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
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