Combinatorial Realizability Of The New Exact Sequence

Abstract

In this work we introduce and study the generalization of
combinatorial realizability of J.H.C.Whitehead and a new notion
analogous of Jm-complex , which we call Jm,q-complex .
Consider the following sequence :
    
   

 

 

 

 

ï‚® ( , ) ( ) ( , ) ( ) ( , ) !
1
!
1
1 1
1
1, ,
, , p p
p q
p
p q
p p
p q
p
p q
p p
p q
p q p q
k k k k k
j
k k k p q p q ï° ï° ï° ï° ï°
ï³ ï³
ï¢
p q
p
p q p q
p p
p q k k by C and k by A , ,
1 ( , ) ( ) 

 Denote ï° ï°
lg log ( , ) 0 ( ) 0 0}
{ , 0, , ,
  
   
a braic topo ists to assume X A and X if n
In case q p then C A by the convevtion a mong
n n
p q p q
ï° ï°
The above sequence { Tq } is called a composite chain
system , if the following tow conditions are satisfied ;
( 1 ) Cp,q = Ap,q = 0 if p < 2 , and
( 2 ) each Cp,q is a free abelain group .
Denote
p q p q
p
p q
p q p q
p q p q p q
p q p q p q
k by
j by
by
j by
1, ,
, ,
, 1, ,
1, , ,
( ) Im
ker
ker Im
ï“
ï‡
ïˆ
 


ï° ï¢
ï³ ï³
ï¯ï¢ ï³
and
ïŒ ï‚® ïˆ ï‚¾ï‚¾ï‚¾ï‚® ï‡ ï‚¾ï‚¾ï‚¾ï‚® ï“ ï‚¾ï‚¾ï‚¾ï‚® ïˆ ï‚® ïŒ ï€«
q p q p q p q p q
p q p q p q E 1, , , ,
1, , , :
ï® ïª ï­
Let
be an exact sequence in whi ch the groups are abelain .
A composite chain system Tq will be called a combinatorial
realization of if and only if is isomorphic to .
We obtain some results , which are ;
Under certain condition the "a new exact sequence"
has a combinatorial realization .
If K is a complex and , then
, maps onto 0 ,
Jm,q-complex .
If K is a Jm,q-complex , then