Partial Euler Characteristic, Normal Generations and the stable D(2) problem

We study the interplay among Wall’s $D(2)$ problem, normal generation conjecture (the Wiegold Conjecture) of perfect groups and Swan’s problem on partial Euler characteristic and deficiency of groups. In particular, for a 3-dimensional complex $X$ of cohomological dimension 2 with a finite fundamental group, assuming the Wiegold conjecture holds, we prove that X is homotopy equivalent to a finite 2-complex after wedging a copy of sphere $S^2$.