The Sheaf of The H-Cogroups

S Ü Fen Ed Fak Fen Derg
Sayı 24 (2004) 57- 61, KONYA
The Sheaf of The H-Cogroups
Ahmet Ali ÖÇAL1
Abstract: In this study we show that the sheaf which is constructed in [1] is a sheaf of
H -cogroups and it is an H -cospace if we equip the set of sections of this sheaf with the
compact-open topology. Finally we give some characterizations.
Key Words: Cospace, Sheaf, Cogroup, Homotopy
H-Kogrupların Demeti
Özet: Bu çalışmada [1] deki metod kullanılarak H-kogrupların demeti inşa edilmiş ve bu
demetin her bir kesitlerinin kümesi kompakt açık topoloji ile donatılmış ise bir H-kouzayı
olduğu gösterilmiştir. Sonuç olarak bazı karakterizasyonlar verilmiştir.
Anahtar Kelimeler: Kouzay, Demet, Kogrup, Homotopi
Introduction
Let C be the category of topological spaces X satisfying the property that all pointed
topological spaces ( X , x ) , x Î X have the same homotopy type. This category includes for
example all topological vector spaces. Let us take X Î C as a base set if Q is any abelian H cogroup, then there exists a sheaf (H , p ) over the topological space X which is formed by a Q
H -cogroup. For each x Î X , p -1 (x ) = [Q; ( X , x )] = H x is the stalk of the sheaf which has a discrete
topology. (where [Q; ( X , x )] = H x is a set of homotopy classes of homotopic maps preserving the
base points from (Q, q 0 ) to ( X , x ) ) [1,7].
2. The Sheaf of the H -Cogroups
Definition 2.1. Let ( X , x ) be a pointed topological space. The diagonal map
DX :X ® X ´X
is defined by D X (x ) = (x, x ) and the dual of the diagonal map D X , denoted by Ñ X , with
ÑX :X Ú X ® X
defined by
Definition 2.2. Let (H , p )
Ñ X (x, x 0 ) = Ñ X (x 0 , x ) = x. [2].
be a sheaf over X and s Î G( X , H ) [1]. The folding map
Ñ Hx : H x Ú H x ® H x
is defined by
Ñ H x ( [ f ]x , (s o p ) [ f ]x
1
) = [ f ]x
Gazi University, Faculty of Arts and Sciences, Department of Mathematics, 06500 Ankara, Turkey
The Sheaf of The H-Cogroups
and
Ñ H x ( (s o p ) ([ f ]x ) , [ f ]x ) = [ f ]x .
Now let us define a comultiplication on H x as follows: if [ f ]x , [c ]x Î H x then
vx : H x ® H x Ú H x
v x ( [ f ]x ) = [ Ñ X o ( f , c ) o v ]x ,
where v is an operation of a Q H -cogroup and
c :Q ® ( X , x )
is a constant map.
It follows from this definition that the comultiplication v x is well-defined, closed, continuous
and with this comultiplication H x is an H -cogroup. In fact, the constant map
cx . H x ® H x
which satisfies
c x ( H x ) = [ c ]x
is a homotopy identity of a pointed topological space ( H x , [ c ]x ) . That is, each of the
composites
( 1H x Ú cx )
vx
®
Hx
Hx ÚHx
is a homotopic to 1 H x , that is,
(
®
( cx Ú 1H x )
Ñx
Hx ÚHx
)
®H
(
x
)
Ñ x 1H x Ú c x v x ~
- 1H x ~
- Ñ H x c x Ú 1H x v x .
The continuous map
fx :H x ® H x
is a homotopy inverse for H x . That is, each of the composites
( 1H x Ú f x )
vx
Hx
®
Hx ÚHx
®
( f x Ú 1H x )
Ñx
Hx ÚHx
®H
x
is homotopic to c x , that is
Ñ x 1H x Ú f x v x ~
- cx ~
- Ñ H x f x Ú 1H x v x ,
(
)
(
)
where f (x ) is defined by
fx
( [ f ]x ) = [ f o f ]x
and f is homotopy inverse of a Q H -cogroup.
It can be shown also that v x is a homotopy associative.
Now we have the following results:
Result: 1. H x is an H -cogroup.
Result: 2. The sheaf which is constructed in [1] is a sheaf of H -cogroups.
