Partial Euler characteristic, normal generations and the stable \(D(2)\) problem (Q721486)

Define the \textit{deficiency} \(def(G)\) for a finitely presentable group \(G\) to be the maximum of the number of generators minus the number of relators over all finite presentations of \(G\). Let \(F\) be a resolution \[ \cdots\rightarrow F_2\rightarrow F_1\rightarrow F_0\rightarrow \mathbb Z\rightarrow 0 \] of the trivial \(\mathbb ZG\)-module \(\mathbb Z\) where each \(F_i\) is \(\mathbb ZG\)-free on \(f_i\) generators. If \(f_0,f_1,f_2\) are finite, define \(\mu_2(F)=f_2-f_1+f_0\) and \(\mu_2(G)\) to be the infimum of \(\mu_2(F)\) over all such resolutions \(F\). The Wiegold conjecture: Let \(G\) be any finitely generated perfect group. Then \(G\) can be normally generated by a single element. The authors show: Theorem 1.2. Assume that the Wiegold-Conjecture is true. Let \(X\) be a finite 3-dimensional CW complex of cohomological dimension 2 with finite fundamental group. Then: {\parindent=6mm \begin{itemize}\item[(1)] The complex \(X\) is homotopy equivalent to a finite 3-dimensional complex with just one 3-cell. \item[(2)] The wedge \(X\vee S^2\) is homotopy equivalent to a finite 2-dimensional complex. \item[(3)] \(1-\mu_2(G)\geq def(G)\geq -\mu_2(G)\) for any finite group \(G\). \end{itemize}}