Orbifolds and stable homotopy groups

TL;DR

This paper explores the relationship between orbifolds and equivariant stable homotopy theory through Lie groupoids, introducing new definitions of stable and extended unstable orbifold homotopy groups.

Contribution

It provides a novel connection between orbifolds and stable homotopy theory using groupoids, and defines new types of orbifold homotopy groups.

Findings

Definition of stable orbifold homotopy groups using twisted sectors and tom Dieck's theorem

Introduction of extended unstable orbifold homotopy groups

Establishment of a natural framework linking orbifolds with equivariant stable homotopy theory

Abstract

Lie groupoids generalize transformation groups, and so provide a natural language for studying orbifolds and other noncommutative geometries. In this paper, we investigate a connection between orbifolds and equivariant stable homotopy theory using such groupoids. A different sort of twisted sector, along with a classical theorem of tom Dieck, allows for a natural definition of stable orbifold homotopy groups, and motivates defining extended unstable orbifold homotopy groups generalizing previous definitions.