Combinatorial Differential Topology and Geometry
Robin Forman · 1999
A variety of questions in combinatorics lead one to the task of analyzing the topology of a simplicial complex, or a more general cell complex. However, there are few general techniques to aid in this investigation. On the other hand, the subjects of differential topology and geometry are devoted to precisely this sort of problem, except that the topological spaces in question are smooth manifolds. In this paper we show how two standard techniques from the study of smooth manifolds, Morse theory and Bochner's method, can be adapted to aid in the investigation of combinatorial spaces. Introduction A variety of questions in combinatorics lead one to the task of analyzing a simplicial complex, or a more general cell complex. For example, a standard approach to investigating the structure of a partially ordered set is to instead study the topology of the associated order complex. However, there are few general techniques to aid in this investigation. On the other hand, the subjects of differential topology and differential geometry are devoted to precisely this sort of problem, except that the topological spaces in question are smooth manifolds, rather than combinatorial complexes. These are classical subjects, and numerous very general and powerful techniques have been developed and studied over the recent decades. A smooth manifold is, loosely speaking, a topological space on which one has a well-defined notion of a derivative. One can then use calculus to study the space. I have recently found ways of adapting some techniques from differential topology and differential geometry to the study of combinatorial spaces. Perhaps surprisingly, many of the standard ingredients of differential topology and differential geometry have combinatorial analogues.