Framed combinatorial topology

TL;DR

This paper introduces framed combinatorial topology, a new framework that combines classical combinatorial topology with framings, enabling better computability and representation of topological phenomena.

Contribution

It develops a novel theory of framed combinatorial spaces that improves upon classical notions by addressing computability and combinatorializability of topological structures.

Findings

Framed combinatorial spaces exhibit unexpectedly good behavior.

The theory overcomes classical obstructions to computable topological representations.

Enhanced ability to algorithmically recognize and classify combinatorial structures.

Abstract

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.