Quillen’s plus construction and the D(2) problem

Abstract

Given a finite connected 3-complex with cohomological dimension 2, we show it may be constructed up to homotopy by applying the Quillen plus construction to the Cayley complex of a finite group presentation. This reduces the D(2) problem to a question about perfect normal subgroups.

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

“…We prove (ii) as follows. By a result of Mannan [13], X is the plus construction of a finite 2-complex Y with respect to a perfect normal subgroup P ≤ π 1 (Y ). Therefore, we have a short exact sequence of groups…”

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).…”

Explicit generators for the relation module in the example of Gruenberg–Linnell

“…The relation gap problem for a finite set of generators of a finitely presented group asks if the number of relations needed to present the group, exceeds the number needed to generate the relation module (in which case the group is said to have a relation gap) [4]. This problem is closely related to major questions in low dimensional topology, most directly Wall's D (2) problem [6], as a group with relation gap would likely lead to a cohomologically 2 dimensional finite cell complex, not homotopy equivalent to a finite 2-complex [7,8].…”