Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial topology with a new combinatorial approach to framings. The resulting notion of framed combinatorial spaces has unexpectedly good behavior when compared to classical, nonframed combinatorial notions of space. In discussing this behavior and its contrast with that of classical structures, we emphasize two broad themes, computability in combinatorial topology and combinatorializability of topological phenomena. The first theme of computability concerns whether certain combinatorial structures can be algorithmically recognized and classified. The second theme of combinatorializability concerns whether certain topological structures can be faithfully represented by a discrete structure. Combining these themes, we will find that in the context of framed combinatorial topology we can overcome a set of fundamental classical obstructions to the computable combinatorial representation of topological phenomena.
The idea of the book is provide an entry point to (combinatorial) stratified geometry in directed or ‘framed’ spaces (see, for instance, manifold diagrams) from a perspective of classical combinatorial topology. That is, the book starts out with well-known classical combinatorial-topological concepts (such as simplices and regular cells) and builds on these to define framing and stratification structures.
The book roughly contains:
Ch. 1 : discussion of framed combinatorial-topological structures based on the idea of “affine framed combinatorics”
Ch. 2 - 3 : classification of framed regular cells via the theory of trusses