Minimal 2-complexes and the D(2)-problem

Abstract

Abstract. We show that when n ≥ 5 there is a minimal algebraic 2-complex over the quaternion group Q(2 n ) which is not homotopy equivalent to the Cayley complex of the standard minimal presentation. This raises the possibility that Wall's D(2)-property might fail for Q(2 n ).

Generalised Swan modules and the D(2) problem

“…We say that weak cancellation holds for G if all stably free Swan modules are free. Johnson has shown [9], that if weak cancellation holds for G, then one may construct modules which are stably equivalent to π 2 (X G ) for some presentation G , but which are not isomorphic to π 2 (X G ) ⊕ Z[G] m . We refer the interested reader to the recently published Beyl and Waller [2] for an explicit construction of such a module.…”

Minimal algebraic complexes overD4n

“…Subsequently it was shown by Swan [16] and Stallings [15] The problem may be phrased in terms of realizing algebraic complexes geometrically (Johnson [6], Mannan [9; 13]), or in terms of group presentations (Mannan [12]). Cell complexes that potentially offer a counterexample to Wall's D(2)-problem have been postulated (Bridson and Tweedale [1] and Johnson [7]), though proving that they are counterexamples would appear to require some new obstruction. Where progress has been made is extending Wall's result to the case n D 2 for all finite cell complexes with a specified fundamental group.…”

A commutative version of the group ring

“…There are also many potential counterexamples which could solve Wall's D2 problem and / or the Relation Gap problem [2,5,10]. We mention a candidate due to recently published work by Gruenberg and Linnell [6]: Potential counterexample 1.2.…”