If we are given a
G-manifold
M,
G a finite abelian group of odd order, by taking fixed submanifolds
MH for various subgroups
H of
G, we obtain a family (
MH)
H≤G of submanifolds of
M. The Euler characteristics χ(
MH) of such submanifolds satisfy some congruence relations. Conversely, in this paper we show that if we are given a family (
Ni) of submanifolds
Ni of a manifold
N and if the Euler characteristics χ(
Ni) satisfy some congruence relations, then, after adding some family, (
Ni) can be cut and pasted into a family obtained from a
G-manifold.
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