Lemma 2. Under the hypotheses of Theorem 3.1 ([The presentation comparison theorem]), including the existence of the two-dimensional K(Γ,1) , there is a presentation Γ=⟨S, T (ρ_z)_ z∈ D , (r)_ r∈ R ⟩ with the following properties: every …

Lemma 2. Under the hypotheses of Theorem 3.1 ([The presentation comparison theorem]), including the existence of the two-dimensional K(Γ,1) , there is a presentation Γ=⟨S, T (ρ_z)_ z∈ D , (r)_ r∈ R ⟩ with the following properties: every letter of T has value 1 in Γ ; the complete relator boundary list is a Λ -basis of the cycle module of the generator graph on S T ; [ρ_z]=[z] as edge chains, for each z∈ D ; projecting the ordinary boundaries [r] , r∈ R , to the T -edge coordinates gives an isomorphism Λ^ ( R) ≅ Λ^ ( T) . No finiteness assumption is imposed on T or R .