Coverings and crossed modules of topological groups with operations

0.16 MB·

by  Osman Mucuk

Coverings and crossed modules of topological
groups with operations
Osman Mucuk∗a and Tunçar S¸ahan†b
6 a
DepartmentofMathematics,ErciyesUniversity,Kayseri,TURKEY
1
0 bDepartmentofMathematics,AksarayUniversity,Aksaray,TURKEY
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a
J

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Abstract

]
T ItisawellknownresultinthecoveringgroupsthatasubgroupGofthefundamental
A
groupattheidentityofasemi-locally simplyconnectedtopologicalgroupdeterminesa
.
h
covering morphismoftopologicalgroupswithcharacteristic groupG. Inthis paperwe
t
a
m generalize this result to a large class of algebraic objects called topological groups with
[ operations, including topological groups. We also give the cover of crossed modules

1 withintopologicalgroupswithoperations.
v
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9 KeyWords: Coveringgroup,universalcover,crossedmodule,groupwithoperations,topo-
0
7 logical group with operations
0
Classification: 18D35, 22A05,57M10
.
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0
6
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1 Introduction
:
v
i
X
Thetheoryofcoveringspacesisoneofthemostinterestingtheoriesinalgebraictopology.
r
a e
It is well known that if X is a topological group, p: X → X is a simply connected covering
e e
map, e ∈ X is the identity element and e˜ ∈ X is such that p(e˜) = e, then X becomes a
topological group with identity e˜such that p is a morphism of topological groups (see for
example [8]).
The problem of universal covers of non-connected topological groups was first stud-
ied in [23]. He proved that a topological group X determines an obstruction class k in
X
∗E-mail: [email protected]
†E-mail: [email protected]
1
H3(π (X),π (X,e)), and that the vanishing of k is a necessary and sufficient condition for
0 1 X
the lifting of the group structure to a universal cover. In [17] an analogous algebraic result
was given in terms of crossed modules and group-groupoids, i.e., group objects in the cat-
egory of groupoids (see also [6] for a revised version, which generalizes these results and
shows the relation with the theory of obstructions to extension for groups and [16] for the
recently developed notion ofmonodromy for topological group-groupoids).
In [7, Theorem 1] Brown and Spencer proved that the category of internal categories
withinthegroups, i.e.,group-groupoids, isequivalenttothecategoryofcrossedmodulesof
groups. Thenin[21,Section3],Porterprovedthatasimilarresultholdsforcertainalgebraic
C
categories , introduced by Orzech [19], which definition was adapted by him and called
category ofgroups with operations. ApplyingPorter’s result, the studyofinternalcategory
C
theory in was continued in the works of Datuashvili [10] and [11]. Moreover, she devel-
opedcohomology theoryofinternalcategories, equivalently,crossed modules,incategories
of groups with operations [9, 12]. In a similar way, the results of [7] and [21] enabled us
to prove that some properties of covering groups can be generalized to topological groups
with operations.
If X is a connected topological space which has a universal cover, x ∈ X and G is a
0
subgroup of the fundamental group π (X,x ) of X at the point x , then by [22, Theorem
1 0 0
e
10.42] we know that there is a covering map p: (X ,x˜ ) → (X,x ) of pointed spaces, with
G 0 0
characteristic group G. InparticularifGissingleton, thenpbecomestheuniversalcovering
e
map. Further if X is a topological group, then X becomes a topological group such that
G
p is a morphism of topological groups. Recently in [2] this method has been applied to
topologicalR-modulesandobtainedamoregeneralresult(seealso[3]and[18]forgroupoid
setting).
The object of this paper is to prove that this result can be generalized to a wide class
of algebraic categories, which include categories of topological groups, topological rings,
topological R-modules and alternative topological algebras. This is conveniently handled
TC
by working in a category . The method we use is based on that used by Rothman in [22,
Theorem 10.42]. Further we give the cover of crossed modules within topological groups
with operations.
