I dimly recall from my student days coming across textbooks on combinatorial topology, which seemed to me to be a somewhat old fashioned term, and perhaps even a bit outdated. When I subsequently took a topology class there seemed to be nothing combinatorial about it, and there was certainly no mention of the term, but perhaps I would have come across it had I been more scrupulous about following a well-defined curriculum in this discipline. As it was, with my specialty being number theory, I never approached topology as other than a marvelous curiosity, even if it meant that in graduate school I sat in on some terrific courses by Max Karoubi. As fate would have it, however, now, in my (later and later) middle age topology looms everywhere in my work, specifically algebraic topology, and it is a pleasure to be called to study the subject at this time — better late than never, I guess. (If only I knew then what I know now!)
Well, what is combinatorial topology, then? Here’s what the redoubtable Paul Alexandroff says on p.12 of his Elementary Concepts of Topology: “[what] today is known as combinatorial topology … essentially amounts to the consideration of manifolds as polyhedra” and we immediately see looming on the horizon such things as simplicial complexes and CW-complexes: the path is clearly lit. And it appears that I was not too far off target in regarding the term as a bit outmoded: checking my five favorite introductory-level algebraic topology books reveals no mention of “combinatorial topology” — a propos, these books are, in alphabetical order, Bott-Tu, Dold, Maunder, May, and Spanier. Well, well.
But what about a terminological inversion like topological combinatorics? Here’s something truly brand-new — this term I certainly never heard of in my younger days (and as a number theory student I did of course come across a goodly portion of combinatorics). Is it something like graph theory on steroids? What is it? The back of the book under review gives a sketch of an answer: “the field of topological combinatorics [is] a subject that has become an active and innovative research area in mathematics over the last thirty years with growing applications in math[ematics], computer science, and other applied areas.” And then we get the clincher: “Topological combinatorics is concerned with solutions to combinatorial problems by applying topological tools …” So there we have it.
The present book, a text “well suited for advanced undergraduate or beginning graduate mathematics students,” accordingly presents a sequence of combinatorial themes which have shown an affinity for topological methods, naturally including a good deal of graph theory. However, there’s a lot more: fair-division problems, embedding and mapping problems, and so on. This material, i.e. the book’s subject-matter proper, takes up the first four chapters; there are five appendices added so as to mitigate the condition that the book should be accessible to kids who know no graph theory or introductory topology (as the back cover continues to advertise); in fact, de Longueville provides a Leitfaden and four “suggested course outlines” (corresponding to the four possible cases of classes lacking in preparation vis a vis graph theory and “the basics of algebraic topology including simplicial homology theory”).
This book is filled with extremely attractive mathematics (so typical of problems in graph theory and combinatorics), and bringing topology into the play of combinatorics and graph theory is a wonderfully elegant manoeuvre. Here it is carried out coherently, and on a pretty grand scale, and we are thus afforded the opportunity to encounter (algebraic) topology in a very seductive uniform context. What a marvelous thing!
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.