). We do not consider infinitely dimensional spaces.
mailto:pawel.razny@student.uj.edu.pl
https://doi.org/10.3842/SIGMA.2017.098
2 P. Raźny
2 Preliminaries
2.1 Some topology
In order to make this short paper as self-contained as possible, we start by recalling some basic
topological notions and properties which are used in subsequent sections. We begin with some
well known properties of quotient maps (identifications). Recall that a continuous surjective
map p : X → Y is a quotient 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).
Proposition 2.1. A surjective continuous map which is either closed or open is a quotient map.
Definition 2.2. Given a quotient map p : X → Y a subset U of X is called saturated if x ∈ p(U)
implies p−1(x) ⊂ U .
Proposition 2.3. Given a quotient map p : X → Y the restriction p|U : U → p(U) to a closed
or open saturated subset U of X is also a quotient map.
Proposition 2.4. 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 f̄ : Y → Z such that the
following diagram commutes:
X Z
Y.
p
f
f̄
All of the above results along with a good revision of quotient maps can be found in [3]. We
also wish to recall the following two versions of the closed mapping theorem:
Theorem 2.5 ([3, Lemma 4.50]). A continuous map f : X → Y , where X is compact and Y is
Hausdorff, is a closed map.
Theorem 2.6 ([3, Theorem 4.95]). A proper continuous map f : X → Y , where X and Y are
locally compact and Hausdorff, is a closed map.
We now recall the notion of sequence covering which will be used extensively throughout this
paper. More information on this subject can be found in [10].
Definition 2.7. A continuous surjective map f : X → Y is said to have the sequence covering
property if for each sequence {yn}n∈N convergent to y in Y there exists a sequence {xn}n∈N
convergent to x in X satisfying xn ∈ f−1(yn) and x ∈ f−1(y).
Proposition 2.8. Any continuous open surjective map has the sequence covering property.
Finally, we recall the following well known fact:
Proposition 2.9 ([1, Theorem 1.6.14 and Proposition 1.6.15]). Let f : X → Y be a function
from a first countable topological space X to a topological space Y . Then f is continuous if and
only if it is sequentially continuous.
On the Generalization of Hilbert’s Fifth Problem to Transitive Groupoids 3
2.2 Groupoids
We give a brief recollection of some basic notions concerning groupoids. Let us start by giving
the definition:
Definition 2.10. 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 2.11. Note that the identity map and the target map restricted to the image of the
identity map are inverses and so the identity map is an embedding. Hence, we can identify G0
with the image of the identity map. This in particular means that we can treat G0 as a subspace
of G1.
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. If V and U are subsets of G0 then we denote by GVU the set of all morphisms with
source in U and target in V . We present a couple of examples which will be of further interest
to us:
Example 2.12. Given a smooth manifold (resp. topological space) M there is a structure of
a Lie groupoid (resp. topological groupoid) with base M on M ×M . For this structure the
source and target maps are projections on to the first and second factor. We call the groupoid
M ×M over M the pair groupoid.
Example 2.13. 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.
There are many more examples of groupoids most of which are much more complicated then
the ones just shown. The above examples will be used later on as they already exhibit some of
the problems one faces when dealing with groupoids that don’t arise for groups. In what follows
we are also going to need a notion of morphism of groupoids:
4 P. Raźny
Definition 2.14. A continuous (resp. smooth) morphism of topological (resp. Lie) groupoids is
a pair (F, f) : G → H where F : G1 → H1 and f : G0 → H0 are continuous (resp. smooth) maps
which commute with the structure maps. If in addition both F and f are homeomorphisms
(resp. diffeomorphisms) then (F, f) is a continuous (resp. smooth) isomorphism. 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:
Definition 2.15. 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.
Definition 2.16. A topological groupoid is said to be proper if the map (s, t) : G1 → G0×G0 is
proper.
Definition 2.17. 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 2.18. 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 : Gx → Gy since it has
an inverse (composition with h−1) and conjugation by h (hgh−1 for g ∈ Gxx) gives a continuous
isomorphism 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. Moreover,
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.
Definition 2.19. A topological groupoid is locally trivial if it is transitive and each point x ∈ G1
has an open neighbourhood U such that GUU is base preserving continuously isomorphic to the
trivial groupoid U × Gxx × U .
2.3 Cartan principal bundles
In this section we give a brief recollection of principal bundles, Cartan principal bundles, how
they relate to one another and their connection to transitive topological groupoids. A more
detailed exposition of this subject can be found in [4, 5, 9].
