Higher order orbifold Euler characteristics for compact Lie group actions

TL;DR

This paper extends the concept of orbifold Euler characteristics to spaces with compact Lie group actions, using Euler characteristic integration, and confirms that existing generating series formulas hold in this broader context.

Contribution

It introduces a generalized definition of higher order orbifold Euler characteristics for compact Lie group actions using Euler characteristic integration.

Findings

The generalized orbifold Euler characteristic framework applies to compact Lie group actions.

The generating series formula for higher order orbifold Euler characteristics remains valid.

The approach unifies finite and Lie group cases through Euler characteristic integration.

Abstract

We generalize the notions of the orbifold Euler characteristic and of the higher order orbifold Euler characteristics to spaces with actions of a compact Lie group. This is made using the integration with respect to the Euler characteristic instead of the summation over finite sets. We show that the equation for the generating series of the k-th order orbifold Euler characteristics of the Cartesian products of the space with the wreath products actions proved by H. Tamanoi for finite group actions and by C. Farsi and Ch. Seaton for compact Lie group actions with finite isotropy subgroups holds in this case as well.