A combinatorial analysis of topological dissections

AbstractFrom a topological space remove certain subspaces (cuts), leaving connected components (regions). We develop an enumerative theory for the regions in terms of the cuts, with the aid of a theorem on the Möbius algebra of a subset of a distributive lattice. Armed with this theory we study dissections into cellular faces and dissections in the d-sphere. For example, we generalize known enumerations for arrangements of hyperplanes to convex sets and topological arrangements, enumerations for simple arrangements and the Dehn-Sommerville equations for simple polytopes to dissections with general intersection, and enumerations for arrangements of lines and curves and for plane convex sets to dissections by curves of the 2-sphere and planar domains