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The fundamental group in discrete homotopy theory

Chris Kapulkin, Udit Mavinkurve

TL;DR

The paper builds a robust foundation for A-homotopy theory of graphs by developing a discrete fundamental groupoid $\Pi_1$ and its monoidal behavior with the box product, enabling a graph-wide enrichment over groupoids.It develops a full covering-graph theory, including local isomorphisms, the category of coverings, universal covers, and a Galois correspondence that mirrors the classical theory for spaces.A discrete Seifert–Van Kampen theorem is established to compute fundamental groupoids from pushouts, and the framework is used to construct graphs with a prescribed fundamental group, highlighting computational and combinatorial capabilities.Overall, the work provides both theoretical foundations and practical tools for calculating and prescribing fundamental groups of graphs, with potential impact on combinatorial topology and related computational applications.

Abstract

We develop a robust foundation for studying the fundamental group(oid) in discrete homotopy theory, including: equivalent definitions and basic properties, the theory of covering graphs, and the discrete version of the Seifert-van Kampen theorem.

The fundamental group in discrete homotopy theory

TL;DR

The paper builds a robust foundation for A-homotopy theory of graphs by developing a discrete fundamental groupoid $\Pi_1$ and its monoidal behavior with the box product, enabling a graph-wide enrichment over groupoids.It develops a full covering-graph theory, including local isomorphisms, the category of coverings, universal covers, and a Galois correspondence that mirrors the classical theory for spaces.A discrete Seifert–Van Kampen theorem is established to compute fundamental groupoids from pushouts, and the framework is used to construct graphs with a prescribed fundamental group, highlighting computational and combinatorial capabilities.Overall, the work provides both theoretical foundations and practical tools for calculating and prescribing fundamental groups of graphs, with potential impact on combinatorial topology and related computational applications.

Abstract

We develop a robust foundation for studying the fundamental group(oid) in discrete homotopy theory, including: equivalent definitions and basic properties, the theory of covering graphs, and the discrete version of the Seifert-van Kampen theorem.
Paper Structure (15 sections, 47 theorems, 104 equations, 11 figures)

This paper contains 15 sections, 47 theorems, 104 equations, 11 figures.

Table of Contents

  1. Preliminaries
  2. The category of graphs
  3. Monoidal structure on the category of graphs
  4. Paths in a graph
  5. Homotopy theory of graphs
  6. Homotopy groups of graphs
  7. A finite formulation of the homotopy groups
  8. The fundamental groupoid
  9. Monoidality
  10. Covering graphs
  11. Local isomorphisms and covering maps
  12. The category of coverings over a fixed base graph
  13. Universal covers
  14. Seifert--Van Kampen theorem for graphs
  15. Application: graphs with prescribed fundamental group

Key Result

proposition 4

The functor ${(-)}_{V} \colon \mathsf{Graph} \to \mathsf{Set}$ that maps a graph to its underlying vertex set admits both adjoints. \begin{tikzcd} \Graph \arrow[rr,"\V{(-)}" description] && \arrow[ll, bend right,"\bot","\mathsf{discrete}"'] \arrow[ll, bend left,"\bot"',"\mathsf{complete}"] \

Figures (11)

  • Figure :
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  • ...and 6 more figures

Theorems & Definitions (127)

  • definition 1
  • definition 2
  • example 3
  • proposition 4
  • proposition 5
  • definition 6
  • definition 7
  • proposition 8
  • definition 9
  • definition 10
  • ...and 117 more