Thesis Format
Integrated Article
Degree
Doctor of Philosophy
Program
Mathematics
Supervisor
Kapulkin, Krzysztof R.
Abstract
Discrete homotopy theory is a homotopy theory designed for studying graphs and for detecting combinatorial, rather than topological, “holes”. Central to this theory are the discrete homotopy groups, defined using maps out of grids of suitable dimensions. Of these, the discrete fundamental group in particular has found applications in various areas of mathematics, including matroid theory, subspace arrangements, and topological data analysis.
In this thesis, 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–van Kampen theorem, generalizing a previous result of H. Barcelo et al. We then use it to solve the realization problem for the discrete fundamental group through a purely combinatorial construction.
One of the biggest open problems in the subject currently is determining whether the cubical nerve functor provides an equivalence between the discrete homotopy theory of graphs and the classical homotopy theory of spaces. We propose a new line of attack towards this open problem, by breaking it into more tractable problems comparing the homotopy theories of the respective n-types, for each integer n ≥ 0. We also solve this problem for the first nontrivial case, n = 1.
Summary for Lay Audience
An important way to study the functioning of a large-scale network — for instance, with a view towards fixing glitches in its behaviour — is to study its “shape”. A specific example of such a large-scale network might be the human connectome, a network where the nodes are individual neurons in the human brain and the edges are synapses between these neurons.
Topology is the area of mathematics concerned with the study of global properties that are invariant under continuous deformations. However, networks are not continuous objects; they are discrete. Thus, it becomes important to adapt techniques from topology and make them suitable for studying discrete objects such as networks. There are many ways to do this. In this thesis, we follow the approach known as discrete homotopy theory, initiated by Barcelo, Laubenbacher, and collaborators in [KL98, BBdLL06]. In this approach, associated to each network, there is an algebraic object called the discrete fundamental group that essentially counts the number of functional “holes” in the network. We study a generalized version of this object, called the discrete fundamental groupoid. Working in this generalized setting allows us to develop several robust techniques to calculate these algebraic objects.
For instance, if we know the number of holes in two halves of a network (e.g. in the left and the right hemispheres of the human brain) as well as the number of holes that are common to both halves (e.g. in the corpus callosum), we should ideally be able to count the total number of holes in the network by adding up the numbers for the two halves and then subtracting the number common to both halves, since we counted these holes twice. For this counting principle to work, we might want to impose the condition that the entirety of any hole in the network lies neatly in one or both of the two halves, as opposed to a situation where a portion of a hole lies in one half and the rest in the other half. This is the content of the discrete Seifert–van Kampen theorem proved in [BKLW01]. However, this condition is found to be too restrictive and is not met in several examples we care about. We prove a refinement of the discrete Seifert–van Kampen theorem where we relax the imposed condition and require instead that every hole in the network can be subdivided into several smaller holes such that each of these smaller holes lies neatly in one or both of the two halves. This relaxed condition is found to be met in a larger class of examples.
We develop several such results that aid in computing these network invariants.
Recommended Citation
Mavinkurve, Udit Ajit, "The Fundamental Groupoid in Discrete Homotopy Theory" (2024). Electronic Thesis and Dissertation Repository. 10271.
https://ir.lib.uwo.ca/etd/10271
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