Seminar
Speaker: Udit Mavinkurve
Host: Swapneel Mahajan
Title: The Fundamental Groupoid in Discrete Homotopy Theory
Time, day and date: 2:25:00 PM, Monday, September 08
Venue: Room 105
Abstract: 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 Barcelo et al., building on the work of 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 more recently, in topological data analysis.

In this talk, based on joint work with Chris Kapulkin, we introduce the discrete fundamental groupoid, a multi-object generalization of the discrete fundamental group, and use it as a starting point to develop some robust computational techniques. A new notion of covering graphs allows us to extend the existing theory of universal covers to all graphs, and to prove a classification theorem for coverings. We also prove a discrete version of the Seifert¨Cvan Kampen theorem, generalizing a previous result of Barcelo et al. We then use it to solve the realization problem for the discrete fundamental group through a purely combinatorial construction.

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 results like the Blakers-Massey theorem from classical homotopy theory to the discrete realm. We propose a new line of attack, by breaking it into more tractable problems comparing the homotopy theories of the respective n-types, for each integer n ¡Ý 0. We also solve this problem for the first nontrivial case, n = 1.