2 Preliminaries on groupoids and covering groups
As it is defined in [4, 15] a groupoid G has a set G of morphisms, which we call just
elementsof G,a setG ofobjectstogether with mapsd ,d : G → G and ǫ: G → G such that
0 0 1 0 0
d ǫ = d ǫ = 1 . The maps d , d are called initial and final point maps respectively and the
0 1 G0 0 1
2
map ǫ is called object inclusion. If a,b ∈ G and d (a) = d (b), then the composite a ◦ b exists
1 0
such thatd (a◦b) = d (a)andd (a◦b) = d (b). Sothere existsapartialcomposition defined
0 0 1 1
by G × G → G,(a,b) 7→ a◦b, where G × G is the pullback of d and d . Further, this
d1 d0 d1 d0 1 0
partial composition is associative, for x ∈ G the element ǫ(x) acts as the identity, and each
0
element a has an inverse a−1 such that d (a−1) = d (a), d (a−1) = d (a), a◦a−1 = ǫd (a) and
0 1 1 0 0
a−1 ◦a = ǫd (a). The mapG → G, a 7→ a−1 is calledthe inversion.
1
In a groupoid G for x,y ∈ G we write G(x,y) for the set of all morphisms with initial
0
point x and final point y. According to [4] for x ∈ G the star of x is defined as {a ∈ G |
0
d (a) = x} and denoted asSt x.
0 G
Let G and H be groupoids. A morphism from H to G is a pair of maps f: H → G and
f : H → G such that d f = f d , d f = f d , fǫ = ǫf and f(a ◦ b) = f(a) ◦ f(b) for all
0 0 0 0 0 0 1 0 1 0
(a,b) ∈ H × H. For such a morphism we simplywrite f: H → G.
d1 d0
e e
Let p: G → G be a morphism of groupoids. Then p is called a covering morphismand G a
e
coveringgroupoid of G ifforeach x˜ ∈ G0 the restriction StGex˜ → StGp(x˜) isbijective.
We assume the usual theory of covering maps. All spaces X are assumed to be locally
path connected and semi-locally 1-connected, so that each path component of X admits a
e
simply connected cover. Recall thata covering mapp: X → X ofconnected spacesiscalled
e
universal if it covers every covering of X in the sense that if q: Y → X is another covering
e e
of X then there exists a map r: X → Y such that p = qr (hence r becomes a covering). A
e e
covering map p: X → X is called simply connected if X is simply connected. Note that a
simply connected covering isauniversal covering.
A subset U of a space X, which has a universal cover, is called liftable if it is open, path
e
connected and lifts to each covering of X, that is, if p: X → X is a covering map, ı: U → X
e
is the inclusion map and x˜ ∈ X such that p(x˜) = x ∈ U, then there exists a map (necessarily
e
unique)ˆı: U → X such that pˆı = ı andˆı(x) = x˜. It is an easy application that U is liftable if
and only if it is open, path connected and for all x ∈ U, the fundamental group π (U,x) is
1
mapped to the singleton by the morphism ı : π (U,x) → π (X,x) induced by the inclusion
⋆ 1 1
ı: (U,x) → (X,x).
A space X is called semi-locallysimplyconnected if each point hasa liftable neighborhood
and locally simply connected if it has a base of simply connected sets. So a locally simply
connected space isalsosemi-locally simplyconnected.
e
For a covering map p: (X,x˜ ) → (X,x ) of pointed topological spaces, the subgroup
0 0
e
p (π (X,x˜ ))ofπ (X,x )iscalledcharacteristicgroupofp,wherep isthemorphisminduced
⋆ 1 0 1 0 ⋆
by p (see [4, p.379] for the characteristic group of a covering map in terms of covering mor-
e e
phism of groupoids ). Two covering maps p: (X,x˜ ) → (X,x ) and q: (Y,y˜ ) → (X,x ) are
0 0 0 0
calledequivalentiftheircharacteristic groups areisomorphic, equivalentlythereisahomeo-
3
e e
morphism f: (X,x˜ ) → (Y,y˜ ) such thatqf = p.
