On the construction of a covering map

Abstract

Abstract Let Y = | X | {Y=\lvert X\rvert} be the geometric realization of a path-connected simplicial set X, and let G = π 1 ⁢ ( X ) {G=\pi_{1}(X)} be the fundamental group. Given a subgroup H ⊂ G {H\subset G} , let G / H {G/H} be the set of cosets. Using the combinatorial model 𝛀 ⁢ X → 𝐏 ⁢ X → X {\boldsymbol{\Omega}X\to\mathbf{P}X\to X} of the path fibration Ω ⁢ Y → P ⁢ Y → Y {{\Omega}Y\to{P}Y\to Y} and a canonical action μ : 𝛀 ⁢ X × G / H → G / H {\mu\colon\boldsymbol{\Omega}X\times G/H\to G/H} , we construct a covering map G / H → Y H → Y {G/H\to Y_{H}\to Y} as the geometric realization of the associated short sequence G / H → 𝐏 ⁢ X × μ G / H → X {G/H\to\mathbf{P}X\times_{\mu}G/H\to X} . This construction, in particular, does not use the existence of a maximal tree in X. For a 2-dimensional X and H = { 1 } {H=\{1\}} , it can also be viewed as a simplicial approximation of a Cayley 2-complex of G.