Definition 2.20. 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:
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 2.21. 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)
On the Generalization of Hilbert’s Fifth Problem to Transitive Groupoids 5
for p ∈ P and g ∈ G. A morphism of Cartan principal bundles is said to be base preserving if
B = B′ and f = IdB.
A stronger and better known object is the following:
Definition 2.22. 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 sometimes refer to principal bundles as continuous principal bundles. It is known that
a principal bundle is a Cartan principal bundle and that a Cartan principal bundle which
is locally trivial is principal (cf. [4] and [9]). We also present the following important result
from [9]:
Theorem 2.23. A Cartan principal bundle (P,B,G, π) with G a Lie group and P Tychonoff
is locally trivial.
We also wish to recall the following well known equivalence (cf. [8]):
Theorem 2.24. A principal bundle (P,B,G, π) with G a Lie group and B a smooth manifold
is continuously isomorphic through a (base preserving isomorphism) to a unique (up to smooth
isomorphism of smooth principal bundles) smooth principal bundle.
We are now going to present important constructions from [4] and [5] which relate the above
notions of principal and Cartan principal bundles to locally trivial and principal groupoids re-
spectively. 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 iso-
morphic by use of translations (cf. [4]). On the other hand given a Cartan principal bun-
dle (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 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 in-
duces a base preserving morphism of vertex bundles. We give the following 3 theorems which
were proven in [4] and [5]:
Theorem 2.25. 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.
Theorem 2.26. The constructions above give a one to one correspondence between continuous
isomorphism classes of principal bundles and continuous isomorphism classes of locally trivial
groupoids.
6 P. Raźny
Theorem 2.27. The constructions above give a one to one correspondence between smooth iso-
morphism classes of smooth principal bundles and smooth isomorphism classes of locally trivial
Lie groupoids.
Remark 2.28. In the previous theorem one can weaken local triviality to transitivity since in
the case of Lie groupoids these notions are equivalent (cf. [4, Corollary 1.9]).
3 Hilbert’s fifth problem for transitive groupoids
3.1 Main results
In the following section by “unique Lie groupoid” we mean unique up to a smooth base preserving
isomorphism. We start with the following observation.
Theorem 3.1. Let G be a principal groupoid with a smooth base G0 and Tychonoff source
fibers Gx for which the topological groups Gxx are locally Euclidean. Then G is continuously
isomorphic to a unique Lie groupoid through a base preserving isomorphism.
Proof. By the correspondence in Theorem 2.25 we can associate to G a Cartan principal bundle.
Due to Hilbert’s fifth problem Gxx is a Lie group. Since the source fibers are Tychonoff we can
see by Theorem 2.23 that this bundle is locally trivial and hence its gauge groupoid (which
is continuously isomorphic to G through a base preserving isomorphism) is locally trivial as
well thanks to the correspondence in Theorem 2.26. Hence, G is locally trivial. Furthermore,
since this bundle is a continuous principal bundle of Lie groups over a smooth manifold it is
continuously isomorphic to a unique smooth principal bundle. This continuous isomorphism
induces a continuous isomorphism between the gauge groupoids of these two bundles. It now
suffices to note that by Theorem 2.27 the gauge groupoid of a smooth principal bundle is a Lie
groupoid. �
Note that if Gx is assumed to be a topological manifold then it is in fact Tychonoff and hence
the above theorem can be applied. In order to further generalize this theorem we present an
important technical result:
Proposition 3.2. 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) tx has the sequence covering property,
4) δx is a quotient map,
5) δx is open.
Furthermore, if (s, t) : G1 → G0 × G0 is a quotient map, then the above properties hold.
Proof. The equivalence of conditions (1) and (2) as well as the equivalence of conditions (4)
and (5) can be found in [4]. That (2) implies (3) is immediate from Proposition 2.8.
For (3) implies (1) let us denote by t′x : Gx → Gx/Gxx the quotient map of the group action
of Gxx on Gx. Using Proposition 2.4 we get the following commutative diagram:
Gx G0
Gx/Gxx ,
t′x
tx
f (3.1)
On the Generalization of Hilbert’s Fifth Problem to Transitive Groupoids 7
where f is a continuous bijection. By Proposition 2.9 it is sufficient to prove that f−1 is sequen-
tially continuous (since G0 is first countable as a subspace of a first countable space G1). Let us
take a sequence {yn}n∈N convergent to y in G0. We prove that its image f−1(yn) is convergent
to f−1(y). Using the sequence covering property of tx we get a sequence {gn : x→ yn}n∈N con-
vergent to g : x → y. Due to the continuity of t′x the sequence t′x(hn) converges to t′x(h). This
sequence and its limit are by the commutativity of the diagram and bijectivity of f the image
of {yn}n∈N and y respectively through f−1. Hence, f−1 is continuous which in turn implies
that f is a homeomorphism and so tx is a quotient map.