Let G ( X , H ) denote the set of global section of H with the compact-open topology. Then
the mapping
v ¢: G ( X , H )® G ( X , H )Ú G ( X , H )
which is given by
v ¢ (s )(x ) = v x ( s (x ) ) = v x ( [ f ]x ) = [ Ñ x o ( f , c ) o v ]x
( s Î G ( X , H ),
x Î X ) defines a comultiplication on G ( X , H ) such that v ¢ is well-defined,
closed and continuous [3]. In fact, if U is an open neighborhood of v ¢ (s ) in G ( X , H ) Ú G ( X , H )
58
Ahmet Ali ÖÇAL
for
every s Î G ( X , H
{M ( C i , Oi ) }iÎJ ( J
),
then there exists is finite collection of open sets in the subbasis,
is finite) such that
v ¢ (s ) Îæç I M ( Ci , Oi ) Ú I M ( Ci , Oi )ö÷ Ì U .
iÎJ
è iÎJ
ø
Since the comultiplications v x are continuous in H x , we can choose neighborhoods U i of
s (x ) such that if si ( x ¢ )ÎU i , then
v ¢ ( s i )(x ¢) = v x¢ ( s i ( x ¢ ) )Î ( Oi Ú Oi )
Thus, s ÎI M ( C i , U i ) and if s ¢Î M ( C i , U i ) then
V ¢ (s )Î( I M ( Ci , Oi ) Ú I M ( Ci , Oi ) )
since
v ¢ ( s ¢ )(x ) = v x ( s ¢ (x ) ) £ I ( Oi , Oi )
for all x ÎI C i .
Let I : X ® H be the section (p o I =1x ) satisfying I (x )ÎC ( c x ) Ì H x . Such an
I
exists
continuous [4], where C ( c x ) is the path component
of c x
in H x . Thus,
I is the identity of G ( X , H ) . Hence, we have the following result:
Result: 3. Since H x is an H -cogroup, G ( X , H ) is an H -cospaces with comultiplication
v¢ .
Let S be family of supports on X and V Ì X . Then GS V ( V , H ) is the collection of
and
sections s ÎG ( V , H
GS V ( V , H
)
satisfying s = { x Î X : s (x ) ÏC (c x ) } where S V = { A Ì V : AÎ S } . The collection
)
is closed under the comultiplication of G ( X , H
s ÎGS V ( V , H
) then
which restricted to GS V ( V , H
)
for if
s Î S V . Since
v ¢S V (s )
it follows that v ¢S V (s ) Ì s
c
{
= xÎ X : v S
V
( V , H )ÎC ( c x )
and so v ¢S V (s ) Î S V
{ xÎ X : I (x )ÏC ( c x ) } = Æ ) and hence
g
U
V
( S )= S
c
,
. Also I ÎGS V ( V , H ) (because I =
) is an H -cospace.
X , G ( X , H ) is an H -cospace (Result 3). If V
open subset of X such that V Ì U we can define a map
restricted map, that is
}É s
GS V ( V , H
Thus for any open subset U is of
If U ,V
)
g
U
V
: G ( U , H ) ® G (V , H
)
is another
which is called
V.
and W are any three open subsets of X such that W Ì V Ì U then one can
U
V
U
U
ì
ü
, X ý is called direct system [5].
and so í G ( U , H ),
W
W
V
V
î
þ
3. The Characterizations
Let Q be any H -cogroup and X 1 , X 2 be two topological spaces in Category C . Let
observe that
g =g og
g
H 1 , H 2 be the corresponding sheaves which is given Result 2, respectively. Let us denote these as
the pairs ( X 1 , H 1 ) and ( X 2 , H 2 ) .
Definition 3. Let the pairs ( X 1 , H 1 ) and ( X 2 , H 2 ) be given. We say that there is a
homomorphism between these pairs and write
F b * , b : ( X 1 , H 1 ) ® ( X 2 , H 2 ),
(
(
*
if there exist a pair F b , b
) such that
)
1. b : X 1 ® X 2 is a surjective and continuous map.
59
The Sheaf of The H-Cogroups
2. b * : H 1 ® H 2 is a continuous map.
3. b * preserves the stalks with respect to b . That is, the following diagram is commutative
b*
®
H1
p1 ¯
H2
¯p2
b
®
X1
X2
4. For every x1 Î X 1 the restricted map b * H x1 : H 1x1 ® H 2 b ( x1 ) is a homomorphism.
Theorem 3.
( X 1 , H1 )
Let the pairs
and
( X 2 , H2 )
be given. If the map b : X 1 ® X 2 is
surjective and continuous, then there exists a homomorphism between the pairs ( X 1 , H 1 ) and
( X 2 , H2 ).
