On the Non-Existence of Model Structures and the Existence of Ring Scores: Two Studies in Combinatorial Topology

This thesis makes two independent contributions at the interface of combinatorics and topology.

Part I studies obstructions to Quillen model structures on categories equipped with intrinsic discrete homotopy theories. The central example is x-homotopy on graphs, which admits an explicit homotopy category but has resisted all attempts to produce a compatible model structure. We identify three distinct obstructions: the Campion--Goodwillie obstruction in cocomplete categories, where iterated wedge constructions force any candidate structure into a contradiction; and the Goyal--Santhanam and Carranza--Kapulkin--Kim obstructions in the finitely cocomplete setting, which derive contradictions from lifting properties and homotopy pushout computations, respectively. By introducing the notion of segmented categories, the Campion--Goodwillie argument extends to many familiar homotopy theories, including simplicial, cubical, x-, A-, and digital homotopy. The emerging picture is that Hurewicz-type model structures are fundamentally incompatible with the discrete nature of these settings.

Part II develops the theory of ring scores: numerical invariants of compact metric spaces that measure the prominence of circular structure, constructed in analogy with the role $H_0$ persistent homology plays for hierarchical clustering. A ring score satisfies four axioms (isometry-invariance, scale-invariance, maximality on the unit circle, and vanishing on tree-like spaces) and encodes how closely a metric space resembles S^1. We prove a stability theorem, construct explicit persistence-based ring scores, and show that scale-invariance forces any meaningful ring score to be discontinuous in the Gromov--Hausdorff topology, a phenomenon analogous to Kleinberg's impossibility theorem for clustering. The framework is applied to single-cell RNA sequencing data, where ring scores serve as screening statistics for detecting circular gene-set projections in transcriptomic datasets.