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On The Extension of Topological Local Groups with Local Cross Section

H. Sahleh¹, A. Hosseini²

^1,2Department 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 cross-section, Topological local group extension.

MSC 2000: Primary 22A05, Secondary 22A10.

1.Introduction:

Let  and  be topological groups,  abelian. We consider  as a G-module 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 Baire-sum is a group [2].

A cross-section 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 crossed-sections.

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: 

1. a distinguished element , the identity element,
2.  a continuous product map  defined on an open subset

.

1.  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; 

1.  : since  is Hausdorff, there are disjoint neighborhood ,  containing , , respectively. Then  and  and . Hence, . So  is continuous.
2.  : .

 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 p-map 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 p-maps  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 p-map is called a local p-cochain of  to .

Let  and  be topological local groups and  abelian (written additively). Let  be the set of equivalence classes of local p-maps, 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 p-maps. 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 p-cochain  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 p-map. 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 p-cochain  and .

Definition 2.11  A local p-cochains  such that  is called a local p-cocycles.

We denote the set of all p-cocyles by .

Definition 2.12  The image of a coboundry operator in  is called a local p-coboundry. We denote the set of all p-coboundries 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 p-th 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 crossed-sections.

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 cross-section 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 cross-section  of  is called normalized if .

In this section we assume all pair factor sets and local cross-sections 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 cross-section  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 cross-section.

Proposition 3.13   Let  and  be topological local groups and  abelian. Suppose  is a topological local extension of  by . Each local cross-section  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 cross-section . Then,  determines uniquely an element of .

Proof. Suppose  is a topological local extension with a local cross-section  where  is a symmetric neighborhood of identity. Let  be another local cross-section 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 cross-section.□

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 semi-direct 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 cross-section  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 cross-section .□

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 cross-section 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 crossed-sections.

Proof.By [4,Theorem 2.9],is a group. Then, it is immediate by, Propositions 3.13 and  3.16.□

References:

1. Goldbring,  Hilbert's fifth problem for local groups, Annales. Math, Vol. 172, No.2, pp 1269-1314, 2010.
2. S.T. Hu,  Cohomology theory in Topological Groups, Michigan Math. J. Vol. 1, pp 11-59, 1952.
3. H. Sahleh, A. Hosseini,  A Note on topoogical local groups, Journal of Mathematical Sciences: Advances and Applications, Vol. 14, No .2, pp 101-127, 2012.
4. H. Sahleh, A. Hosseini,The Group of Extensions of a Topological Local Group ,General Mathematics Notes (GMN), to appear.