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