0 0
We recall a construction from [22, p.295] as follows: Let X be a topological space with a
basepointx andGasubgroupofπ (X,x ). LetP(X,x )bethesetofallpathsofαinX with
0 1 0 0
initial point x . Then the relation defined on P(X,x ) by α ≃ β if and only if α(1) = β(1)
0 0
and [α ◦ β−1] ∈ G, is an equivalence relation. Denote the equivalence relation of α by hαi
G
e
and define X as the set of all such equivalence classes of the paths in X with initial point
G
e
x . Define a function p: X → X by p(hαi ) = α(1). Let α be the constant path at x and
0 G G 0 0
e
x˜ = hα i ∈ X . Ifα ∈ P(X,x )andU isanopenneighbourhoodofα(1),thenapathofthe
0 0 G G 0
formα◦λ,whereλisapathinU withλ(0) = α(1),iscalledacontinuationofα. Foranhαi ∈
G
e
X and an open neighbourhood U of α(1), let (hαi ,U) = {hα ◦λi : λ(I) ⊆ U}. Then the
G G G
e e
subsets (hαi ,U) form a basisfor a topology on X such that the mapp: (X ,x˜ ) → (X,x )
G G G 0 0
is continuous.
InTheorem3.7wegeneralizethefollowingresulttotopological groupswithoperations.
Theorem 2.1. [22, Theorem 10.34] Let (X,x ) be a pointed topological space and G a subgroup
0
of π (X,x ). If X is connected, locally path connected and semi-locally simply connected, then
1 0
e
p: (X ,x˜ ) → (X,x ) isa coveringmap withcharacteristicgroup G .
G 0 0
Remark2.2. LetX beaconnected,locallypathconnectedandsemilocallysimplyconnected
e
topologicalspaceandq: (X,x˜ ) → (X,x )acoveringmap. LetGbethecharacteristicgroup
0 0
e
of q. Then the covering map q is equivalent to the covering map p: (X ,x˜ ) → (X,x )
G 0 0
corresponding to G.
Sofrom Theorem 2.1the following result isobtained.
Theorem2.3. [22,Theorem10.42]SupposethatX isaconnected,locallypathconnectedandsemi-
e
locallysimplyconnectedtopologicalgroup. Lete ∈ X betheidentityelementandp: (X,e˜) → (X,e)
e e
a covering map. Then the group structure of X lifts to X, i.e., X becomes a topological group such
e
that e˜isidentity and p: (X,e˜) → (X,e) isamorphismof topological groups.
3 Universal covers of topological groups with operations
In this section we apply the methods of Section 2 to the topological groups with opera-
tions and obtain parallelresults.
The idea of the definition of categories of groups with operations comes from [14] and
[19](seealso[20])andthedefinitionbelowisfrom[21]and[13,p.21],whichisadaptedfrom
[19].
4
Definition 3.1. Let C be a category of groups with a set of operations Ω and with a set E of
E
identities such that includes the group laws, and the following conditions hold for the set
Ω ofi-ary operations inΩ:
i
(a)Ω = Ω ∪Ω ∪Ω ;
0 1 2
(b)Thegroup operationswritten additively0,−and+aretheelementsofΩ ,Ω andΩ
0 1 2
′ ′ ′ ◦
respectively. Let Ω = Ω \{+}, Ω = Ω \{−} and assume that if ⋆ ∈ Ω , then ⋆ defined by
2 2 1 1 2
◦ ′
a⋆ b = b⋆ais alsoin Ω . Alsoassume thatΩ = {0};
2 0
(c)For each ⋆ ∈ Ω′, Eincludesthe identity a⋆(b+c) = a⋆b+a⋆c;
2
(d) For each ω ∈ Ω′ and ⋆ ∈ Ω′, E includes the identities ω(a + b) = ω(a) + ω(b) and
1 2
ω(a)⋆b = ω(a⋆b).
C
Thecategory satisfyingtheconditions(a)-(d)iscalledacategoryofgroupswithoperations.
C
Inthe paperfrom nowon will denote the category ofgroups with operations.
C
A morphism between any two objects of is a group homomorphism, which preserves
′ ′
the operations ofΩ and Ω .
1 2
Remark 3.2. The set Ω contains exactly one element, the group identity; hence for instance
0
the category of associative rings with unit isnota category ofgroups with operations.
Example3.3. Thecategoriesofgroups,ringsgenerallywithoutidentity,R-modules,associa-
tive, associative commutative, Lie, Leibniz, alternative algebras are examples of categories
✷
of groups with operations.
Remark 3.4. The set Ω contains exactly one element, the group identity; hence for instance
0
associative ringswith unitare not groups with operations
Thecategory of topological groups with operations is definedin [1]asfollows:
Definition 3.5. A category TC of topological groups with a set Ω of continuous operations
E E
and with a set of identities such that includes the group laws such that the conditions
(a)-(d) in Definition 3.1 are satisfied, is called a category of topological groups with operations
TC
and the object of are called topologicalgroups with operations.
