  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 $S^2$.
