TheD(2) property forD₈

Abstract

W H MANNANWall's D.2/ problem asks if a cohomologically 2-dimensional geometric 3-complex is necessarily homotopy equivalent to a geometric 2-complex. We solve part of the problem when the fundamental group is dihedral of order 2 n and give a complete solution for the case where it is D 8 the dihedral group of order 8.

Two remarks on Wall's D2 problem

“…This result is an application of [11, Theorem 2.1]. The result was known for cyclic and dihedral groups (see [23], [28], [26]), but the argument given here is more uniform and the tetrahedral, octahedral and isosahedral groups do not seem to have been covered before. Remark 1.2.…”

TheD(2)–problem for dihedral groups of order 4n

“…Now take J D ker @ 2 in (3); the following proposition is due to Mannan [9]: Proposition 3.1 J has minimal Z -rank in 3 .Z /.…”

“…Johnson [7] has shown that the D.2/-property holds for the groups D 4nC2 for any n 1; however his result relies on the fact that D 4nC2 has periodic cohomology, a property not shared by D 4n . Mannan [9] has shown that the D.2/-property holds for D 8 . We say that torsion-free cancellation holds for a group ring Z OEG if…”

Minimal algebraic complexes overD4n

“…The first finite non-Abelian non-periodic group shown to have the D (2) property was the dihedral group of order 8 (Mannan [10]). The methodology adopted in [10] was an instance of the approach developed and laid out in [6]. This approach for verifying the D (2) property for a given group G involves two steps.…”