Proof. Let x1 Î X be arbitrarily fixed point. Then b ( xi )Î X 2 and
[ Q : ( X , x1 ) ] = H 1x
1
Ì H 1 , [ Q : ( X , b (x1 )) ] = H 2 b ( x1 ) Ì H 2
are corresponding H -cogroups or stalks.
Since ( X 1 , x1 ) , ( X 2 , b ( x1 ) ) are pointed topological spaces and f1 , g1 are base-points
preserving continuous
from ( Q, q 0 )
maps
( X 1 , x1 ) then there exists f 2 , g 2 base-points
( X 2 , b (x1 ) ) can be defined as f 2 = b o f1 , g 2 = b o g1 ,
to
preserving continuous maps from ( Q, q 0 ) to
respectively. Moreover, if f1 ~ g1 rel.q 0 , then it can be easily shown that f 2 ~ g 2 rel.q 0 . Thus the
correspondence
[ f ]x
1
®[ b o f
]b (x )
1
is well-defined [6] and it maps homotopy classes of base-
points preserving continuous maps from ( Q, q 0 ) to
points preserving continuous maps from ( Q, q 0 ) to
there corresponds a unique element
( X 1 , x1 ) , to the homotopy classes of base( X 2 , b (x1 ) ) . That is, to each element [ f ]x
1
[ b o f ]b (x ) .
1
Since the point x1 Î X is arbitrarily fixed, the above correspondence gives us a map
b * : H 1 ® H 2 such that b * ( [ f
1)
b
* -1
b
*
] ) = [ b o f ]Î H 2 , for every [ f ]Î H1 .
is continuous. Because if U 2 Ì H 2 is any open set, then it can be shown that
(U 2 ) = U 1 Ì H 1 is
if U 2 Ì H 2 is an open set, then U 2 = Ú s i2 (Wi ) and
an open set. In fact,
iÎI
p 2 (U 2 ) = Ú Wi , where the Wi ’ s are open neighborhoods and the
si2
iÎI
are sections over Wi . Thus
U Wi Ì X 2 is an open set and b -1 æç Ú Wi ö÷ Ì X 1 is an open set since b is a surjective and
è iÎI ø
iÎI
continuous map. Furthermore, since b -1 (Wi ) , i Î I are open in X 1 , there exist sections
s1i : b -1 (Wi ) ® H 1
such that
(
)
Ú s 1i b -1 (Wi ) Ì H 1
iÎI
is an open set. It can be shown that
(
).
U 1 = Ú s 1i b -1 (Wi )
iÎI
2. b * preserves the stalks with respect to b . In fact, for any
(p
60
2
[ f ]x Î H 1x
1
1
(
(b
op 1 ) [ f
ob
*
)( [ f ]
x1
Ì H1 ,
)= b (p ( [ f ] ) )= b (x )= x
)= p (b [ f ] )= p ( [b o f ] )= x .
]x
1
1
1
x1
2
*
2
x1
2
x2
2
Ahmet Ali ÖÇAL
3.It can be easily shown that b * H 1x1 is a homomorphism.
(
)
As a result F = b * , b is a homomorphism.
References
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spaces, Commun. Fac. Sci. Univ. Ank. Serias A1, 37: 1-4 (1988).
[2] Switzer, R.M., Algebraic Topology-Homotopy and Homology, Springer-Verlag 1975.
[3] Yıldız, C. And Öcal, A.A., The sheaf of the H-groups, Jour. Of the Inst. Of Sci. And Tech. Of Gazi
Univ. 10 (2): 223-230 (1997).
[4] Yıldız, C., “Kompleks Analitik Manifoldların Homoloji Grubu üzerine” Erciyes Univ. Fen-Bil.
Enst. Dergisi, 2: 1008-1015 (1990).
[5] Yıldız, C., The Sheaf of the Homology Groups of the Complex Manifolds, Commun Fac. Sci
Univ. Ank. Series A1, 46: 135-141 (1997).
[6] Yıldız, C., Some Theorems on the sheaf of the groups Formed by H-groups over topological
spaces, Pure and Applied Math. Sci. XXIX (1-2) March: 141-147 (1989).
[7] Yıldız, C., On the sheaf cohomology, Gazi Univ. Fen-Edb. Fak. Fen Bil. Dergisi 7: 117-124
(1997).
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The Sheaf of The H-Cogroups
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