By definition (on p. 8 of the book under review), a ribbon graph is a graph, in the usual sense of the word (vertices plus edges), together with a cyclic ordering — just what it sounds like — on the so-called star of each of the vertices of the graph; here a star is defined, for a given vertex v of the graph, as the set of all edges than have v as initial point. One observes that any planar graph is a ribbon graph and that “[o]ne can always represent a ribbon graph in the plane as the projection of a graph in space … so that the cyclic ordering of the edges coincides with the cyclic ordering arising from the orientation in the plane.” More generally, “[e]very ribbon graph can be embedded into an oriented surface so that its cyclic orderings are induced by the orientations of the surface.” The point of all this is that it is possible to turn the tables, so to speak, and construct surfaces from ribbon graphs.
Going on in this way, one gets that graphs embedded in surfaces are said to be “filling” (p. 11) iff each connected component of the graph complement is diffeomorphic to the disc, and then there is a bijection between the faces of such a filling graph and the connected components of the graph complement (in the ambient surface). One gets that (as always, up to homeomorphisms) ribbon graphs associate to unique compact oriented surfaces in which they fit as a filling graph, and the converse is true, too. Next, and very elegantly, we in fact get nothing less than the following classification theorem: There is a family of “petal graphs” (again, just what it sounds like) such that every compact oriented surface is homeomorphic to one of the indicated corresponding surfaces (having the petal graphs as filling graphs); this correspondence is bijective via the distinctness of the fundamental groups of the surfaces indexed on the aforementioned graphs.
So, right off, we’re dealing here with a very nice bit of interplay between combinatorial graph theory and basic algebraic topology, and, to be sure, before long the author gets down to the business of describing “combinatorial versions of the fundamental group” (cf. p. 22ff.). He quickly gets to de Rham theory (in its usual form) and then does something very nice indeed: he establishes that there is an isomorphism between de Rham cohomology for a surface S and the so-called cohomology of S with respect to its associated filling (ribbon) graph. This marvelous correspondence even extends to, e.g., the yoga surrounding the (symplectic) intersection form in de Rham cohomology.
Thus, what we have here is an alternative route to the cohomology of surfaces that does not involve the model of (or the skeletons provided by) simplicial cohomology. However, that said, the book is not about combinatorial topology per se. Labourie notes on p. 36 that his “main objects of study [are] vector bundles, connections and [the] representation variety.” The representation variety is also called the character variety and is associated to the fundamental group of a closed connected surface of genus at least 3 with values in some Lie group. Labourie notes in the book’s Introduction that these varieties occur with gusto “in the context of gauge theories … and hyperbolic geometry, when [the Lie group] is … PSL(2,R) or PSL(2,C).” He then offers as the raison d’être of the book “the topology of these varieties and their symplectic structure, which was discovered by Atiyah, Bott, and Goldman, without touching upon their interpretation in the theory of Riemann surfaces.”
Thus, the book under review presents a good deal of background for the enterprise mentioned, and Labourie certainly rolls up his sleeves here, including in the mix such things as a very extensive discussion of connections, a combinatorial approach to connections, local systems, an entire chapter on twisted cohomology, and right after that a full chapter on moduli spaces (with Zariski heavily featured).
The last two chapters of the book are, as it were, pretty avant garde, seeing that they deal with symplectic structure (well, that’s not all that unusual, I guess, but he does hit some very new stuff like spin networks and Ed Witten’s gorgeous formula on the volume of a moduli space relative to a symplectic form — see p. 113) and with low-dimensional topology (3-manifolds and integrality questions: Chern-Simons).
The European Mathematical Society has been publishing compact books like this one for a number of years now, and it is indeed a great service to all mathematicians. The books (at least the ones I’ve reviewed) are of a high quality and are eminently readable, modulo the right preparation. This book is no exception: it’s very well-written and the topics covered are wonderful and deep. Furthermore, Labourie takes a fascinating approach to all this very sexy differential geometry by working in the graph theoretic and combinatorial angles, as indicated. It is an excellent book.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.