Historically, combinatorial topology was a precursor to what is now the field of algebraic topology, and this book gives an elementary introduction to the subject, directed towards the beginning student of topology or geometry. Due to its importance in applications, the physicist reader who is intending eventually to specialize in elementary particle physics will gain much in the perusal of this book.
Combinatorial topology can be viewed first as an attempt to study the properties of polyhedra and how they fit together to form more complicated objects. Conversely, one can view it as a way of studying complicated objects by breaking them up into elementary polyhedral pieces. The author takes the former view in this book, and he restricts his attention to the study of objects that are built up from polygons, with the proviso that vertices are joined to vertices and (whole) edges are joined to (whole) edges.
He begins the book with a consideration of the Euler formula, and as one example considers the Euler number of the Platonic solids, resulting in a Diophantine equation. This equation only has five solutions, the Platonic solids. The author then motivates the concept of a homeomorphism (he calls them "topological equivalences") by considering topological transformations in the plane. Using the notion of topological equivalence he defines the notions of cell, path, and Jordan curve. Compactness and connectedness are then defined, along with the general notion of a topological space.
Elementary notions from differential topology are then considered in chapter 2, with the reader encountering for the first time the connections between analysis and topology, via the consideration of the phase portraits of differential equations. Brouwer's fixed point theorem is proved via Sperner's lemma, the latter being a combinatorial result which deals with the labeling of vertices in a triangulation of the cell. Gradient vector fields, the Poincare index theorem, and dual vector fields, which are some elementary notions in Morse theory, are treated here briefly.
An excellent introduction to some elementary notions from algebraic topology is done in chapter 3. The author treats the case of plane homology (mod 2), which is discussed via the use of polygonal chains on a grating in the plane. Beginning students will find the presentation very understandable, and the formalism that is developed is used to give a proof of the Jordan curve theorem. Then in chapter 4, the author proves the classification theorem for surfaces, using a combinatorial definition of a surface.
The author raises the level of complication in chapter 5, wherein he studies the (mod 2) homology of complexes. A complex is defined somewhat loosely as a topological space that is constructed out of vertices, edges, and polygons via topological identification. He proves the invariance theorem for triangulations of surfaces by showing that the homology groups of the triangulation are same as the homology groups of the plane model of the surface. This is an example of the invariance principle, and the author briefly details some of the history of invariance principles, such as the Hauptvermutung, its counterexample due to the mathematician John Milnor, and Heawood's conjecture, the latter of which deals with the minimum number of colors needed to color all maps on a surface with a given Euler characteristic. Integral homology is also introduced by the author, and he shows the origin of torsion in the consideration of the "twist" in a surface.
In the last part of the book, the author returns to the consideration of continuous transformations, tackling first the idea of a universal covering space. Algebraic topology again makes its appearance via the consideration of transformations of triangulated topological spaces, i.e. simplicial transformations. He shows how these transformations induce transformations in the homology groups, thus introducing the reader to some notions from category theory. The elaboration of the invariance theorem for homology leads the author to studying the properties of the group homomorphisms via matrix algebra, and then to a proof of the Lefschetz fixed point theorem. The book ends with a brief discussion of homotopy, topological dynamics, and alternative homology theories.
The beginning student of topology will thus be well prepared to move on to more rigorous and advanced treatments of differential, algebraic, and geometric topology after the reading of this book. There are still many unsolved problems in these areas, and each one of these will require a deep understanding and intuition of the underlying concepts in topology. This book is a good start.