4 R. E.STONG

equivalent if there is a 6-tuple (V^V4*V0,Vi,$ ,F), where:

1) (V, § ) is an 7 -free action and F: V — X is an

equivariant map;

2) 8V is the union of

M,Mf

, and

V+,

with

MnV+

= 3M,

Mf^V+ « oM', MAM 1 =^ , V+n(M^M! ) = 9V"*"; V+ is invariant

under the G action $ which restricts to 9 on M, ? ' on M! ; F

restricts to f on M, ff on M!; and

3) V+ is the union of G invariant submanifolds V0,V^ with

intersection a submanifold with boundary V~ in their boundaries,

such that BV± = IA± ^ V" ^ M|, M ^ V = 91^, M[ n V = SM|,

satisfying:

a) (V*0, 3?IGXV ) *s an ^f-free action, and

b) F(V1) C A.

Under the operation induced by disjoint union, the set of

(?,

7f)-free

bordism elements of (X,A,^) forms an abelian group

denoted flG(7, ?!)(X,A, r), which is the direct sum of its

subgroups Tl ( 7 , ^*)(X,A,V ) consisting of those classes

(M,M0,Mj,9,f) for which the dimension of M is n. If N is a

closed manifold and (M,M0,M^, p ,f) is an (7, /7I )-free bordism

element of (X,A,^), one lets N. (M,!^,!^,? ,f)

= (NxM,N^M0,NxM1,l x ? ,fo-nrM) making fljj(¥ , 7!)(X,A,y ) a

module over the unoriented bordism ring ^*. It should be noted

that all operations made will be compatible with this module

structure.

Being given an equivariant map *;(X,A,^) — (Y,B, %) one