On The Extension of Topological Local Groups with Local Cross Section
H. Sahleh^{1}, A. Hosseini^{2}
^{1,2}Department of Mathematics, Faculty of Mathematical Sciences
University of Guilan, P.O.BOX 1914, Rasht – IRAN.
Corresponding Address:
[email protected] , [email protected]
Research Article
Abstract: In this paper, we introduce the cohomology of topological local groups and topological local extensions. We show that the second cohomology of a local topological group is in one to one correspondence with the class of topological local extensions with local cross sections.
keywords: Cohomology of topological local group, Strong homomorphism, Local crosssection, Topological local group extension.
MSC 2000: Primary 22A05, Secondary 22A10.
1.Introduction:
Let and be topological groups, abelian. We consider as a Gmodule with a continuous action of on , that is, a continuous function from into , carrying onto .
By a topological extension of by , , we mean a short exact sequence
with an open continuous homomorphism and a closed normal subgroup of .
Two extensions, , and , of by are said to be equivalent, , if there exists a continuous isomorphism such that and . The set of equivalence classes of extensions of by , denoted by , with the Bairesum is a group [2].
A crosssection of a topological group extension of by is a continuous map such that for each . There is a one to one correspondence between , and [2].
In this paper we show a similar result for topological local groups [3]. In section 1 we give some primarily definitions which will be needed in sequel. In section 2, we introduce the local extension on topological local groups and prove that the second cohomology of topological local group is isomorphic with the group of equivalence classes of topological local extensions with local crossedsections.
We use the following notations:
• " 1 " is the identity element of .
• "" : , a sublocal group (subgroup) of a local group (group) .
• where is a local group.
2.Primary Definitions:
We recall the following definition from [5]:
A local group is like a group except that the action of group is not necessarily defined for all pairs of elements, The associative law takes the following form: if and are defined, then if one of the products , is defined, so is the other and the two products are equal. It is assumed that each element of has an inverse.
Definition 2.1 [3] Let be a local group, if there exist:
 a distinguished element , the identity element,
 a continuous product map defined on an open subset
.
 a continuous inversion map
satisfying the following properties:
(i)Identity: for every
(ii) Inverse: for every
(iii) Associativity: If , , and all belong to , then
then is called a topological local group.
Example 2.2Let be a Hausdorff topological space and be the diagonal of , and . Define by:
Now , by the action of , is a local group.
If , , we have . If is a neighborhood of , then . There are two cases;
 : since is Hausdorff, there are disjoint neighborhood , containing , , respectively. Then and and . Hence, . So is continuous.
 : .
If and is a closed neighborhood of in then . Hence, is continuous. Therefore, , and , are continuous. So is a topological local group.
Definition 2.3 Let and be topological local groups. We say that operates on the left of if:
There is a neighborhood of the identity in and a neighborhood of the identity in such that for every , there exists with the following condition:
1., is continuous;
2. for all ;
3. If and and is defined for all then ;
4. If , are so that and are defined in and respectively, then .
Definition 2.4 A continuous map of topological local groups, is called a homomorphism if ;
1. where ;
2. and ;
3. if then exists in and .
With these morphisms topological local groups form a category which contains the subcategory of topological groups.
Definition 2.5A homomorphism of topological local groups is called strong if for every , the existence of implies that .
A morphism is called a monomorphism (epimorphism) if it is injective (surjective).
Lemma 2.6 [1, Lemma 2.5] Let be a symmetric neighborhood of the identity in a topological local group . There is a neighborhood of identity in such that for every , .
We denote the product of p copies of by .
Definition 2.7Let and be topological local groups. A local pmap of into is a continuous map where is a symmetric neighborhood of identity in such that whenever .
Definition 2.8Let and be topological local groups. Two local pmaps and of into , where are symmetric neighborhoods of the identity in , are said to be equivalent if there is a neighborhood with such that
whenever for all .
Definition 2.9 The equivalence class of a local pmap is called a local pcochain of to .
