Let $G$ be a finitely presentable group. If we assume $H_2(G,Z/pZ) =0$, $p$ is a prime, then can we always find a finite presentation $\mathcal{P}$ of $G$ so that its presentation complex $K_{\mathcal{P}}$ satisfies $H_2(K_{\mathcal{P}},Z/pZ)=0$?
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