The proof of (3) implies (4) is similar to the previous implication. Let us first note that it
suffices to prove that the multiplication map restricted to Gx×Gx is a quotient map (since δx is
a composition of the multiplication map and the inverse map which is a homeomorphism). Let
us also note that Gxx acts on Gx × Gx by
(h1, h2)g =
(
h1g, g
−1h2
)
.
We prove that the orbits of this action are precisely the fibers of the multiplication map restricted
to Gx × Gx. It is apparent that an orbit of this action is contained in a fiber of the restricted
multiplication map (since the composition h1gg
−1h2 is equal to h1h2). On the other hand given
two elements (f, g) and (f ′, g′) in a single fiber of the multiplication map we have fg = f ′g′.
This implies that Idx = f−1f ′g′g−1 and hence g′g−1 is inverse to f−1f ′. This means that by
acting on (f, g) with f−1f ′ we get (f ′, g′) which in turn implies that (f, g) and (f ′, g′) belong to
the same orbit of the group action. Hence, each fiber of the restricted multiplication is contained
in some orbit of the group action (and so the fibers and orbits coincide). Let us denote by ◦x
this restricted multiplication map and by ◦′x : Gx × Gx → (Gx × Gx)/Gxx the quotient map of the
group action. As in the previous case we have the following commutative diagram:
Gx × Gx G1
(Gx × Gx)/Gxx .
◦′x
◦x
f (3.2)
The map f is again bijective and continuous and we prove the continuity of f−1. Since G1 is
first countable it is sufficient to prove sequential continuity by Proposition 2.9. Let us take
a sequence {gn : y′n → yn}n∈N convergent in G1 to g : y′ → y. This implies in particualr that
the sequence {yn}n∈N converges to y in G0. Hence, we can use the sequence covering property
of tx to produce a sequence {hn : x → yn}n∈N convergent to h : x → y in Gx. This allows us to
define a sequence (hn, h
−1
n gn) which by continuity of multiplication is convergent to (h, h−1g)
in Gx × Gx. As before, using the commutativity of the diagram, bijectivity of f and continuity
of ◦′x we conclude that the classes of this sequence represent the image of the initial sequence gn
and converge to g in (Gx × Gx)/Gxx . This implies that f is a homeomorphism and that ◦x is in
fact a quotient map.
For (5) implies (3) let us take a sequence {yn}n∈N convergent to y. Since δx is open, so is
the multiplication ◦x. Hence, ◦x has the sequence covering property. Let {(gn, hn)}n∈N be the
sequence covering Idyn (which by continuity of the identity structure map converges to Idy)
through ◦x. This implies that {gn}n∈N is a sequence covering {yn}n∈N through tx.
Finally, if (s, t) is a quotient map then (s, t)|Gx = (x, tx) : Gx → {x} × G0 is also a quotient
map (since Gx is a saturated closed subset of G1). This implies that tx is a quotient map. �
Remark 3.3. This in particular means that a transitive groupoid G with G1 first countable is
a principal groupoid if and only if tx is a quotient map.
8 P. Raźny
Remark 3.4. One could add a sixth condition: “δx has the sequence covering property” to the
previous theorem and prove its equivalence in a similar fashion. However, this condition is of
little importance to this paper and hence we skip the proof for the sake of brevity.
Using the two previous results we immediately arrive at the following conclusion:
Theorem 3.5. 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.
It is interesting that this result makes only the first countability assumptions on the topology
of G1. We proceed to give a number of corollaries of this seemingly simple result:
Theorem 3.6. 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.
Proof. Apply the previous theorem. �
We present the following two forms of the Hilbert’s fifth problem for proper transitive
groupoids (a stronger one and an elegant one):
Theorem 3.7. Let G be a proper transitive topological groupoid with a smooth base G0, Ty-
chonoff locally compact source fibers Gx, a locally compact Hausdorff first countable space of
morphisms G1 for which the topological groups Gxx are locally Euclidean. Then G is continuously
isomorphic to a unique Lie groupoid through a base preserving isomorphism.
Proof. Under these assumptions using the closed mapping theorem to (s, t) we conclude that
it is in fact closed and hence a quotient map. This implies that tx is a quotient map as (by
Proposition 3.2) and hence we can apply Theorem 3.5. �
Theorem 3.8. 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.