Let and be topological local groups and abelian (written additively). Let be the set of equivalence classes of local pmaps, with the usual addition of functions. The set is an abelian group. Therefor, we define an addition on . Suppose and are symmetric neighborhoods of the identity in where , are local pmaps. Let be a neighborhood of identity in . By Lemma 2.6, there is a symmetric neighborhoods in such that is defined when . Since are continuous, then there exists a neighborhood of 1 in such that . Now define a local map by
for every and . It is clear that the local pcochain does not depend on the choice of the representations and . Hence, we define an addition in , by
.
Definition 2.10 Suppose and are topological local groups and abelian. We define a coboundary operator
.
Let be a neighborhood of the identity in . By Lemma 2.6, there is a neighborhood of the identity in , such that is defined whenever . Suppose is a neighborhood of 1 in such that whenever and . Suppose , a local pmap. By continuity of , we choose a symmetric neighborhood in such that . By Lemma 2.6, there is a symmetric neighborhood in such that , for all .
Define a local (p+1)map , for each point by
It is easy to show that the local (p+1)cochain depends only on the given local pcochain and .
Definition 2.11 A local pcochains such that is called a local pcocycles.
We denote the set of all pcocyles by .
Definition 2.12 The image of a coboundry operator in is called a local pcoboundry. We denote the set of all pcoboundries by .
It is easy to show that ,since . Then is a subgroup of .
Definition 2.13 Let and be topological local groups and abelian. Then is called the pth cohomology topological local group.
3.Second Cohomology and Topological Local Group Extensions:
In this part we prove that the second cohomology of a topological local group is isomorphic with the group of the equivalence classes of topological local extensions with local crossedsections.
Definition 3.1 Let be topological local groups and a symmetric neighborhood in . The continuous map is an open continuous local homomorphism of onto if
1. there exists a symmetric neighborhood in which , ;
2. , ;
3. for every symmetric neighborhood , , is open in .
The map is called an open continuous local isomorphism of to if can be chosen that so is one to one.
Definition 3.2 A topological local group extension of the topological local group by a topological local group is a triple where is a topological local group, is an open continuous local homomorphism of to , and is an open continuous local isomorphism of onto the kernel of [2].
Remark 3.3If is a topological local group extension of by , where a strong homomorphism and , then is a closed normal topological subgroup of .
Definition 3.4 Let be a topological local group extension of by . A continuous map where is a neighborhood of 1 in is called a local crosssection if for every .
Definition 3.5A topological local group extension of is said to be fibered if it has a local cross section.
Definition 3.6A topological local group extension of is said to be essential if it has a local homomorphism of to .
Definition 3.7 Let and be topological local extensions of an abelian topological local group by a topological local group . If there exists a strong isomorphism of onto such that and .
then and are equivalent, .
Note 3.8 Let be a topological local group. The set is called the center of if
.
In this paper, we replace by in Definition 3.2. In this case, is a subset of and .
Definition 3.9 Let be an abelian topological local group. A pair factor set on topological local group is a pair where is continuous, and is a neighborhood of the identity in . Suppose there exist a neighborhoods of the identity in and such that , for every . Let be a symmetric neighborhood in such that , for every and .
Now, there is an action of on , such that , for every is a local inner automorphism,
(3.1)
whenever and
(3.2)
for .
Definition 3.10 The pair factor set is normalized if
,
where is the identity automorphism.
Remark 3.11Let be an extension of by and a pair factor set on to where is locally isomorphism with the kernel . It is clear that is a pair factor set on to . We call the extension of by .
Let be an extension of by . The local crosssection of is called normalized if .
In this section we assume all pair factor sets and local crosssections are normalized.
Definition 3.12 Let be topological local groups. Suppose a topological local group extension of by where is locally isomorphism with the kernel with a local crosssection where is a neighborhood of the identity in . There are a symmetric neighborhood , and a neighborhood of the identity in such that and are defined for every and . Then, the action of on is defined by
(3.3)
In Definition 3.12, since is the center of , we can easily see that for each and , the element dose not depend on the choice of the local crosssection.
Proposition 3.13 Let and be topological local groups and abelian. Suppose is a topological local extension of by . Each local crosssection of determines a pair factor set on to .
