What is the usefulness of computing the fundamental group of a graph embedded in $R^3$?

Question

Recently I have learned about computing the fundamental group of a graph embedded in $R^3$. The procedure is to find a minimal spanning tree and then consider the free group on all the edges of graph not in this tree.

My question is, what is the usefulness or what are possible applications of associating this group with a graph that tells me something about graphs in general that I can't determine without using algebraic topology?

For instance, it is possible by computing minimal spanning trees of two different graphs and then computing their fundamental groups we can rule out these graphs as isomorphic. But this isn't much more than counting and comparing the number of edges.

So what does knowing the fundamental group of a graph spur us on to?

Answer 1

The computation of the fundamental group of a graph is the first step of the computation of many, many other fundamental groups. For example, for any simplicial complex, or more generally a CW complex $X$, the first step in computing its fundamental group $\pi_1(X)$ is to compute the fundamental group of the 1-skeleton $X^{(1)}$ which is a graph. The final outcome is the calculation of an explicit presentation of the fundamental group of $X$. Intermediate steps steps involve applications of Van Kampen's Theorem, but you can't get off the ground without knowing that the fundamental group of the 1-skeleton is a free group.

Also, the computation of the fundamental group of a graph gives us a powerful topological tool for studying algebraic properties of the free group itself. One example is the theorem that every subgroup of a free group is free. The first step is to take a graph whose fundamental group is the given free group. Subsequent steps use covering space theory to show that every subgroup of the fundamental group is isomorphic to the fundamental group of some other graph (namely a covering space), and hence is a free group.

There is also a whole theory of automorphisms and outer automorphisms of free groups which starts by representing the free group as the fundamental group of a graph, and then goes on to study automorphisms and outer automorphisms by representing those as homotopy equivalences of the graph.

I don't know if that's enough evidence to convince you, but it's just a few tips of a lot of icebergs.

Answer 2

The fundamental group of a graph is not an extremely interesting object - it is just a free group as you describe. And it has nothing to do with any particular embedding of the graph in $R^3$ or anywhere else.

If a graph is embedded in $R^3$ then the really interesting object is the fundamental group of the complement. It depends not just on the graph but also on the embedding, and in the simplest case of cycles it is a knot invariant.

Answer 3

The question is a bit confusing as it refers to a "graph embedded in $R^3$": this is relevant to Michael Adamszek's answer.

Here is Figure 9.7 from Topology and Groupoids illustrating relations at a vertex or crossing for such complements of a graph or link:

But the general question of using fundamental groups of graphs themselves also has problems, because sticking to groups, and correspondingly pointed, connected spaces, throws away a lot of structure, namely the structure of a graph as pieces, i.e. edges, stuck together at vertices. We may also need at some time any symmetry of the original graph, a symmetry lost by choosing a base point and tree,

We even need the structure of the pieces!

Thus, using groupoids, we have an important example, the groupoid say $\mathcal I$, which has two vertices $0,1$ and two nonidentity arrows $\iota: 0 \to 1$ and $\iota^{-1}: 1 \to 0$. This seemingly trivial groupoid plays a role in the category of groupoids analogous to that of the integers in the category of groups! Notice that if in the category of groupoids you identify $0,1$ in $\mathcal I$ you get the group of integers. This "explains" the fundamental group of the circle!

So the natural concept for the algebraic topology of graphs is that of free groupoids. You can read about them in Philip Higgins' downloadable book Categories and Groupoids, first published in 1971.

I feel this broader use "spurs us on".