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).