Generalized orbifold Euler characteristics for general orbifolds and wreath products

TL;DR

This paper introduces generalized orbifold Euler characteristics for a broad class of orbifolds using $ ext{Gamma}$-sectors, extending previous concepts to non-global quotient orbifolds and analyzing their behavior under various operations.

Contribution

It defines $ ext{Gamma}$-Euler-Satake characteristics for general orbifolds and explores their properties and relationships to existing invariants, extending formulas to more general group actions.

Findings

Defined $ ext{Gamma}$-Euler-Satake characteristics for orbifolds.

Analyzed behavior under product operations for groups and orbifolds.

Extended formulas for wreath products to orbifolds with Lie group actions.

Abstract

We introduce the $\Gamma$-Euler-Satake characteristics of a general orbifold $Q$ presented by an orbifold groupoid $\mathcal{G}$, generalizing to orbifolds that are not necessarily global quotients the generalized orbifold Euler characteristics of Bryan-Fulman and Tamanoi. Each of these Euler characteristics is defined as the Euler-Satake characteristic of the space of $\Gamma$-sectors of the orbifold where $\Gamma$ is a finitely generated discrete group. We study the behavior of these characteristics under product operations applied to the group $\Gamma$ as well as the orbifold and establish their relationships to existing Euler characteristics for orbifolds. As applications, we generalize formulas of Tamanoi, Wang, and Zhou for the Euler characteristics and Hodge numbers of wreath symmetric products of global quotient orbifolds to the case of quotients by compact, connected Lie groups acting almost freely.