Algebraic topology was once very combinatorial in nature. Classically, spaces were studied by computing invariants of cell complex decompositions. Of course, they still are. However, the subject now involves many tools, and the link with discrete mathematics is not at the forefront.
Combinatorial algebraic topology arises from this classical tradition. As Dmitry Kozlov makes clear in the introduction to his text, however, while the roots of the subject are classical, “the aspects of the theory that we consider here… are far from classical, and have been brought to the attention of the general mathematical public fairly recently.” The author goes on to explain that combinatorial algebraic topology concerns itself with the intersection and interplay of discrete mathematics and algebraic topology.
Kozlov’s Combinatorial Algebraic Topology is an introduction to the field, intended for graduate students and beyond. As an algebraic topologist, my interest in it was to learn a bit of how this subject fits into the broad field of topology, and what applications it has.
I got sidetracked by the second part of the text, and never did quite manage to get something concrete from the last part of the text. This was my own fault, not that of the author. I did, in fact, come away with some understanding of how the word “combinatorial” modifies the phrase “algebraic topology”, and my impression is that had I offered this text enough time, I would have been rewarded on my second goal as well.
The book has three parts. The first part is a brief primer of some basic algebraic topology: cell complexes, some homology and some bundle theory, for example. Those with a working knowledge of algebraic topology will have no difficulty beginning directly with Part II. This part begins by describing many particular complexes and moves on to discuss various tools of the subject. Part III is entitled Complexes of Graph Homomorphisms and Kozlov has chosen this topic he says, “as a source of illustrations of various techniques developed in the second part” of the text.
Overall, the text is very readable, and feels like it covers a lot of ground even as the author makes clear that he specifically limited his exposition of applications. Of particular note, each chapter ends with remarks which put the chapter in context, and with further references. I appreciated this. Kozlov does provide a brief introduction, and most chapters do begin with a brief paragraph of introduction as well. In my opinion, these sorts of remarks are some of the most valuable that an introductory text has to offer, and at the beginning of each chapter, I could have used more than was provided here.
Michele Intermont is an associate professor of mathematics at Kalamazoo College in Kalamazoo, MI.