Proof. Apply the previous theorem. �
Remark 3.9. This in particular means (when combined with the results of [12]) that any
transitive topological groupoid which satisfies the assumptions of Theorem 3.8 is continuously
isomorphic to a real analytic transitive groupoid.
Finally, Theorem 3.5 also solves the problem for groupoids with compact source fibers (and
in particular for groupoids with G1 compact):
Theorem 3.10. Let G be a transitive topological groupoid with a smooth base G0, Tychonoff
compact source fibers Gx for which the topological groups Gxx are locally Euclidean and G1 is first
countable. Then G is continuously isomorphic to a unique Lie groupoid through a base preserving
isomorphism.
Proof. By Theorem 3.5 it is sufficient to prove that tx is a quotient map. Using the closed
mapping theorem we conclude that tx is closed and hence a quotient map. �
Theorem 3.11. Let G be a transitive topological groupoid with a smooth base G0, for which the
spaces Gxx and G1 are topological manifolds and Gx is a compact topological manifold. Then G is
continuously isomorphic to a unique Lie groupoid through a base preserving isomorphism.
Proof. Apply the previous theorem. �
On the Generalization of Hilbert’s Fifth Problem to Transitive Groupoids 9
3.2 Some remarks considering assumptions
We start by showing why it is necessary for G0 to be smooth. 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 even though in this example Gxx , G1,
Gx and Gx are Hausdorff second-countable topological manifolds and the map tx is a quotient
map it is not continuously isomorphic to a Lie groupoid (since this would imply that G0 is
homeomorphic to some smooth manifold). The above groupoid is a counterexample to all the
theorems in the previous section if the smoothness assumption on the base is omitted.
Remark 3.12. We also wish to note that this assumption is natural in the following sense.
One can consider a Lie group G as a Lie groupoid with G0 = {x} and G1 = Gxx = G. So if we
state the Hilbert’s fifth problem in the language of groupoids our assumptions would be G1 is
a topological manifold and G0 is a point (and hence a smooth manifold).
Assumption that Gxx is a topological manifold might not be necessary in Theorems 3.6, 3.8
and 3.11. However, for the time being it seems hard to get rid of it. We note that this assumption
is superfluous assuming the validity of Hilbert–Smith conjecture (cf. [11]).
In the previous section we explored an approach to Lie groupoids presented in [5]. We wish
to address a different approach which is found in [6]:
Definition 3.13 (Moerdijk [6]). A topological groupoid G which satisfies the following condi-
tions:
1) G0 is a smooth manifold,
2) G1 is a possibly non-Hausdorff and not second countable smooth manifold,
3) the structure maps are smooth,
4) the source map is a surjective submersion with Hausdorff fibers,
is called a Lie groupoid.
This definition is somewhat weaker than the one previously considered. Theorems 3.5, 3.6
and 3.11 fit nicely with this definition. However, there is a slight problem with Theorem 3.8
as it uses the assumption which states that G1 is Hausdorff which is somewhat restricting when
considering the former definition. However, in this context one usually uses a stronger definition
of transitivity:
Definition 3.14. (Moerdijk [6]) A Lie groupoid is called transitive if (s, t) : G1 → G0 × G0 is
a surjective submersion.
Taking this into consideration we arrive at a version of our main Theorem which is more
fitting in this case:
Theorem 3.15. Let G be a transitive topological groupoid with a smooth base G0, Tychonoff
source fibers Gx for which the topological groups Gxx are locally Euclidean, the space G1 is first
countable and the map (s, t) : G1 → G0×G0 is a quotient map. Then G is continuously isomorphic
to a unique Lie groupoid through a base preserving isomorphism.
Proof. This follows from Proposition 3.2 and Theorem 3.5. �
Theorem 3.16. Let G be a transitive topological groupoid with a smooth base G0, for which the
group Gxx is a topological manifold, Gx is a (possibly not second countable) topological manifold,
G1 is a (possibly not second countable and non-Hausdorff) topological manifold and the map
(s, t) : G1 → G0 × G0 is a quotient map. Then G is continuously isomorphic to a unique Lie
groupoid through a base preserving isomorphism.
10 P. Raźny
Proof. Apply the previous theorem. �
Remark 3.17. Note that in the above theorems the resulting groupoids are Lie with respect
to both definitions.
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https://doi.org/10.1090/gsm/153
https://arxiv.org/abs/1612.09012
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 Main results
3.2 Some remarks considering assumptions
References
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