Homotopy sequence of a topological groupoid with a basegroup and an obstruction to presentability of proper regular Lie groupoids

TL;DR

This paper investigates the homotopy groups of K-pointed topological groupoids, establishes their relation to ordinary homotopy groups via a long exact sequence, and identifies an obstruction to presenting proper regular Lie groupoids.

Contribution

It introduces a framework for understanding homotopy groups of K-pointed topological groupoids and applies it to identify obstructions in the presentation of proper regular Lie groupoids.

Findings

Homotopy groups of K-pointed topological groupoids are characterized.

A long exact sequence relates these groups to ordinary homotopy groups.

An obstruction criterion for the presentability of proper regular Lie groupoids is established.

Abstract

A topological groupoid G is K-pointed, if it is equipped with a homomorphism from a topological group K to G. We describe the homotopy groups of such K-pointed topological groupoids and relate these groups to the ordinary homotopy groups in terms of a long exact sequence. As an application, we give an obstruction to presentability of proper regular Lie groupoids.