URegina Topology & Geometry Seminar: Udit Mavinkurve

In classical homotopy theory, graphs are treated as 1-dimensional CW complexes. But since the classical notions of continuous maps and their homotopies do not respect the discrete nature of graphs, this fails to capture the full combinatorial richness of graph theory. Discrete homotopy theory, introduced around 20 years ago by H. Barcelo and collaborators, building on the work of R. Atkin from the mid-seventies, is a homotopy theory specifically designed to study discrete objects like graphs. This theory has found a wide range of applications, including in matroid theory, hyperplane arrangements, and topological data analysis.

Currently, a central open problem in the field is to determine whether the cubical nerve functor, which associates a cubical Kan complex to a graph is a DK-equivalence of relative categories. If true, this would allow the import of important results like the Blakers-Massey theorem from classical homotopy theory to the discrete realm. In this talk, based on joint work with C. Kapulkin (arXiv:2408.05289), we will describe a new line of attack towards this problem by establishing the theory of discrete homotopy n-types.