Geometric interpretation of the fundamental groupoid

Question

Motivation

The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy groups are how higher dimensional spheres behave (up to homotopy in both cases, of course). Even better, for nice enough spaces the (integral) homology groups count $n$-dimensional holes.

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A groupoid is a category where all morphisms are invertible. Given a space $X$, the fundamental groupoid of $X$, $\Pi_1(X)$, is the category whose objects are the points of $X$ and the morphisms are homotopy classes of maps rel end points. It's clear that $\Pi_1(X)$ is a groupoid and the group object at $x \in X$ is simply the fundamental group $\pi_1(X,x)$. My question is:

Is there a geometric interpretation $\Pi_1(X)$ analogous to the geometric interpretation of homotopy groups and homology groups explained above?

Answer 1

I'm not sure how to answer this, because it already seems pretty geometric to me. So let me answer a slightly different question: what is the fundamental groupoid good for? Since one knows that the fundamental group and groupoid are equivalent as categories for path connected spaces, it's tempting to view the groupoid as giving nothing new. But in fact, there are situations when it seems more natural. For example, if a group $G$ acts continuously on a space $X$, then unless one knows something more, we only get an outer action of $G$ on $\pi_1(X,x)$ i.e. it's only well defined up to inner automorphisms. However, $G$ will act on the fundamental groupoid $\Pi_1(X)$ on the nose, and in fact, the previous statement becomes easier to see from this point of view.

Answer 2

A very geometric approach to the fundamental groupoid can be found in Ronald Brown's Topology and Groupoids. Since EVERYTHING is expressed from the beginning in terms of the category of equivelence classes of paths,the formulation is very straightforward and simple. I highly recommend the book to all mathematicians:I have seen the future of point-set topology courses and Brown's text is the crystal ball.

Answer 3

For an even more geometric application, the fundamental groupoid tells you all about parallel transport in bundles, as long as the transport is independent of the actual path and only relies on the homotopy class of the path. This is the case for flat connections (in particular on vector bundles). For appropriate spaces, one can define a flat connection on a line bundle as a character of the fundamental group, but is more natural to define it as a functor from the fundamental groupoid to the groupoid $core(1dVect)$ of 1-dimensional spaces and isomorphisms between them (this latter groupoid is equivalent to $GL_1$ considered as a groupoid with one object). A point is sent to the fibre over that point, and a (homotopy class of paths) to the isomorphism between the fibres induced by the flat connection. This has manifest extensions to vector bundles of higher rank $n$, replacing $core(1dVect)$ by the groupoid $core(ndVect)$.

There are versions of this for covering spaces, but vector bundles are perhaps more geometric.

Answer 4

Thanks Andrew for the nice comments!

In relation to the comment on G-spaces by Donu, I should point out that Chapter 11 of "Topology and groupoids" is on "Orbit spaces, orbit groupoids". But I doublt many topologists are aware of the latter concept!

My new jointly authored book `Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids' is published in August 2011 by the European Math. Soc and distributed in the Amercas by the AMS from October 2011. It is a kind of sequel to the above book, exploring some major uses of groupoids in higher homotopy theory. A kind of new foundation for algebraic topology at the border between homotopy and homology, and applications of some higher order Seifert-van Kampen Theorems. In particular, the 2-dimensional version allows some quite explicit calculations of homotopy 2-types.

See my web site for more infomration.