Cutting and pasting of manifolds into $G$-manifolds

Abstract

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