Foundations of Combinatorial Topology

This is a concise and polished introduction to combinatorial topology at the graduate level. It is clearly not for novices. Although the prerequisites are modest (a little real analysis, linear algebra and some group theory), a generous dose of mathematical maturity is necessary to use the book successfully. The treatment is so concise that, without benefit of any examples, figures, or exercises, readers have to dig in, struggle through the development, draw their own figures and create their own examples. Pontryagin’s writing is always clear, his proofs are complete and nothing important is omitted, but he’s not much inclined to elaboration.

It is, nonetheless, a remarkably elegant presentation. Beginning with the idea of a simplex and a simplicial complex, Pontryagin defines the Betti groups and then proves that they are topologically invariant. With a basic homology theory established, he then goes on to establish homology invariants for continuous mappings from one polyhedron to another and to use those invariants to prove fixed point theorems. Along the way he offers a couple of digressions into dimension theory. The first digression proves that a compact metric space of topological dimension r can be mapped homeomorphically into some subset of Euclidean space of dimension 2r + 1. The second proves that the dimension of the polyhedron associated with an r-complex is r.

Pontryagin meticulously assembles the basic tools. First comes the groundwork with simplexes, complexes and polyhedra. Next are the foundations for simplicial homology: Betti groups, Betti numbers and the Euler-Poincaré formula. Then, to get to the invariance of the Betti groups, Pontryagin introduces simplicial mappings and approximations with the attendant tools of the barycentric subdivision and cone construction. In the end all the pieces fit together and it looks a little bit like magic.

This book, first published in 1952 and newly reprinted, retains the fresh and distinct voice of the author. This would be a tough place to start learning combinatorial topology. Readers looking for a gentler introduction — really at the other end of the spectrum — might consider Frechét and Fan’s Invitation to Combinatorial Topology, for example. 

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.