Proof.Let be a neighborhood of the identity in . By Lemma 2.6, there is a symmetric neighborhood in such that for all . Consider the continuous map , . Let and . Hence, and are defined in . We define
Let , , where is a symmetric neighborhood and . Then, is a local automorphism. We show that is a pair factor set. We verify the first condition of Definition 3.9. Let and and are defined in , since and are in . Hence, the relation implies that
We have
The right hand side of (1) is in , since , , and are defined in . Similarly, we obtain
(2)
By comparing (1) and (2),
for which proves (3.1). Now, for and :
.
So, (3.2) holds.□
Proposition 3.14 Let and be topological local groups and abelian. Suppose is a topological local extension of by which has a continuous local crosssection . Then, determines uniquely an element of .
Proof. Suppose is a topological local extension with a local crosssection where is a symmetric neighborhood of identity. Let be another local crosssection of , a symmetric neighborhood in . Suppose the pair factor set corresponds to .
We define a local cochain by , for .
By Lemma 2.6, there is a symmetric neighborhood , such that for . The map is continuous and by Lemma 2.6, there is a neighborhood , such that for .
Let and be defined in , for all . Consider the continuous map ,
.
;
;
;
by the action (3.3)
where . Since is abelian, then
.
Therefore, the pair factor set is independent of the choice of the continuous local crosssection.□
Definition 3.15 Let and be topological local groups with the action on . Suppose and are neighborhoods of the identities in and respectively such that all products are defined . Let , be a continuous strong homomorphism where and . We define by
for every and . The space is called the semidirect product of topological local groups and with respect to , denoted by .
Proposition 3.16 Let and be topological local groups and abelian. Let be a pair factor set on to . There exists an extension of by with a continuous local crosssection which corresponds to .
Proof. Let and be topological local groups and abelian. Let is a neighborhood of the identity in . By Lemma 2.6, there is a symmetric neighborhood in such that for all . Suppose is a neighborhood of the identity in . By Lemma 2.6, there is a symmetric neighborhood in , such that for all .
Suppose . By [3, Theorem 2.28], is a topological local group with the product
for every and ;
where is a local automorphism.
The identity of is and the inverse is given by
for and .
The map , is a strong homomorphism, since is the projection of onto . It is clear that is open and continuous. The kernel = consists of the elements with the product whenever is defined. Since is normalized then , for every . So can be identified with by the correspondence .
Let . If , we define
(3.4)
We have
We can verify that
(3.5)
Then, by (3.4) and (3.5), is a pair factor set of with the local crosssection .□
Remark 3.17Let be a pair factor set of to . By Propositions 3.13, 3.16, 3.14, every pair factor set corresponds to an element of . Therefore, determines , . Now, let , be another local extension and .
By Proposition 3.13, there is a pair factor set which corresponds to . By Definition 3.7, choose an arbitrary open continuous strong isomorphism such that and for . Then, is a continuous local crosssection of . As in Proposition 3.14, corresponding to is identical with corresponding to . Hence, equivalent local extensions of by determine the same element of .
Theorem 3.18 Let and be topological local groups and abelian. There is a one to one corresponding between the second cohomology of a topological local group and the group of equivalence classes of topological local extensions with continuous local crossedsections.
Proof.By [4,Theorem 2.9],is a group. Then, it is immediate by, Propositions 3.13 and 3.16.□
References:
 Goldbring, Hilbert's fifth problem for local groups, Annales. Math, Vol. 172, No.2, pp 12691314, 2010.
 S.T. Hu, Cohomology theory in Topological Groups, Michigan Math. J. Vol. 1, pp 1159, 1952.
 H. Sahleh, A. Hosseini, A Note on topoogical local groups, Journal of Mathematical Sciences: Advances and Applications, Vol. 14, No .2, pp 101127, 2012.
 H. Sahleh, A. Hosseini,The Group of Extensions of a Topological Local Group ,General Mathematics Notes (GMN), to appear.
S. Swierczkowski, Embedding Theorems for Local Analytic Groups, Acta . Math. Vol.114, pp 207235, 1965.
