6 R. E. STONG
Being given families ^
!
c ? ,
q.y
c ¥ , with ?e ^
c
-* »
o o o o
one has a homomorphism
j:tt*(¥,
?f)(X,A,V)—
T\(G, *Q, ^)(X,A,V)
obtained byconsidering an(%,
^!)-free
bordism element as an
(? ,
^f)-free
bordism element. One also has a boundary
homomorphism
a;* ^(7,
?!)(X,A,V)
n*( ?',* )(X,A,¥0
of degree -1 obtained bysending (M,MQ,M^, 9 ,f)to
(M0, * ,
3MC,tP|GxM0'f IMO)- 0nethenhas:
Proposition 2.2; Being given families ^" c 7
!
c 7 , the
sequence
n°(
7-1,
7»)(X,A,y) -2l»n*(7, 7»)(X,A,V)
is exact, where jQ : ( qM , 9 ")c( 7, 7"),jx : (7, 7 ")c ( 7, 7-')
and j
2
: ( 7-' , 4 ) c (
71,
7")are the inclusion morphisms.
Notes; 1) These propositions are basically trivial, but
writing out all details is extremely messy. Partial proofs appear
in [5,9].
p
2) If A = i , M, must beempty and one writes TZ*(7, ? f ) (X, ^ ),
If
%x
is the empty family, one must have MQ = & , and one writes
Tt*( y- )(X,A,T ). If A = 4 and
?!
= 4 , then aM=* , and one
writes f£(7 )(X,Y).
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