On One Application of Asphericity of Presentations

I present a direct proof of Lemma 3(a) from O. V. Kulikova’s work on torsion in the group $F\mathrm{/}[M,N]$, using only Proposition 1.2 of Chiswell-Collins-Huebschmann on combinatorially aspherical presentations. In particular, I show that if two presentations satisfy the RC condition and are combinatorially aspherical, then the quotient $N\mathrm{/}[F,N]$ is free abelian on the defining relators and central in $F\mathrm{/}[F,N]$, whence the extension $F\mathrm{/}[M,N]$ is torsion-free.