Generalised Swan modules and the D(2) problem

Abstract

We give a detailed proof that, for any natural number n, each algebraic two complex over C_n \times C_\infty is realised up to congruence by a geometric complex arising from a presentation for the group.Comment: This is the version published by Algebraic & Geometric Topology on 24 February 200

On CW-complexes over groups with periodic cohomology

“…The work of W. J. Browning [3] can be applied in these cases and this has led to proofs of the D2 property for finite abelian groups [3], [5], [11], dihedral groups [25], the polyhedral groups T , O, I [15] (exhausting the finite subgroups of SO(3)) and various metacyclic groups [21], [33], [46]. It has also been shown for various infinite abelian groups [12], [13] and free groups [20].…”

Minimal algebraic complexes overD4n

“…At present, the D(2) property in known to hold only in comparatively few cases. These include cyclic groups, products of the form C 1 C n (Edwards [2]) and, more relevantly to the present paper, the dihedral groups D 4nC2 (Johnson [5; 6]).…”

“…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].…”

“…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.…”

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

“…For connected X with certain fundamental groups, it has shown been shown that X must be homotopy equivalent to a finite 2-complex (see for example Johnson [7], Edwards [4] and Mannan [9]). However no general method has been forthcoming.…”