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

Abstract

We obtain relations among normal generation of perfect groups, Swan’s inequality involving partial Euler characteristic, and deficiency of finite groups. The proof is based on the study of a stable version of Wall’s D(2) problem. Moreover, we prove that a finite 3-dimensional CW complex of cohomological dimension at most 2 with fundamental group G is homotopy equivalent to a 2-dimensional CW complex after wedging n copies of the 2-sphere S 2 , where n depends only on G.