The Universal Euler Characteristic of $V$-Manifolds

The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with certain finiteness properties. A generalization of the notion of a manifold is the notion of a $V$-manifold. We discuss a universal additive topological invariant of $V$-manifolds, the universal Euler characteristic. It takes values in the ring freely generated (as a ${\mathbb Z}$-module) by isomorphism classes of finite groups. We also consider the universal Euler characteristic on the class of locally closed equivariant unions of cells in equivariant $CW$-complexes. We show that it is a universal additive invariant satisfying a certain “induction relation.” We give Macdonald-type identities for the universal Euler characteristic for $V$-manifolds and for cell complexes of the described type.