UNORIENTED BORDISM AND ACTIONS OF FINITE GROUPS 5
ha s induce d homomorphisms
* * = n^(7 , ?')(*): nliv,
?
!
)(X,A,^ )
_nj(?,
*«)(Y,B,X
sending (M,MQ,Mi,9 ,f) to (M,M0,M]_, 9 , * « f)• One has a boundary
homomorphism
3 * : n J ( 7 ,
7 « ) ( X , A , V )
—i njj(* ,
7 ) ( A , * ,
y|GjcA)
of degree -1, sending (M,M0,Mlf P ,f) to (M1, 3 Mj,4 , P |GvM ,f|M1).
One thenhas:
Proposition 2,1; The covariant functor H
#
(^,
/ t
) and
natural transformation 9^ define an equivariant homology theory
on the category of pairs with G action to the category of Tl#
modules. Specifically:
1) If * ,0^: (X,A,V) (Y,B,X) are equivariantly
opic then 71*( ^
2) The sequence
homotopic, , the n 7l$(^, ^'M"^)
= T1*(^,
^'M^).
n j t * , *
')(A,*
,v|GxA) ~ n j ( 7 , r ) ( x , M ? )
where i,j are the inclusions, is exact.
3) If AcX is closed and UcA is an open G invariant set
with closure contained in the interior of A, then the homomorphism
f^C?,
^f)(i)
induced by the inclusion i:
(X~U A-U
^lGx/x „))
(X,A,V) is an isomorphism.
(Note: See Bredon [2,§2].)
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