8 - Polish Groupoids

Summary

Introduction

This chapter has two themes. The primary one is closely related to the topic of the workshop, being the idea of groupoids as generalizing transformation groups and as a tool for thinking about transformation groups. Questions about orbit spaces or ergodicity, for example, are natural for groupoids. The secondary theme is minor from the viewpoint of the workshop, but it seems worthwhile to mention some other situations where groupoids can be used.

The first section gives some basic information about Polish spaces. The second one provides a definition of groupoid and a few examples. Then we continue with a third section about groupoids related to foliations of manifolds and a fourth section about groupoids in which the unit space has a cover by open invariant sets on which the groupoid structure is given by a transformation group. For the final section we concentrate on dichotomies.

Polish Spaces

Although many readers may know good sources for information on Polish spaces, it won't hurt to mention books by Bourbaki [Bou], Kechris [Ke], Kuratowski [Ku], and Srivistava [Sr], for those not familiar with them.

Many spaces that arise naturally in mathematics are Polish, and this is certainly true for groupoids in the analytic setting. Besides analysis itself, we include in this class anything involving spaces of functions, most problems having a physical motivation, and most questions deriving primarily from geometry.