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].)