TC
Inthe rest of the paper will denote a category oftopological groups with operations.
TC
Amorphismbetweenanytwoobjectsof isacontinuousgrouphomomorphism,which
′ ′
preserves the operations in Ω and Ω .
1 2
Thecategoriesoftopologicalgroups,topologicalrings,topological R-modulesandalter-
nativetopologicalalgebrasareexamplesofcategoriesoftopologicalgroupswithoperations.
Proposition 3.6. If X is a topological group with operations, then the fundamental group π (X,0)
1
becomesagroup with operations.
5
Proof: Let X be an object of TC and P(X,0) the set of all paths in X with initial point 0
asdescribed in Section 1. There are binaryoperations on P(X,0)definedby
(α⋆β)(t) = α(t)⋆β(t) (1)
for ⋆ ∈ Ω andt ∈ I, unitinterval, and unaryoperations definedby
2
(ωα)(t) = ω(α(t)) (2)
for ω ∈ Ω . Hencethe operations (1) induce binary operations on π (X,0)defined by
1 1
[α]⋆[β] = [α⋆β] (3)
for [α],[β] ∈ π (X,0). Since the binary operations ⋆ in Ω are continuous it follows that
1 2
the binary operations (3) are well defined. Similarly the operations (2) reduce the unary
operations definedby
ω[α] = [ωα]. (4)
By the continuity of the unary operations ω ∈ Ω , the operations (4) are also well defined.
1
The other details can be checked and so π (X,x ) becomes a group with operations, i.e., an
1 0
C ✷
object of .
WenowgeneralizeTheorem2.1totopologicalgroupswithoperations. Wefirstmakethe
following preparation:
LetX beatopological group withoperations. Bytheevaluationofthecompositions and
operations of the paths in X such that α (1) = β (0) and α (1) = β (0) at t ∈ I , we have the
1 1 2 2
following interchangelaw
(α ◦β )⋆(α ◦β ) = (α ⋆α )◦(β ⋆β ) (5)
1 1 2 2 1 2 1 2
for ⋆ ∈ Ω , where◦ denotes the composition of paths, and
2
(α⋆β)−1 = α−1 ⋆β−1 (6)
for α,β ∈ P(X,0) where, say α−1 is the inverse path defined by α−1(t) = α(1−t) for t ∈ I.
Further wehave that
(ωα)−1 = ωα−1 (7)
6
ω(α◦β) = (ωα)◦(ωβ) (8)
when α(1) = β(0).
Parallelto Theorem 2.1,in the following theorem we prove a general resultfor topologi-
cal groups with operations.
Theorem 3.7. Let X be a topological group with operations, i.e., an object of TC and let G be the
subobject of π (X,0). Suppose that the underlying space of X is connected, locally path connected
1
e
andsemi-locallysimplyconnected. Letp: (X ,˜0) → (X,0)bethecoveringmapcorrespondingtoG
G
as a subgroup of the additive group of π (X,0) by Theorem 2.1. Then the group operations of X lift
1
e e e
to X , i.e.,X isatopological group withoperations and p: X → X isamorphismof TC.
G G G
e e
Proof: BytheconstructionofX inSection1,X isthesetofequivalenceclassesdefined
G G
via G. The binaryoperations on P(X,0)definedby (1) induce binaryoperations
hαi ⋆hβi = hα⋆βi (9)
G G G
and the unary operations on P(X,x ) defined by(2)induce unary operations
0
ωhαi = hωαi (10)
G G
e
on X .
G
We now prove that these operations (9) and (10) are well defined: For ⋆ ∈ Ω and the
2
paths α,β,α ,β ∈ P(X,0)with α(1) = α (1) and β(1) = β (1), we havethat
1 1 1 1
[(α⋆β)◦(α ⋆β )−1] = [(α⋆β)◦(α−1 ⋆β−1)] (by6)
1 1 1 1
= [(α◦α−1)⋆(β ◦β−1)] (by5)
1 1
= [α◦α−1]⋆[β ◦β−1] (by3)
1 1
So ifα ∈ hαi andβ ∈ hβi , then[α◦α−1] ∈ Gand [β ◦β−1] ∈ G. Since G isa subobject of
1 G 1 G 1 1
π (X,0), we have that [α◦α−1]⋆[β ◦β−1] ∈ G. Therefore the binary operations (9) are well
1 1 1
defined.
