The universal Euler characteristic of V-manifolds

TL;DR

This paper introduces a universal Euler characteristic for V-manifolds, extending the classical Euler characteristic to a broader class of spaces with a focus on additive invariants valued in a ring generated by finite groups.

Contribution

It defines and explores the properties of a universal additive topological invariant for V-manifolds, generalizing the Euler characteristic and satisfying an induction relation.

Findings

Universal Euler characteristic takes values in a ring generated by finite groups.

It satisfies a specific induction relation.

Macdonald type equations are established for the invariant.

Abstract

The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with some finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. Here we discuss a universal additive topological invariant of V-manifolds: the universal Euler characterictic. It takes values in the ring generated (as a 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 equations for the universal Euler characteristic for V-manifolds and for cell complexes of the described type.