UNORIENTED BORDISM AND ACTIONS OF FINITE GROUPS 3

§ 2. Equivariant Bordism Groups

In order to analyze the actions of finite groups, one

needs the machinery of equivariant bordism, extending the

notions introduced in Conner and Floyd [5; § 5].

Let G be a finite group, A family ^ in G is a collection

of subgroups of G such that:

a) if H £ 7 and KcH, then K £ 7 , and

b) if H £ 7 and g £ G, then gHg""1 £ 7 .

Being given a family 7 in G, consider a pair (M,p ) consisting

of a compact manifold with boundary M and a differentiable G

action ?: G^M — M satisfying: if x £ M, the isotropy group

Gx = [9 £ Gl ^(9?x) = xl belongs to 7* Such a pair is called an

7 -free action.

If (X,A,W) is a pair of topological spaces with G action

(given by V: G^X — X with V(6xA)cA) and 7f c 7 are

families in G, an ( 7 , " ?

!

)-free bordism element of (X,A, W ) is an

equivalence class of 5-tuples (M,M0,Mi,9,f), where:

1) (M, p) is an ^7 -free action and f: M — X is an

equivariant map (i.e. f P(g,x) = P(g,f(x))), and

2) M0,MJL are compact submanifolds of the boundary of M,

denoted 9M, with 9M = MQ ^ A/^, MQ rs lA^ = aMQ = ^M^, which

are invariant under the G action ( 9 (Gx M^)c M^) and satisfy:

a) (MQ, 9|GXM ) ^s an "^'-free action, and

o

b) f(Mx)CA.

The 5-tuples

(M,MQ,M1,

P ,f) and (M!,NP,MJ, 9!,f!) are