Groupoids in Topology

The tutorial met during the Spring 2012 term on Tuesdays from 4pm to 5:30pm in room SC 507 and on Fridays also from 4pm to 5:30pm but in room 304.

Problem sets and /un/polished lecture notes will be posted here.

This tutorial is meant as propaganda for the underrated notion of groupoid, and particulary for the fundamental groupoid of a space. Here’s what Grothendieck had to say about fundamental groupoids:

… people are accustomed to work with fundamental groups and generators and relations for these and stick to it, even in contexts when this is wholly inadequate, namely when you get a clear description by generators and relations only when working simultaneously with a whole bunch of base-points chosen with care - or equivalently working in the algebraic context of groupoids, rather than groups. Choosing paths for connecting the base points natural to the situation to one among them, and reducing the groupoid to a single group, will then hopelessly destroy the structure and inner symmetries of the situation, and result in a mess of generators and relations no one dares to write down, because everyone feels they won’t be of any use whatever, and just confuse the picture rather than clarify it. I have known such perplexity myself a long time ago, namely in Van Kampen type situations, whose only understandable formulation is in terms of (amalgamated sums of) groupoids.

Most of what we’ll cover can be found in Ronald Brown’s textbook Topology and Groupoids. For more information about it, visit its website.

We will focus on the fundamental gropoid of a topological space. The main topics we will discuss are:

• A van Kampen theorem for fundamental groupoids.
• The theory of covering spaces.
• The Jordan curve theorem.
• The fundamental groupoid of the orbit space of a group action.

This list can be adjusted depending on time and participants’ interests.

From basic point set topology: topological spaces, paths, connectedness, compactness, etc. Really, if you know what continuous functions between subsets of \(\mathbb{R}^n\) are, you’ll probably be able to follow with a little effort.

From category theory: just familiarity with basic definitions, and even that is not necessary as we will cover all the stuff we need.

Groupoids are used in many fields of mathematics beyond topology, so there is probably some groupoid-related topic for every taste. Here is a brief list, more details about these topics can be found on this page:

In topology:

• Groupoids model homotopy 1-types.
• Higher groupoids and van Kampen type theorems for them.

A small sample of groupoid-related topics in other fields of mathematics:

• Joyal’s theory of combinatorial species. (Or, more generally, Baez and Dolan’s theory of Stuff Types.)
• Groupoidification, a way to do linear algebra with groupoids instead of vector spaces.
• Applying groupoids to group theory.
• The algebra of groupoids themselves.
• Lie groupoids.
• Convolution algebras of groupoids.

Ronald Brown has a useful webpage on groupoids. It is mostly about groupoids in topology, but does have links to two survey papers: From groups to groupoids by Brown himself, and Groupoids: relating internal and external symmetry by Alan Weinstein. The references in those two papers are extensive and could be useful to find more about uses of groupoids in your favorite area of mathematics.