Making the Discrete Continuous: How Combinatorics Becomes Topology

At first, topology and combinatorics seem far removed from each other. Combinatorics deals with finite sets and “rigid” structures whereas in topology you often have spaces which you are allowed to deform continuously (hence the classic phrase that a coffee cup and a donut are the same thing to a topologist). Yet the last several decades have seen beautiful applications of topology in combinatorics and discrete geometry. A recent textbook, [10], focuses entirely on combinatorial uses of the Borsuk-Ulam theorem! So how is it that topology finds such a useful place in combinatorics? How do we get from our rigid objects to continuous ones? And most importantly, why should we expect this transition from the discrete to the continuous to be successful? The purpose of this essay is to find some answers to these questions. Our goal is not for the reader to understand lots of broad technical details; in fact, some of the ideas in the final section are very difficult and take years to understand. We will instead highlight some specific situations in combinatorics where topology arises, hopefully enticing the reader into this interesting area of mathematics. This essay therefore has three sections: a gentle introduction to graph coloring, an introduction to simplicial complexes, and a discussion of situations where topology plays a role in graph theory and group theory.