Stable Modules and the D(2)-Problem

Abstract

This 2003 book is concerned with two fundamental problems in low-dimensional topology. Firstly, the D(2)-problem, which asks whether cohomology detects dimension, and secondly the realization problem, which asks whether every algebraic 2-complex is geometrically realizable. The author shows that for a large class of fundamental groups these problems are equivalent. Moreover, in the case of finite groups, Professor Johnson develops general methods and gives complete solutions in a number of cases. In particular, he presents a complete treatment of Yoneda extension theory from the viewpoint of derived objects and proves that for groups of period four, two-dimensional homotopy types are parametrized by isomorphism classes of projective modules. This book is carefully written with an eye on the wider context and as such is suitable for graduate students wanting to learn low-dimensional homotopy theory as well as established researchers in the field.

Generalised Swan modules and the D(2) problem

“…The class of such modules is tame in the sense of Johnson [8,Chapter 4], so that projective modules are relatively injective and Ext…”

“…Accordingly, we say that the D(2) property holds for if every three dimensional complex with fundamental group isomorphic to , satisfying the hypothesis of the D(2) problem, is homotopy equivalent to a two dimensional complex. In [8], Johnson relates the D(2) problem to the realization problem as follows:…”

“…Indeed, in all confirmed cases of the D(2) property, the proof is dependent on a proof that stably free modules are free. At present, the class of groups to which this applies is limited to finite abelian groups (Latiolais and Browning [11; 3]), free groups (Johnson [8]), the dihedral groups of order 4n C 2 (Johnson [7]) and the dihedral group of order 8 (Mannan [13]). In the appendix to this paper we give a proof that the D(2) property holds for the group C 1 C 1 .…”

Examples of exotic free 2–complexes and stably free nonfree modules for quaternion groups

“…Johnson [8,Theorem IV,p 220] showed that the chain homotopy classes of algebraic 2-complexes of minimal Euler characteristic over Q 4n correspond to the isomorphism classes of rank 1 stably free ZQ 4n -modules. It is unknown whether the algebraic 2-complexes corresponding to stably free nonfree modules are geometrically realizable.…”

“…Major techniques for classifying the homotopy types of presentation complexes actually classify chain homotopy types of algebraic 2-complexes; see eg, Beyl-Latiolais-Waller [2], Browning [4], Johnson [8] and Sieradski-Dyer [16]. As long as Wall's D(2)-problem remains unresolved, a complete homotopy classification of 2-complexes requires that an actual presentation complex be constructed for each chain homotopy type.…”