Similarlyfor the paths α,α ∈ P(X,0)with α(1) = α (1)and ω ∈ Ω we havethat
1 1 1
[(ωα)◦(ωα )−1)] = [(ωα)◦(ωα−1)] (by7)
1 1
= [(ω(α◦α−1)] (by8)
1
= ω[α◦α−1] (by4)
1
7
Since G isasubobject ofπ (X,0),if[α◦α−1] ∈ G andω ∈ Ω then ω[α◦α−1] ∈ G. Hencethe
1 1 1 1
unary operations (10)are also welldefined.
e e
The axioms (a)-(d) of Definition 3.1 for X are satisfied and therefore X becomes a
G G
e
group with operations. Further by Theorem 2.1, p: (X ,˜0) → (X,0) is a covering map,
G
e
X is a topological group and p is a morphism of topological groups. In addition to this
G
e
we need to prove that X is an object of TC and p is a morphism of TC. To prove that the
G
′ e
operations(9)for⋆ ∈ Ω arecontinuouslethαi ,hβi ∈ X and(W,hα⋆βi )beabasicopen
2 G G G G
neighbourhood of hα ⋆ βi . Here W is an open neighbourhood of (α ⋆ β)(1) = α(1)⋆ β(1).
G
Since the operations ⋆: X × X → X are continuous there are open neighbourhoods U and
V of α(1)and β(1)respectively in X such thatU ⋆V ⊆ W. Therefore (U,hαi ) and (V,hβi )
G G
are respectively baseopen neighbourhoods of hαi andhβi ,and
G G
(U,hαi )⋆(V,hβi ) ⊆ (W,hα⋆βi ).
G G G
Therefore the binary operations (9) are continuous
′
Wenowprovethattheunaryoperations(10)forω ∈ Ω arecontinuous. Forif(V,hωαi)is
1
abaseopenneighbourhoodofhωαi,thenV isanopenneighbourhoodofωα(1)andsincethe
unary operations ω: X → X are continuous there isan open neighbourhood U of α(1) such
thatω(U) ⊆ V. Therefore(U,hαi)isanopenneighbourhoodofhαiandω(U,hαi) ⊆ (V,hωαi).
e
Moreover the map p: X → X defined by p(hαi ) = α(1) preserves the operations of Ω
G G 2
✷
and Ω .
1
From Theorem 3.7 the following result can be restated.
Theorem 3.8. Suppose that X is a topological group with operations whose underlying space is
e
connected, locally path connected and semi-locally simply connected. Let p: (X,˜0) → (X,0) be a
e
covering map such that X is path connected and the characteristic group G of p is a subobject of
e
π (X,0). Thenthe group operations of X liftstoX.
1
e
Proof: Byassumption thecharacteristic group Gofthe covering mapp: (X,˜0) → (X,0)
e e
is a subobject of π (X,0). So by Remark 2.2, we can assume that X = X and hence by
1 G
Theorem 3.7, the group operations ofX liftto Xe asrequired. ✷
In particular, in Theorem 3.7 if the subobject G of π (X,0) is chosen to be the singleton,
1
then the following corollary isobtained.
Corollary 3.9. Let X be a topological group with operations such that the underlying space of X is
e
connected, locally path connected and semi-locally simply connected. Let p: (X,˜0) → (X,0) be a
e
universal coveringmap. Thenthegroup structuresof X liftstoX.
Thefollowing proposition isuseful for Theorem 3.12.
8
Proposition 3.10. Let X be atopologicalgroup with operations and V a liftableneighbourhood of0
inX. Thenthereisa liftableneighbourhood U of 0 inX suchthatU ⋆U ⊆ V for ⋆ ∈ Ω .
2
Proof: Since X is a topological group with operations and hence the binary operations
⋆ ∈ Ω are continuous, there is an open neighbourhood U of 0 in X such that U ⋆ U ⊆ V.
