Homology roses and the D(2)-problem

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

“…A finitely presentable group G is said to have D(2) property if any finite 3-dimensional CW complex X, of cohomological dimension 2 with fundamental group G, is homotopy equivalent to a 2-dimensional CW complex. For the status of D(2) problem, see Johnson [8,9] (see also [5,7] for some recent work).…”

Two remarks on Wall's D2 problem

“…The D2 problem for a finitely-presented group G asks whether every finite complex X with fundamental group G which satisfies the D2-conditions is homotopy equivalent to a finite 2-complex. The D2 problem has been actively studied for finite groups, but answered affirmatively only in a limited number of cases (see [18,21] for references to the literature on 2-complexes and the D2-problem, and compare [19,20,24] for some more recent work).…”