{"id":"palomar/PALOMAR-2026-08-27-000004","url":"https://github.com/Arthur742Ramos/finite-graph-fundamental-group/tree/dd57e3ab8bc5a7042fcc5498e778b008d87ac032","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-08-27-000004-v1.json","repo":"https://github.com/Arthur742Ramos/finite-graph-fundamental-group","tags":["math.AT","math.LO","msc:05C25","msc:18B40","msc:20F65","algebraic-topology","lean","graph-theory","fundamental-group"],"score":0,"title":"Arthur742Ramos/finite-graph-fundamental-group","trust":"high","domain":"math.AT","source":"palomar","authors":["Arthur Freitas Ramos","David Hulak","Ruy de Queiroz"],"reviews":0,"subject":"math.AT","summary":"Formalizes the fundamental group of a finite connected graph, represented as a finite weakly connected directed multigraph with its standard interval-cell quotient realization. The main result identifies the topological fundamental group at a chosen vertex with `FreeGroup (Fin (E + 1 - V))`. The proof compares the topological group with the endomorphism group at the root in Mathlib's free groupoid, whose explicit spanning-tree basis supplies the non-tree-edge generators. The development also proves the cycle-rank identities, root-change equivalences, the tree/triviality criterion, and the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-08-27","reactions":{},"formalized":true,"formalizes":[{"title":"Mathlib GroupTheory.FreeGroup.NielsenSchreier","identifier":"","relationship":"adapts"},{"title":"Mathlib CategoryTheory.Groupoid.FreeGroupoid","identifier":"","relationship":"adapts"},{"title":"Mathlib AlgebraicTopology.FundamentalGroupoid.FundamentalGroup","identifier":"","relationship":"adapts"},{"title":"Mathlib Combinatorics.Quiver.Arborescence","identifier":"","relationship":"adapts"},{"title":"Mathlib Combinatorics.Quiver.Covering","identifier":"","relationship":"adapts"}]}