Graphs, Surfaces and Homology

Peter Giblin’s Graphs, Surfaces, and Homology is a particularly tantalizing member of the collection of introductory level texts on the subject of homology, its obvious distinguishing feature being that graphs are given prominent airplay from the outset. Judging from Giblin’s comments in his prefaces, this was already a dominant feature of the book’s first edition in the 1970s. Indeed, in view of what most homology texts looked like in those days, this was truly something of a departure from the norm. This approach is still something of a novelty nowadays, so to go the route he did was a prescient move on Giblin’s part.

Additionally, the timing for this third edition may be particularly fortuitous. If we consider, for instance, such relatively recent phenomena as Timothy Gowers’ application of combinatorics to functional analysis (leading to his Fields Medal), Giblin’s ecumenical approach seems particularly apt. Ours appears to be an age exceptionally friendly to cross fertilization: certainly a prominently featured connection to graphs can only add to the appeal of what might otherwise be perceived by a broader audience as hard-core topology.

So it is, then, that in this liberal spirit the climactic ninth chapter of Graphs, Surfaces, and Homology is titled “Graphs in surfaces” and contains such fare as John Horton Conway’s “Brussels Sprouts.” (Does it get any more ecumenical than that?) Additionally, and perhaps more along main-street lines, the ninth chapter sports, on p.196, quite a beautiful derivation of the theorem that states that real 3-space does not allow the realization of a closed non-orientable surface inside it. These examples are in themselves already a marvelous advertisement for Giblin’s focus on graphs in the indicated manner.

Beyond this, Graphs, Surfaces, and Homology is a well-crafted and thorough text. It is very readable and very well-written, and the aforementioned fact that the book is now in its third edition testifies to its recognized efficacy. The book’s layout is logical and streamlined, given Giblin’s goals, taking the reader from graphs and closed surfaces to simplicial complexes in the span of around 125 pages. Thereafter invariance is briefly addressed, after which homology sequences for pairs, excision, the Euler characteristic, and Mayer-Vietoris are covered, pretty much in short order. Finally, after a brief discussion of homology mod 2, the aforementioned ninth chapter arrives, closing the orbit, so to speak.

I find this book very appealing on a number of grounds: it is compact and easy to read, the motivating examples and applications are well-chosen and treated beautifully (with excellent illustrations where needed), and homology, as such, is developed effectively and painlessly. I wish I had had this book as part of my own mathematical upbringing.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.