2
Further if V is liftable, then U can be chosen as liftable. For if V is liftable, then for each
x ∈ U,thefundamentalgroupπ (U,x)ismappedtothesingletonbythemorphisminduced
1
by the inclusion map ı: U → X. Here U is not necessarily path connected and hence not
necessarily liftable. But since the path component C (U) of 0 in U is liftable and satisfies
0
theseconditions, U canbereplacedbythethepathcomponentC (U)of0inU andassumed
0
that U isliftable. ✷
Definition 3.11. Let X and Y be topological groups with operations and U an open neigh-
bourhood of 0 in X. A continuous map φ: U → S is called a local morphism in TC if
φ(a⋆b) = φ(a)⋆φ(b) when a,b ∈ U such thata⋆b ∈ U for ⋆ ∈ Ω .
2
e e
Theorem 3.12. Let X and X be topological groups with operations and q: X → X a morphism of
TC,whichisacoveringmap. LetU beanopen,pathconnectedneighbourhoodof0inX suchthatfor
each ⋆ ∈ Ω , the set U ⋆U is contained in a liftable neighbourhood V of 0 in X. Then the inclusion
2
e
map i: U → X liftstoalocal morphismˆı: U → X inTC.
e e e
Proof: Since V lifts to X, then U lifts to X by ˆı: U → X. We now prove that ˆı is a
local morphism oftopological groups with operations. Weknowbythe liftingtheorem that
e
ˆı: U → X is continuous. Let a,b ∈ U be such that for each ⋆ ∈ Ω , a⋆b ∈ U. Let α and β be
2
the paths from 0 to a and b respectively in U. Let γ = α ⋆ β. So γ is a path from 0 to a ⋆ b.
e ˜
Since U ⋆U ⊆ V, the paths γ is in V. So the paths α,β and γ lift to X. Suppose that α˜, β and
e
γ˜ are the liftings ofα, β and γ inX respectively. Then wehave
e
q(eγ) = γ = α⋆β = q(αe)⋆q(β).
But q isamorphism of topological group with operations and sowe have,
e e
q(αe⋆β) = q(αe)⋆q(β)
for⋆ ∈ Ω . Sincethepathsγ˜ andα˜⋆β˜havetheinitialpoint˜0 ∈ Xe,bytheuniquepathlifting
2
eγ = αe⋆β˜
On evaluatingthese pathsat 1 ∈ I wehave
ˆı(a⋆b) =ˆı(a)⋆ˆı(b).
9
✷
4 Covers of crossed modules within topological groups with
operations
IfAand B are objects of C, an extensionofAby B isanexact sequence
ı p
0 −→ A −→ E −→ B −→ 0 (11)
in which p is surjective and ı is the kernel of p. It is split if there is a morphism s: B → E
such that ps = ıd . A split extension of B by A is called a B-structure on A. Given such a
B
B-structure on A we get actions of B on A corresponding to the operations in C. For any
′
b ∈ B, a ∈ Aand⋆ ∈ Ω wehave the actions called derivedactions byOrzech [19, p.293]
2
b·a = s(b)+a−s(b)
(12)
b⋆a = s(b)⋆a.
In addition to this we note that topologically if an exact sequence (11) in TC is an split ex-
tension,thenthederivedactions(12)arecontinuous. SowecanstateTheorem[19,Theorem
2.4]in topological case, which isuseful for the proof of Theorem 4.6,asfollows.
Theorem 4.1. A setof actions(one for eachoperationinΩ )isasetofcontinuous derivedactions if
2
and only ifthe semidirectproductB ⋉Awith underlyingsetB ×Aand operations
′ ′ ′ ′
(b,a)+(b,a) = (b+b,a+(b·a))
′ ′ ′ ′ ′ ′
(b,a)⋆(b,a) = (b⋆b,a⋆a +b⋆a +a⋆b)
TC
is anobjectin .
C
Theinternalcategoryin isdefinedin[21]asfollows. WefollowthenotationsofSection
1 for groupoids.
Definition 4.2. An internal category C in C is a category in which the initial and final point
maps d ,d : C ⇒ C , the object inclusion map ǫ: C → C and the partial composition
0 1 0 0
◦: C × C → C,(a,b) 7→ a◦bare the morphisms in the category C. ✷
d1 d0
Notethatsince ǫisamorphism inC, ǫ(0) = 0andthatthe operation ◦beingamorphism
10